273
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 3. MARCH 1991
Adaptive Control is Enhanced by Background Estimation William Donald Timmons, Student Member, IEEE, Howard J. Chizeck, Senior Member, and Peter G. Katona, Member, IEEE
Abstract-The automated control of physiological variables must often contend with an unknown and time-varying background (i.e., the output level corresponding to no input). To allow for simultaneous realtime identification of background as well as the parameters of an autoregressive moving average model with exogenous inputs (ARMAX model) during adaptive control, a "floating identifier" (FI) approach was developed which may be used with most recursive identification algorithms. This method separates input and output data into low- and high-frequency components. The high-frequency components are used to identify the ARMAX model parameters and the low-frequency components to identify background. This approach was evaluated in computer simulations and animal experiments comparing an adaptive controller coupled to the FI with the same controller coupled to two other standard least squares identifiers. In the animal experiments, sodium nitroprusside was used to control mean arterial pressure of anesthetized dogs in the presence of background changes. Results showed that with the FI, the controller performed satisfactorily, while with the other identifiers, it sometimes failed. It is concluded that the FI approach is useful when applying ARMAX-based adaptive controllers to systems in which a change in background is likely.
w
INTRODUCTION
EN automated control is applied to physiological sys-
tems, the output level corresponding to no input (i.e., the background) is often unknown and time-varying. For example, mean arterial blood pressure is normally near 100 mmHg, but it often deviates from this value even in the absence of an obvious cause [ 151. When applying model-based adaptive controllers to such systems, one should include a background term within the system model; however, estimation of a time-varying background is not straight-forward when using autoregressive moving average models with exogenous inputs (ARMAX models [ 6 ] ) .Indeed, we later show that two popular techniques can result in unstable control and failure in the presence of large swings in background. In this paper, we present thefloating identiJier (FI) approach, a method that estimates time-varying background when using recursive identification algorithms for linear or linearizable systems. It operates by dividing the inputs and outputs of the controlled system into low- and high-frequency components. Background, which is a constant or slowly drifting quantity, is estimated from the low-frequency components. The system dyManuscript received January 26, 1989; revised July 11, 1990. This work was supported by NSF Grants ECS84-00765, BCS-89087 13 and a contract from Eli Lilly and Company. W. D. Timmons is with the Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106. H . J. Chizeck is with the Department of Systems Engineering, Case Western Reserve University, Cleveland, OH 44106. I P. G . Katona is with the Biomedical Engineering and Aiding the Disabled Program, National Science Foundation, Washington, DC, on leave from the Department of Biomedical Engineering, Case Westem Reserve University, Cleveland, OH 44106. IEEE Log Number 9042175.
IEEE,
namics, on the other hand, are estimated from the high-frequency components. For linear systems, superposition applies so that the original parameterization is maintained. For nonlinear systems, the FI technique can be thought of as producing a linear dynamical model about a slowly varying operating point. The choice of bandwidth for the low and high frequency components is related to the frequency spectrum of the background and dynamics, and will be discussed later. Our hypothesis was that when large background disturbances disrupt the operation of the control system, performance can be improved by using the FI. Furthermore, when such disturbances are absent, the FI is transparent, neither helping nor hindering the controller's performance. To test this claim, computer simulations and animal experiments were used to compare the FI to two standard system identifiers during control.
STANDARD IDENTIFICATION TECHNIQUES Adaptive controllers previously used for controlling physiological variables are surveyed in [8], [ 9 ] ,[ 2 2 ] .Typically, these controllers obtain on-line parameter estimates of an ARMAX time series model relating the inputs and outputs of the system by a recursive least squares ident$er ( L S I ) [6]. Control is then based on the past inputs and outputs, using a control law modified by the identified model parameters. In this paper, an exponentially weighted recursive LSI is used (see [ 2 0 ] for a further description of our control system). Consider the recursive least squares identification of the following ARMAX model [ 6 ] :
A(q-')Y(t)= B(q-')U(t)
+ e(?)
(1)
where Y ( r ) , U ( t ) , and e ( r )are, respectively, the output, input, and zero mean noise at time t . A ( 4 - ) and B ( q- ) are polynomials in q - ', the backwards time shift operator, and are defined by
'
A(q-') = 1
-
'
a , q - ' - . . . - a,,q-"
B(q-I) = b,q-'
+ . . . + b,,,q-"'.
(2)
(3)
T o apply an LSI, a format is used in which the parameters and data are represented as
~ ( t=) X r ( t
-
110
(4)
where Y ( t ) is the current output, 0 is the vector of parameters to be estimated, and X ( t ) is an appropriate regression vector. For ( l ) , we have
07 X r ( t - 1)
0018-9294/'9110300-0273$01 .OO 0 1991 IEEE
=
[ a l . . . a,, b l . . . b,,,]
= [ ~ ( -t
U(t
-
(5)
I ) . . . ~ ( -t n )
1 ) . . . U(t
-
m)].
(6)
I E E E TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL 38. NO 3 . MARCH 1991
27J
The parameter vector 0 contains no terms relating to background. If a known background were present, however, one could subtract it from the output at each time, and still use the LSI to identify 0 . Background may be included explicitly in the ARMAX model by letting e ( t ) have a nonzero mean value. Breaking e ( t ) into two terms, such that slowly varying components are lumped into D ( t ) and all else into U ( t ) , (1) may be written as
A(q-')Y(t)
=
B ( q - ' ) U ( f )+ D ( f ) + u ( t ) .
Ut)
U(t)
X7(r
= [al
I)
=
-
. . . a,, b l . . . b,,, D ]
[Y(t - 1 ) . . . Y(t U(t
-
-
b I
LOW PASS
a-'(q-')s(r)
(9)
+
) A ( 9 - I 1Y( f )
=
A ( q - I ) YL, ( f )
G(q-')B(q-')U(t) = B(q-I)UL,(f).
(1 1)
(12)
Now, the separation filter G is chosen so that
G ( q - ' ) D ( t )= D ( t )
(13)
G(q-')zt(t)= 0
(14)
and
i.e., G is chosen so that D ( t ) is unaffected by filtering, but high frequency noise is attenuated. Multiplying both sides of (7) by G results in a low-pass filtered version of (7)
A(q-I)YLP(t)
=
B(q-l)ULP(t)+ D ( t )
(15)
which, when subtracted from (7) results in a high-frequency equation A ( 9-
'
YHP ( I)=
(q
'
UHP ( )
+
(t ) .
(16)
Here YHp(?)= Y(f) - YLP(f)and UHp(t) = U ( t ) - U L P ( t ) . The dynamic parameters A ( q - l ) and B ( q - ' ) are estimated from the relation (16). In this case, (4) involves vectors
O'-=
[ a l . . . a,, h , . . . b,,,]
p+-
-
:_._________... : : A(q-'), ^B(q-')
Fig. 1. Adaptive controller using floating identifier: the adaptive controller uses a control law altered by both the estimated dynamic model parameters and the on-line offset estimates. ( / ( I ) is the input, Y ( f )the output, E ( r ) a noise on the system; subscripts LP and HP, respectively, signify low-pass and high-pass filtered data; arrows indicate signals (solid) and parameters (dashed).
X'(f
~
1)
=
/YHp(?- 1 ) . . . YHp(f UHp(t
-
I) ...
UHp(1
-
/I)
(18)
/TI)].
The low-pass noise component estimate @ ( r ) is calculated from (15):
d(f) = a(q-')YLP(f)- &q-l)UL,(f).
(19)
c,(f)
THE FLOATING IDENTIFIER The idea behind the FI is twstrip dc and low frequency components from the data before parameter identification, effectively "floating" the identifier on the dc signal. After parameter identification, these components are used to calculate background. The overall scheme for the FI is shown in Fig. 1. The low pass filter G ( q - I ) separates low and high frequency components of the input and output signals. Let the subscript L P indicate the low-pass filtered version of each variable by the separation filter G ( q-' ): -I
LOW PASS
(10)
where denotes an estimated quantity. This method is frequently suggested as the way to handle unknown offsets ( e . g . , see [l], [3], [6], [ l l ] ) . As will be shown later, in some cases of background shift, use of the ALSI may lead to incorrect identification of the sign of the system's steadystate gain. This error can cause a stabilizing negative feedback loop to become one of destabilizing positive feedback.
aq
-
(8)
If desired, an estimated background ( f,,(1 ) ) may be calculated using P"(f) =
.4
FILTER
(7)
n)
1 ) . . . U ( f - m ) 11.
Lt) =E Li')U Lt) +E Lt)
CONTROL LAW
The parameters of this model may be estimated by an augmented LSI (ALSI), in which case (4)involves vectors
0'
Y(t)
SYSTEM
A LQ') Y
(17)
An estimated background may be calculated using (10). The FI can be used for multiple-input multiple-output systems if the corresponding filter polynomial matrix G ( q - ' ) commutes with the corresponding polynomial matrices A ( q - ) and B ( q - ' ) . A convenient example is G = g ( q - I ) 1 w h e r e g ( q - ' ) is a low-pass filter transfer function designed to satisfy (13) and (14). and I is the identity matrix.
'
SIMULATIONS Before the identifiers were tested in animals, a series of computer simulations were run to characterize algorithm performance under completely known conditions. Each identifier described above (the LSI, ALSI, and FI) was coupled to the same predictive controller and the identifier/controller combinations (designated according to the identification method as either the LSI, ALSI, or FI) were then evaluated. We chose the control advance moving average controller (CAMAC) as the predictive controller because of its suitability for drug infusion systems and our previous experience with it [16]-[18]. At each sampling time, the CAMAC chooses in input U ( f ) to drive the output to a setpoint at a future time f k . Here, k is the time horizon, and is greater than or equal to the system's input-output delay. The C A M A C assumes that all future inputs will be equal to the current input, but then repeats the computations every sampling interval so that the actual input changes over time. By choice of horizon k , the CAMAC may be tuned between minimum variance control (horizon equal to the system's input-output delay) and steady state control (horizon equal to infinity). T o concentrate attention on the identification algorithm, the CAMAC's automatic horizon selection was disabled. Since simulations using various types of linear plants resulted in similar outcomes, the results from only one type of plant are reported here. The system. a first-order linear ARMAX plant
+
275
TIMMONS et a l . : ADAPTIVE CONTROL ENHANCED BY BACKGROUND ESTIMATION
TABLE I SIMULATION RESULTS FOR FIRSTDISTURBANCE INTERVAL
(%)
Recovery Time (Steps)'
Mean Squared Em02
SteadyState Err02
LSI ALSI
-23.5 -23.5 -24.0
13 13 6
236 237 82
-4.4 -4.4
LSI ALSI
25 25 8 13
499 504
-7.0 -7.0
FI
-31.0 -31.0 -29.5 -29.5 -29.5 -28.5
6
LSI ALSI
-53.5 -53.5
25 25 25
Peak Error
Conditions
-251
.
-
k = L,
No Noise
k -201
25
I
q YO ,
05-52-
0
25
54
73
1W
125
I50
0
Z5
M
75
TIME
TIME
TIME
LSI
ALSI
FI
,
100 125
yo 150
Fig. 2. Simulated behavior of the three identifiers: LSI (left). ALSI (middle). and FI (right). An initial period of control is followed by a background disturbance at the arrow. After recovery, the setpoint is changed and the procedure repeated. The simulation is concluded with a fina! setpoint change. The controller horizon is fixed at 2. Y * , U. Y,,, and Y, are the setpoint, output, background, and estimated background, respectively. Note that only the FI fully recovers from the background disturbance and has zero overshoot.
which was previously used as a benchmark by Clarke [2] and then Voss et al. [ 191, is simulated by y(t) = 0.7y(t - 1)
+ u(r -
1)
+ 2u(t
-
2)
+ e(r)
(20)
w h e r e y ( t ) , u ( r ) , and e ( t ) are, respectively, the output, input, and zero-mean white noise at time t . Baseline was added to the simulated plant as follows:
U t ) = Y(t) +
K,(t)
(21)
where &, ( 1 ) is the plant background, and Y ( t ) is the resultant plant output. All outputs and inputs are in arbitrary units. The protocol for the simulations was designed to test the different closed loop performances of the identifiericontrollers in the presence of various background disturbances. In each case, an initial period of control was followed by a background disturbance. After adequate time was allowed for the output to retum to the setpoint, the setpoint was changed and the procedure repeated. One A-term ( a , ) and two B-terms ( b , and b 2 ) were estimated by the adaptive controllers. A fourth-order ITAE (minimal integral of time-multiplied absolute error) filter [5] was used for the separation filter G in the FI design. The cutoff frequency was set equal to the inverse of the sampling interval, resulting in a normalized filter. The controller performance for a control horizon of two is illustrated in Fig. 2 . The performance of the three identifiers were similar before the first offset disturbance (first arrow). After the disturbance, the LSI and the ALSI exhibited increasingly larger overshoots, input excursions, and slowly decaying offsets, while the FI maintained overshoots and input excursions as at start-up, and rapidly and smoothly eliminated offsets. Also, the ALSI often estimated a background that moved in the wrong direction (as in Fig. 2 after the second disturbance, 100 < t < 125). Table I summarizes the controllers' responses to the first background disturbance when different controller tunings were used, o r when zero-mean noise was added to the
= 4, No Noise
k = 2,
Small Noise (SNR = 25) k
2, Large Noise (SNR = 1.5)
=
FI
FI
LSI ALSI
FI
65.0
13
0.0
151
0.0
215
216 114
-3.0 -3.0 0.8
829 832 938
2.5 2.5 3.8
'Time from onset of disturbance until the output remains within of the setpoint. zErrors calculated as percentage of setpoint,
15%
output. Note how similar the measures for the LSI and the ALSI are (for example, mean square error typically differs by less than 0.5%). More striking, note that in all the simulations except the one with large noise, the FI recovered from the background disturbance in less than half the time needed by the LSI and the ALSI, and that both mean square error and the steadystate error was less for the FI than for the LSI. It was only the simulations in which a large zero mean noise was added to the plant outputs that the FI performed worse than the LSI or the ALSI. Since the performance of the LSI was almost indistinguishable from that of the ALSI throughout our simulations, we decided to use only the LSI in further comparisons with the FI. Therefore, except for a few trials using the ALSI, our animal experiments concentrate on the pairwise comparison of the FI with the LSI.
ANIMALEXPERIMENTS Experimental Procedure Mean arterial pressure (MAP) in five anesthetized dogs was lowered using sodium nitroprusside (SNP), a vasodilator. Phenylephrine, a vasopressor, was used to test the performance of the controllers during a change in the M A P background level. Anesthesia was induced with sodium pentobarbital (30 m g / k g IV bolus) and followed with maintenance doses as needed. As in [lo], catheters were inserted into the femoral vein to administer drugs and into the descending aorta through the femoral artery to measure arterial pressure (Bentley Model 800 blood pressure transducer, Gould/Brush transducer coupler). The pressure signal was low-pass filtered ( 5 s time constant) to derive MAP, which was then sampled every 20 s (or in some trials, 30 s) by a DEC MINC computer. The pressure samples and the calculated SNP infusion rates were stored on disk. An W A C 630 volumetric infusion pump was used to deliver SNP (100 p g / m L in a 5 % dextrose solution [7], [14]) under computer control; another W A C 630 was controlled manually to administer phenylephrine (100 p g / m L in saline). Following the previous example of Stem et al. [17] and Voss et al. [20], two A terms were identified by the controller. The number of B terms was selected to cover a minimum range of
216
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL 38, NO 3 . MARCH 1991 MEAN ARTERIAL PRESSURE
90 s, the longest expected dead time. Three B terms were used for the first dog because the sample/control interval was 30 s; five B terms were used for the last four dogs because the sampleicontrol interval was 20 s . As in the simulations, the FI used a fourth-order ITAE filter with a normalized cutoff for the separation filter G. Therefore, the cutoff was set to 0.033 or 0.05 Hz depending on the sampling interval. In a few of the paired trials, a cutoff of 0.5 Hz was used to test the effects of the cutoff frequency on the FI’s performance. Our experiments were designed to challenge the controllers with a large background disturbance. Since the effects of phenylephrine (our source of background disturbance) varied from dog to dog, we first determined the amount of phenylephrine (between 25 and 83 pg/min) that would require a change in SNP greater than 100 pg/min if the setpoint were to be maintained. Once an appropriate phenylephrine level was determined, the testing procedure involved the sudden administration of this amount after control had been established for at least 14 min. In one case study, we administered a very large dose of phenylephrine (120 p g / m i n ) to try to “break” the controllers (Fig. 3). The initial MAP was between 120 and 140 mmHg in 14 trials, above 140 mmHg in four trials, and below 120 mmHg in five trials. Prior to closed loop control, four to 8 min of open loop probing was used for system identification (typically three pulses of S N P at a rate of 50 pg/min). Two minutes was typically allotted from the last pulse until the start of closed-loop control. During this time, the M A P returned to within 3 % of initial baseline in 3 5 % of the trials. In the worst cases, the MAP returned to within 15.8% below and 13.3% above the initial MAP. The objective upon closing the loop was to lower the MAP to a prescribed level (typically 100 mmHg). Since the initial background level varied, ten trials resulted in lowering the MAP 30 mmHg or more from baseline, and 13 trials resulted in lowering the MAP less than 30 mmHg. To reduce bias from changes in the animal’s condition over time, the order of the tested controllers was randomized. In any of the trials, if it looked like the controller had frozen (stopped infusing S N P for more than four samples when the infusion rate should have increased), the trial was terminated, the phenylephrine stopped, and SNP manually infused.
21
O J
SODIUM NITROPRUSSIDE
DISTURBANCE (120
ug/arn
PHENYLEPHRINE)
El ESTIMATED BACKGROUND
E
1
,2/1“
-
~
0 1
-2
ESTIMATED S T E A D Y STATE P L A N T GAIN
:;lo*
0
:r > ,i /i ;
15
LSI
sb b
ox
7
Is
ALSI
io
;
1;
T I M E IN So H l N U T E S
FI
Fig. 3 . Control of MAP in an anesthetized dog: for each of the three identifiers (LSI, ALSI, and F1. from left to right), control was started and allowed to settle. After steady state was reached, a challenge was given by infusing phenylephrine (120 pg/min. duration indicated by bars across the center of the figure). Note that control was stable only when using the FI.
standard deviation of the error, and standard deviation of the input (SNP infusion rate)-were used to compare their performances during the other time periods. The student’s paired t-test and two nonparametric tests (the sign test and the Wilcoxon rank sum test [12]) were applied to the measures to test for statistical significance. Finally, the Bonferroni method [ 131 was used to test for joint significance of the multiple measures ( n = 11).
Data Analysis
Results
Trials involving the FI and the LSI were broken into four periods of control: startup, predisturbance, disturbance onset, and continued disturbance. Startup was defined as the initial 5 min of control immediately following open loop identification or a change in setpoint. Predisturbance was defined as the time following start-up until the onset of the disturbance or termination of the run, whichever came first. Disturbance onset was defined as the initial 5 min of the disturbance; and continued disturbance as the remaining time until cessation of the disturbance. Only paired runs in which the dog’s response did not change significantly were included in this analysis. Also, since meaningful disturbance data could not be collected from the paired run in which the LSI froze, it was excluded from the analysis, resulting in a slight bias in favor of the LSI. Two measures-overshoot and settling time (defined as the time from start of control for the M A P to stay within five mmHg of the setpoint)-were used to compare the different identifier/ controller performances during startup. Three measures-mean e r r o r (where error is defined as the setpoint minus the MAP),
Fig. 3 shows what can happen in the presence of large swings in background caused by phenylephrine infusion. After the dramatic change in background, the LSI estimated a positive patient gain which resulted in zero S N P infusion rate. Since the controller had frozen, the experiment was terminated. The ALSI also estimated a positive gain, but unlike the LSI, it produced wild oscillations in the S N P infusion rate. This excitation resulted in the eventual reidentification of the plant and subsequent stabilization. As seen often in the simulation studies, the estimated background initially moved in the wrong direction when the disturbance was applied. The FI had little problem with this challenge. The estimated background increased as expected and the gain maintained a negative value throughout the run. The increased S N P rate, showing only moderate oscillations, kept the M A P near the target level. The effectiveness of the FI is further shown in Fig. 4 . During the disturbance in this experiment, the input and output oscillations ceased when the FI was substituted for the LSI. Note
TIMMONS er a l . : ADAPTIVE CONTROL ENHANCED BY BACKGROUND ESTIMATION DISTURBANCE
TABLE IV ANIMAL RESULTS,PERIOD 111: DISTURBANCE ONSET(MEAN SD)
(80 u g l m i n P H E N Y L E P H R I N E )
21
MEAN A R T E R I A L P R E S S U R E
Measure
SODIUM NITROPRUSSIDE
Mean Error (mmHg) STD. Dev. of Error WmHg) Input STD.
:I
-
LSI
211
7.30 f 2.88 5.81
pVALUE3
N
6.22 f 4.34
0.1807
9
6.54 f 1.70
0.4082
9
78.55 f 44.89
0.3423
9
FI
LSI
+ 2.13
65.10 f 24.36
TABLE V A N I M ARESULTS, L PERIOD IV: CONTINUED DISTURBANCE (MEAN SDI
+
+--
~
FI
--D
~
Measure
LSI
FI
pVALUE'
N
2.54 f 2.13
0.92 f 2.85
0.3209
9
4.09 f 2.03
3.68 & 1.70
0.5769
9
69.11 f 27.35
41.00 f 26.82
0.0027
9
0 N
Mean Error (mmHg) STD. Dev. of Error (mmHg) Input STD. Dev . (dmin)
ESTIMATED BACKGROUND mD 1a _ E- '
r42 0
'Student's paired t-test (two-sided)
turbance (Table V) where the FI was better (with more than a 95 % confidence level). Adjustments for eleven joint measures TABLE I1 ANIMAL RESULTS, PERIOD 1: STARTUP (MEAN SD)
Measure
LSI
Overshoot ("Hg! Settling Time (Sec)
-7.79 f 8.96
-4.78
88.33 f 53.06
96.25 f 68.73
FI
3.56
using the Bonferroni method did not change this result.
pVALUE'
N
0.2797
12
0.6872
12
TABLE 111, A N I M ARESULTS, L PERIOD 11: PREDISTURBANCE (MEANi SD)
Measure
LSI
FI
pVALUE3
N
MeanError (mmHg) STD. Dev. of Error ("Hg) Input STD. Dev (dmin)
-1.03 f 3.18
-0.58 rt 1.24
0.5691
II
2.65 f 2.53
2.65 f 2.09
0.9873
I1
27.00 f 36.58
18.27 f 14.63
0.4578
I1
that only a minor adjustment in the background was needed to stabilize the system. Quantitative measures for the paired luns between the LSI and the FI are summarized in Tables 11-V. Only results from the student's paired t-tests are shown because the two nonparametric tests (the sign test and the Wilcoxon rank sum test) produced similar results. Statistically significant differences occurred only for input standard deviation during continued dis-
DISCUSSION Previous investigators who have applied ARMAX based adaptive control to physiological systems (for example, [ 11, [ 171, [21]) either did not consider the background level or used one of the techniques that are shown here to lead to instability. Clarke et al. [4] developed the K-incremental ( K - I N C ) controller to compensate for background. It is identical to the FI when G ( q - ' ) is specified as a Kth-order delay ( q - K ) . It can be shown that such a choice for G can double the variance of the noise on the modified output data used for dynamics estimation if e ( t ) and e ( t - K ) in (1) are uncorrelated. Another approach similar to the FI may be found in Sripada and Fisher [16]. Their approach was to subtract an estimated mean from the input and output data for dynamics estimation. This method may be useful for processes with known means or well conditioned data. However, it is not designed to handle situations where the input and output means are unknown or changing, such as at startup and during setpoint changes. Tham et al. [ 181 and Latawiec and Chyra [ 1 11 independently proposed an algorithm very close to the FI; however, they assumed that the filtered version of the output will be equal to the setpoint. While simplifying the equations of the algorithm, this assumption is incorrect for plants with a time-varying background. Nevertheless, Tham et al. [18] reported that the integrative action provided by the algorithm resulted in zero steadystate error, and that their controller was less sensitive to colored noise than the K-INC of [4]. Interestingly, comparisons of their algorithm with the K-INC and the LSI showed almost no performance differences [ 181. It is possible that their simplifying
I E E t T R A N S A C T I O N S O N B I O M E D I C A L FNGINEERIN(;. VOL. 3X. NO 3. M A R C H 1991
278 Y
x DATA POINT ~
x
- DATA POINT
FIXED b = T R U E b
\ '
TRUE
/
TRUE b ESTIMATED LINE
FIXED b
x 0
OLD DATA POlNT NEW DATA POINT
OLDb
\
Fig. 5 . Linear regression thought experiinent: estiiiiated line U hen (a) thc correct Y-axis intercept is specified to the LSI. (b) a poor intercept i \ specified to the LSI, and ( c ) the intercept changes while using the ALSI. Note that the estimated line is correct only I n (a).
that lowering the cutoff degrades the controller's performance during background changes (probably because the dc bias is removed too slowly), while raising the cutoff increases the controller's susceptibility to noise (probably because too much of the signal is removed from the data, leaving only high frequency noise). In the extreme cases, control approaches that of the ALSI (for low frequency cutoffs) or the K-INC (for high frequency cutoffs). Cutoff frequencies near the inverse of the sampling period generally produced good results for us, although this observation may be application dependent. In the animal experiments, the most important finding was that the LSI could be made to freeze and the ALSI to oscillate under conditions where the FI did not (as demonstrated by Figs. 3 and 4). It was also demonstrated that in cases where both the F1 and the LSI produced stable control, the controller performances, as judged by output measures. were not different for the two identifiers. For the same quality of control, however, the input was smoother when using the FI than when using the LSI. Since the MAPiSNP system is well damped, problems developing in the controller will more likely be seen first in the input rather than in the output, as in Fig. 3.
REFERENCES assumption and choice of filter cutoff limited the performance of their controller. As illustrated by the computer simulations reported in this paper, the standard LSI algorithm must bias the A and B parameter estimates after a change in background in order to eliminate steady state errors. This biasing may cause oscillations in the inputs and/or outputs, and slowly decaying mean errors. If the biasing becomes too large the controller may freeze as in Fig. 3 (left column). This effect is easily demonstrated by a simple thought experiment. Consider the case of estimating the slope m of the line y = t 7 ~ x b where b corresponds to background and in to sensitivity. Applying the LSI to estimate m results in a good estimate when h is known [Fig. 5(a)]: however. an estimate with the wrong sign as in Fig. 3 (left column) may be obtained when h is specified incorrectly [Fig. S(b)l. Indeed, the importance of estimating the background is illustrated in Fig. 4 where with just a small improvement in the background estimate, control becomes stabilized. The ALSI explicitly estimates a low frequency term. However, it may not estimate background correctly since it estimates all of the parameters together and at the same rate even if only the background changes [4]. Referring to Fig. S(c), note that before a disturbance in background, the plant parameters may be well estimated, resulting in satisfactory control. However, after a disturbance new data may be incongruous with old data, so that the ALSI may estimate the wrong background and the wrong sensitivity. Such behavior occurred repeatedly in our simulations and animal trials, as i n Fig. 3 (middle column). In all of our simulations except those with large amplitude zero mean noise ( S N R = 1.5). the controller using the FI performed better than the ones using the LSI or the ALSI. It had faster recovery from offset disturbances, smaller amplitude mean output errors, and smaller overshoots. It is possible that the FI's performance for the large noise cases could be improved by using a lower cutoff frequency and/or a higher order filter for
+
G. From our simulation studies, we have found that increasing the filter order tends to decrease the controller's susceptibility to noise and improves background tracking. We have also found
N. A. Normann. "Adaptive control of blood pressure." IEEE Trans. Biott/ed. 0 7 g . . VOI. BME-30. pp. 168-176. 1983. 121 D. W . Clarke, "Self-tuning control of nonminimum-phase systems." Automuricer. vol. 20. pp. 501-517. 1984. 131 D . W . Clarke and P. J. Gawthrop. "Self-tuning controller." Proc,. I E E .. vol. 122. no, 9. pp. 929-934. 1975. 141 D. W . Clarke, A. J . F. Hodgson. and P . S . Tuffs, "Offset problem and K-incremental predictors in self-tuning control," IEE Proc.. vol. 130, part D. no. 5 . pp. 217-225, 1983. [SI G . F. Franklin and J . D. Powell, Digital Control qf Dynmmic S~sterm. Reading MA: Addison-Wesley. 1980. ch. 3. pp. 6673. [61 G. C. Goodwin and K . W . Sin. Adupfive Filtering. Prediction, mid Conrrol. Englewood Cliffs. N J : Prentice-Hall. 1984, ch. 6. pp. 224-228. 171 A. G . Gilnian er e r / . . Eds. Goodtnuri u r d Gilniari's The Pharmaco/o,qiccrl Busis of 7'herupruric.s. 7th ed.. New York: Macmillan, 1985. (81 H. J. Chizeck. R. W . Jelliffe. and P. W . Cheung. (Eds.). "Special issue on adaptive control and drug delivery." vol. BME-34. IEEE Trans. Biomed. Eng.. pp. 56.5-656. Aug. 1987. 191 P. G. Katona. "On-line control of physiological variables and clinical therapy." CRCRr.1,. Bionrc~/.En?.. vol. 8. pp. 281-310, 1982. I IO] A. J . Koivo. "Microprocessor-ha\ed controller for pharmacodynamical applications. ' ' I E E E Trtrr7.s. Automa. Contr. vol. AC26. no. 5 . pp. 1208-1213. 1981. I I ] K . Latawiec and M. Chyra. "Low frequency drift and long run effects in self-tuning control." presented at 5th IFAC Symp. Identification Syst. Parameter Estimation. Damstadt, 1979. pp. 1169-1 178. 121 J . S . Milton and J . 0. Tsokos. Srutr.\ric~ulMethods iri the Biologi c ~ crncl / Hculth Scierices. New York: McGraw Hill, 1983. 131 J . Neter. W . Wasserman. and M. H . Kutncr. AppliedLinrarSruti.sric,ul Modcl.\: Rrgrc>.s,\io,i, A / i t r l ~ . ~ i .of' s Varicrnce, and E-uperirticnrul Drsigm, 2nd ed. Homewood. IL: Richard D. Irwin. 1985. 141 PhJ,.\ic,icrtr.c' Desk Rrft?rotic.c,.14th ed. Oradell, NJ: Medical Economics. 1990. 151 J . B. Slate and L. C . Sheppard. "Automatic control of blood pressure by drug infusion," IEE Proc., vol. 129, part A. no. 9 , pp. 639-645, Dec. 1982. 161 N . R. Sripada and D. G. Fisher. "Improved least squares identification for adaptive controllers." in Proc. 1987 ACC, Minneapolis. MN 1987. pp. 2027-2037. 171 K . S. Stern. H. J . Chizeck. B. K . Walker, P. S. Krishnaprasad, P. J . Dauchot. and P. G. Katona, "The self-tuning controller: [ I \ J . M. Arnsparger. B. C . Mclnni\. J . R. Glover. Jr.. and
TIMMONS et ol. : ADAPTIVE CONTROL ENHANCED BY BACKGROUND ESTIMATION
Comparison with human performance in the control of arterial pressure,” Ann. Biomed. Eng., vol. 13, pp. 341-357, 1985. [18] M. T. Tham, A. J. Morris, and G. A. Montague, “Self-tuning process control: A review of some algorithms and their applications,” in Proc. 1987 ACC, Minneapolis, MN, 1987, pp. 9961001. [19] G . I. Voss, H. J. Chizeck, and P. G. Katona, “Regarding selftuning controllers for nonminimum phase plants,’’ Auromatica, vol. 23, no. 3, pp. 405-408, 1987. [20] G. I. Voss, H. J. Chizeck, and P. G. Katona, “Self-tuning controller for drug delivery systems,’’ Inr. J. Contr., vol. 47, no. 5, pp. 1507-1520, 1988. [21] G. I. Voss, P. G. Katona, and H. J . Chizeck, “Adaptive multivariable drug delivery: Control of arterial pressure and cardiac output in anesthetized dogs,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 617-623, Aug. 1987. [22] D. R. Westenskow, “Automating patient care with closed-loop control,” M.D. Comput., vol. 3, no. 2, pp. 14-20, 1986.
279
Howard J. Chizeck (S’74-M’79-SM’88) was
born on March 27, 1952 in Columbus, OH. He received the B.S. and M S degrees in systems and control engineering from Case Western Reserve University, Cleveland, OH, i n 1974 and 1976, respectively, and the Sc D degree in electrical engineering and computer science from the Massachusetts Institute of Technology, Cambridge, in 1982. He is an Associate Professor in the Departments of Systems Engineering and Biomedical Engineering, Case Western Reserve University His research interests involve stochastic control theory and the applications of control engineering to biomedical problems Current projects include the design and analysis of controllers for the electrical stimulation of paralyzed muscle, adaptive control of drug delivery, identification and control of cerebral ventricle volume dynamics for hydrocephalus management, algebraic modeling of DNA sequence-geometry relationships and applications of algebraic systems theory to coding Dr. Chizeck is a member of AAAS, Sigma Xi and the Rehabilitation Society of North America (RESNA), the Biomedical Engineering Society, and the International Neural Network Society. He is also a member of the International Federation of Automatic Control (IFAC) Technical Committee on Biomedical Engineering. Peter G. Katona (M’69) was born
William Donald Timmons (S’88) was born in Allentown, PA, on March 25, 1961. He received the B.S. and M.S. degrees in biomedical engineering from Case Western Reserve University, Cleveland, OH, in 1983 and 1989, respectively, where he is currently in the doctoral program. Since 1988, has been the Student Activities Coordinator for the IEEE Engineering in Medicine and Biology Magazine, for which he writes the “Student’s corner.” His research interests include adaptiv e control, modeling and identification, biomedical signal processing, neural nets, and computational neurobiology. Mr. Timmons is a member of the IEEE Engineering in Medicine and Biology Society.
in Budapest, Hungary, in 1937. He received the B S degree from the University of Michigan, Ann Arbor, in 1960, and the M S. and Sc.D degrees from Massachusetts Institute of Technology, Cambridge, in 1962 and 1965, respectively, all in electrical engineering. In 1969 he joined the Department of Biomedical Engineering at Case Western Reserve University, Cleveland, OH, where he is now Professor. He served as chairman of the department during 1980-87 Currently he is on a two-year temporary assignment as director of the Biomedical Engineering and Aiding the Disabled program at the National Science Foundation in Washington, D.C. Dr Katona’s research interest is the natural and artificial control of the cardio-pulmonary system.
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 3. MARCH 1991
Adaptive Control is Enhanced by Background Estimation William Donald Timmons, Student Member, IEEE, Howard J. Chizeck, Senior Member, and Peter G. Katona, Member, IEEE
Abstract-The automated control of physiological variables must often contend with an unknown and time-varying background (i.e., the output level corresponding to no input). To allow for simultaneous realtime identification of background as well as the parameters of an autoregressive moving average model with exogenous inputs (ARMAX model) during adaptive control, a "floating identifier" (FI) approach was developed which may be used with most recursive identification algorithms. This method separates input and output data into low- and high-frequency components. The high-frequency components are used to identify the ARMAX model parameters and the low-frequency components to identify background. This approach was evaluated in computer simulations and animal experiments comparing an adaptive controller coupled to the FI with the same controller coupled to two other standard least squares identifiers. In the animal experiments, sodium nitroprusside was used to control mean arterial pressure of anesthetized dogs in the presence of background changes. Results showed that with the FI, the controller performed satisfactorily, while with the other identifiers, it sometimes failed. It is concluded that the FI approach is useful when applying ARMAX-based adaptive controllers to systems in which a change in background is likely.
w
INTRODUCTION
EN automated control is applied to physiological sys-
tems, the output level corresponding to no input (i.e., the background) is often unknown and time-varying. For example, mean arterial blood pressure is normally near 100 mmHg, but it often deviates from this value even in the absence of an obvious cause [ 151. When applying model-based adaptive controllers to such systems, one should include a background term within the system model; however, estimation of a time-varying background is not straight-forward when using autoregressive moving average models with exogenous inputs (ARMAX models [ 6 ] ) .Indeed, we later show that two popular techniques can result in unstable control and failure in the presence of large swings in background. In this paper, we present thefloating identiJier (FI) approach, a method that estimates time-varying background when using recursive identification algorithms for linear or linearizable systems. It operates by dividing the inputs and outputs of the controlled system into low- and high-frequency components. Background, which is a constant or slowly drifting quantity, is estimated from the low-frequency components. The system dyManuscript received January 26, 1989; revised July 11, 1990. This work was supported by NSF Grants ECS84-00765, BCS-89087 13 and a contract from Eli Lilly and Company. W. D. Timmons is with the Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106. H . J. Chizeck is with the Department of Systems Engineering, Case Western Reserve University, Cleveland, OH 44106. I P. G . Katona is with the Biomedical Engineering and Aiding the Disabled Program, National Science Foundation, Washington, DC, on leave from the Department of Biomedical Engineering, Case Westem Reserve University, Cleveland, OH 44106. IEEE Log Number 9042175.
IEEE,
namics, on the other hand, are estimated from the high-frequency components. For linear systems, superposition applies so that the original parameterization is maintained. For nonlinear systems, the FI technique can be thought of as producing a linear dynamical model about a slowly varying operating point. The choice of bandwidth for the low and high frequency components is related to the frequency spectrum of the background and dynamics, and will be discussed later. Our hypothesis was that when large background disturbances disrupt the operation of the control system, performance can be improved by using the FI. Furthermore, when such disturbances are absent, the FI is transparent, neither helping nor hindering the controller's performance. To test this claim, computer simulations and animal experiments were used to compare the FI to two standard system identifiers during control.
STANDARD IDENTIFICATION TECHNIQUES Adaptive controllers previously used for controlling physiological variables are surveyed in [8], [ 9 ] ,[ 2 2 ] .Typically, these controllers obtain on-line parameter estimates of an ARMAX time series model relating the inputs and outputs of the system by a recursive least squares ident$er ( L S I ) [6]. Control is then based on the past inputs and outputs, using a control law modified by the identified model parameters. In this paper, an exponentially weighted recursive LSI is used (see [ 2 0 ] for a further description of our control system). Consider the recursive least squares identification of the following ARMAX model [ 6 ] :
A(q-')Y(t)= B(q-')U(t)
+ e(?)
(1)
where Y ( r ) , U ( t ) , and e ( r )are, respectively, the output, input, and zero mean noise at time t . A ( 4 - ) and B ( q- ) are polynomials in q - ', the backwards time shift operator, and are defined by
'
A(q-') = 1
-
'
a , q - ' - . . . - a,,q-"
B(q-I) = b,q-'
+ . . . + b,,,q-"'.
(2)
(3)
T o apply an LSI, a format is used in which the parameters and data are represented as
~ ( t=) X r ( t
-
110
(4)
where Y ( t ) is the current output, 0 is the vector of parameters to be estimated, and X ( t ) is an appropriate regression vector. For ( l ) , we have
07 X r ( t - 1)
0018-9294/'9110300-0273$01 .OO 0 1991 IEEE
=
[ a l . . . a,, b l . . . b,,,]
= [ ~ ( -t
U(t
-
(5)
I ) . . . ~ ( -t n )
1 ) . . . U(t
-
m)].
(6)
I E E E TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL 38. NO 3 . MARCH 1991
27J
The parameter vector 0 contains no terms relating to background. If a known background were present, however, one could subtract it from the output at each time, and still use the LSI to identify 0 . Background may be included explicitly in the ARMAX model by letting e ( t ) have a nonzero mean value. Breaking e ( t ) into two terms, such that slowly varying components are lumped into D ( t ) and all else into U ( t ) , (1) may be written as
A(q-')Y(t)
=
B ( q - ' ) U ( f )+ D ( f ) + u ( t ) .
Ut)
U(t)
X7(r
= [al
I)
=
-
. . . a,, b l . . . b,,, D ]
[Y(t - 1 ) . . . Y(t U(t
-
-
b I
LOW PASS
a-'(q-')s(r)
(9)
+
) A ( 9 - I 1Y( f )
=
A ( q - I ) YL, ( f )
G(q-')B(q-')U(t) = B(q-I)UL,(f).
(1 1)
(12)
Now, the separation filter G is chosen so that
G ( q - ' ) D ( t )= D ( t )
(13)
G(q-')zt(t)= 0
(14)
and
i.e., G is chosen so that D ( t ) is unaffected by filtering, but high frequency noise is attenuated. Multiplying both sides of (7) by G results in a low-pass filtered version of (7)
A(q-I)YLP(t)
=
B(q-l)ULP(t)+ D ( t )
(15)
which, when subtracted from (7) results in a high-frequency equation A ( 9-
'
YHP ( I)=
(q
'
UHP ( )
+
(t ) .
(16)
Here YHp(?)= Y(f) - YLP(f)and UHp(t) = U ( t ) - U L P ( t ) . The dynamic parameters A ( q - l ) and B ( q - ' ) are estimated from the relation (16). In this case, (4) involves vectors
O'-=
[ a l . . . a,, h , . . . b,,,]
p+-
-
:_._________... : : A(q-'), ^B(q-')
Fig. 1. Adaptive controller using floating identifier: the adaptive controller uses a control law altered by both the estimated dynamic model parameters and the on-line offset estimates. ( / ( I ) is the input, Y ( f )the output, E ( r ) a noise on the system; subscripts LP and HP, respectively, signify low-pass and high-pass filtered data; arrows indicate signals (solid) and parameters (dashed).
X'(f
~
1)
=
/YHp(?- 1 ) . . . YHp(f UHp(t
-
I) ...
UHp(1
-
/I)
(18)
/TI)].
The low-pass noise component estimate @ ( r ) is calculated from (15):
d(f) = a(q-')YLP(f)- &q-l)UL,(f).
(19)
c,(f)
THE FLOATING IDENTIFIER The idea behind the FI is twstrip dc and low frequency components from the data before parameter identification, effectively "floating" the identifier on the dc signal. After parameter identification, these components are used to calculate background. The overall scheme for the FI is shown in Fig. 1. The low pass filter G ( q - I ) separates low and high frequency components of the input and output signals. Let the subscript L P indicate the low-pass filtered version of each variable by the separation filter G ( q-' ): -I
LOW PASS
(10)
where denotes an estimated quantity. This method is frequently suggested as the way to handle unknown offsets ( e . g . , see [l], [3], [6], [ l l ] ) . As will be shown later, in some cases of background shift, use of the ALSI may lead to incorrect identification of the sign of the system's steadystate gain. This error can cause a stabilizing negative feedback loop to become one of destabilizing positive feedback.
aq
-
(8)
If desired, an estimated background ( f,,(1 ) ) may be calculated using P"(f) =
.4
FILTER
(7)
n)
1 ) . . . U ( f - m ) 11.
Lt) =E Li')U Lt) +E Lt)
CONTROL LAW
The parameters of this model may be estimated by an augmented LSI (ALSI), in which case (4)involves vectors
0'
Y(t)
SYSTEM
A LQ') Y
(17)
An estimated background may be calculated using (10). The FI can be used for multiple-input multiple-output systems if the corresponding filter polynomial matrix G ( q - ' ) commutes with the corresponding polynomial matrices A ( q - ) and B ( q - ' ) . A convenient example is G = g ( q - I ) 1 w h e r e g ( q - ' ) is a low-pass filter transfer function designed to satisfy (13) and (14). and I is the identity matrix.
'
SIMULATIONS Before the identifiers were tested in animals, a series of computer simulations were run to characterize algorithm performance under completely known conditions. Each identifier described above (the LSI, ALSI, and FI) was coupled to the same predictive controller and the identifier/controller combinations (designated according to the identification method as either the LSI, ALSI, or FI) were then evaluated. We chose the control advance moving average controller (CAMAC) as the predictive controller because of its suitability for drug infusion systems and our previous experience with it [16]-[18]. At each sampling time, the CAMAC chooses in input U ( f ) to drive the output to a setpoint at a future time f k . Here, k is the time horizon, and is greater than or equal to the system's input-output delay. The C A M A C assumes that all future inputs will be equal to the current input, but then repeats the computations every sampling interval so that the actual input changes over time. By choice of horizon k , the CAMAC may be tuned between minimum variance control (horizon equal to the system's input-output delay) and steady state control (horizon equal to infinity). T o concentrate attention on the identification algorithm, the CAMAC's automatic horizon selection was disabled. Since simulations using various types of linear plants resulted in similar outcomes, the results from only one type of plant are reported here. The system. a first-order linear ARMAX plant
+
275
TIMMONS et a l . : ADAPTIVE CONTROL ENHANCED BY BACKGROUND ESTIMATION
TABLE I SIMULATION RESULTS FOR FIRSTDISTURBANCE INTERVAL
(%)
Recovery Time (Steps)'
Mean Squared Em02
SteadyState Err02
LSI ALSI
-23.5 -23.5 -24.0
13 13 6
236 237 82
-4.4 -4.4
LSI ALSI
25 25 8 13
499 504
-7.0 -7.0
FI
-31.0 -31.0 -29.5 -29.5 -29.5 -28.5
6
LSI ALSI
-53.5 -53.5
25 25 25
Peak Error
Conditions
-251
.
-
k = L,
No Noise
k -201
25
I
q YO ,
05-52-
0
25
54
73
1W
125
I50
0
Z5
M
75
TIME
TIME
TIME
LSI
ALSI
FI
,
100 125
yo 150
Fig. 2. Simulated behavior of the three identifiers: LSI (left). ALSI (middle). and FI (right). An initial period of control is followed by a background disturbance at the arrow. After recovery, the setpoint is changed and the procedure repeated. The simulation is concluded with a fina! setpoint change. The controller horizon is fixed at 2. Y * , U. Y,,, and Y, are the setpoint, output, background, and estimated background, respectively. Note that only the FI fully recovers from the background disturbance and has zero overshoot.
which was previously used as a benchmark by Clarke [2] and then Voss et al. [ 191, is simulated by y(t) = 0.7y(t - 1)
+ u(r -
1)
+ 2u(t
-
2)
+ e(r)
(20)
w h e r e y ( t ) , u ( r ) , and e ( t ) are, respectively, the output, input, and zero-mean white noise at time t . Baseline was added to the simulated plant as follows:
U t ) = Y(t) +
K,(t)
(21)
where &, ( 1 ) is the plant background, and Y ( t ) is the resultant plant output. All outputs and inputs are in arbitrary units. The protocol for the simulations was designed to test the different closed loop performances of the identifiericontrollers in the presence of various background disturbances. In each case, an initial period of control was followed by a background disturbance. After adequate time was allowed for the output to retum to the setpoint, the setpoint was changed and the procedure repeated. One A-term ( a , ) and two B-terms ( b , and b 2 ) were estimated by the adaptive controllers. A fourth-order ITAE (minimal integral of time-multiplied absolute error) filter [5] was used for the separation filter G in the FI design. The cutoff frequency was set equal to the inverse of the sampling interval, resulting in a normalized filter. The controller performance for a control horizon of two is illustrated in Fig. 2 . The performance of the three identifiers were similar before the first offset disturbance (first arrow). After the disturbance, the LSI and the ALSI exhibited increasingly larger overshoots, input excursions, and slowly decaying offsets, while the FI maintained overshoots and input excursions as at start-up, and rapidly and smoothly eliminated offsets. Also, the ALSI often estimated a background that moved in the wrong direction (as in Fig. 2 after the second disturbance, 100 < t < 125). Table I summarizes the controllers' responses to the first background disturbance when different controller tunings were used, o r when zero-mean noise was added to the
= 4, No Noise
k = 2,
Small Noise (SNR = 25) k
2, Large Noise (SNR = 1.5)
=
FI
FI
LSI ALSI
FI
65.0
13
0.0
151
0.0
215
216 114
-3.0 -3.0 0.8
829 832 938
2.5 2.5 3.8
'Time from onset of disturbance until the output remains within of the setpoint. zErrors calculated as percentage of setpoint,
15%
output. Note how similar the measures for the LSI and the ALSI are (for example, mean square error typically differs by less than 0.5%). More striking, note that in all the simulations except the one with large noise, the FI recovered from the background disturbance in less than half the time needed by the LSI and the ALSI, and that both mean square error and the steadystate error was less for the FI than for the LSI. It was only the simulations in which a large zero mean noise was added to the plant outputs that the FI performed worse than the LSI or the ALSI. Since the performance of the LSI was almost indistinguishable from that of the ALSI throughout our simulations, we decided to use only the LSI in further comparisons with the FI. Therefore, except for a few trials using the ALSI, our animal experiments concentrate on the pairwise comparison of the FI with the LSI.
ANIMALEXPERIMENTS Experimental Procedure Mean arterial pressure (MAP) in five anesthetized dogs was lowered using sodium nitroprusside (SNP), a vasodilator. Phenylephrine, a vasopressor, was used to test the performance of the controllers during a change in the M A P background level. Anesthesia was induced with sodium pentobarbital (30 m g / k g IV bolus) and followed with maintenance doses as needed. As in [lo], catheters were inserted into the femoral vein to administer drugs and into the descending aorta through the femoral artery to measure arterial pressure (Bentley Model 800 blood pressure transducer, Gould/Brush transducer coupler). The pressure signal was low-pass filtered ( 5 s time constant) to derive MAP, which was then sampled every 20 s (or in some trials, 30 s) by a DEC MINC computer. The pressure samples and the calculated SNP infusion rates were stored on disk. An W A C 630 volumetric infusion pump was used to deliver SNP (100 p g / m L in a 5 % dextrose solution [7], [14]) under computer control; another W A C 630 was controlled manually to administer phenylephrine (100 p g / m L in saline). Following the previous example of Stem et al. [17] and Voss et al. [20], two A terms were identified by the controller. The number of B terms was selected to cover a minimum range of
216
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL 38, NO 3 . MARCH 1991 MEAN ARTERIAL PRESSURE
90 s, the longest expected dead time. Three B terms were used for the first dog because the sample/control interval was 30 s; five B terms were used for the last four dogs because the sampleicontrol interval was 20 s . As in the simulations, the FI used a fourth-order ITAE filter with a normalized cutoff for the separation filter G. Therefore, the cutoff was set to 0.033 or 0.05 Hz depending on the sampling interval. In a few of the paired trials, a cutoff of 0.5 Hz was used to test the effects of the cutoff frequency on the FI’s performance. Our experiments were designed to challenge the controllers with a large background disturbance. Since the effects of phenylephrine (our source of background disturbance) varied from dog to dog, we first determined the amount of phenylephrine (between 25 and 83 pg/min) that would require a change in SNP greater than 100 pg/min if the setpoint were to be maintained. Once an appropriate phenylephrine level was determined, the testing procedure involved the sudden administration of this amount after control had been established for at least 14 min. In one case study, we administered a very large dose of phenylephrine (120 p g / m i n ) to try to “break” the controllers (Fig. 3). The initial MAP was between 120 and 140 mmHg in 14 trials, above 140 mmHg in four trials, and below 120 mmHg in five trials. Prior to closed loop control, four to 8 min of open loop probing was used for system identification (typically three pulses of S N P at a rate of 50 pg/min). Two minutes was typically allotted from the last pulse until the start of closed-loop control. During this time, the M A P returned to within 3 % of initial baseline in 3 5 % of the trials. In the worst cases, the MAP returned to within 15.8% below and 13.3% above the initial MAP. The objective upon closing the loop was to lower the MAP to a prescribed level (typically 100 mmHg). Since the initial background level varied, ten trials resulted in lowering the MAP 30 mmHg or more from baseline, and 13 trials resulted in lowering the MAP less than 30 mmHg. To reduce bias from changes in the animal’s condition over time, the order of the tested controllers was randomized. In any of the trials, if it looked like the controller had frozen (stopped infusing S N P for more than four samples when the infusion rate should have increased), the trial was terminated, the phenylephrine stopped, and SNP manually infused.
21
O J
SODIUM NITROPRUSSIDE
DISTURBANCE (120
ug/arn
PHENYLEPHRINE)
El ESTIMATED BACKGROUND
E
1
,2/1“
-
~
0 1
-2
ESTIMATED S T E A D Y STATE P L A N T GAIN
:;lo*
0
:r > ,i /i ;
15
LSI
sb b
ox
7
Is
ALSI
io
;
1;
T I M E IN So H l N U T E S
FI
Fig. 3 . Control of MAP in an anesthetized dog: for each of the three identifiers (LSI, ALSI, and F1. from left to right), control was started and allowed to settle. After steady state was reached, a challenge was given by infusing phenylephrine (120 pg/min. duration indicated by bars across the center of the figure). Note that control was stable only when using the FI.
standard deviation of the error, and standard deviation of the input (SNP infusion rate)-were used to compare their performances during the other time periods. The student’s paired t-test and two nonparametric tests (the sign test and the Wilcoxon rank sum test [12]) were applied to the measures to test for statistical significance. Finally, the Bonferroni method [ 131 was used to test for joint significance of the multiple measures ( n = 11).
Data Analysis
Results
Trials involving the FI and the LSI were broken into four periods of control: startup, predisturbance, disturbance onset, and continued disturbance. Startup was defined as the initial 5 min of control immediately following open loop identification or a change in setpoint. Predisturbance was defined as the time following start-up until the onset of the disturbance or termination of the run, whichever came first. Disturbance onset was defined as the initial 5 min of the disturbance; and continued disturbance as the remaining time until cessation of the disturbance. Only paired runs in which the dog’s response did not change significantly were included in this analysis. Also, since meaningful disturbance data could not be collected from the paired run in which the LSI froze, it was excluded from the analysis, resulting in a slight bias in favor of the LSI. Two measures-overshoot and settling time (defined as the time from start of control for the M A P to stay within five mmHg of the setpoint)-were used to compare the different identifier/ controller performances during startup. Three measures-mean e r r o r (where error is defined as the setpoint minus the MAP),
Fig. 3 shows what can happen in the presence of large swings in background caused by phenylephrine infusion. After the dramatic change in background, the LSI estimated a positive patient gain which resulted in zero S N P infusion rate. Since the controller had frozen, the experiment was terminated. The ALSI also estimated a positive gain, but unlike the LSI, it produced wild oscillations in the S N P infusion rate. This excitation resulted in the eventual reidentification of the plant and subsequent stabilization. As seen often in the simulation studies, the estimated background initially moved in the wrong direction when the disturbance was applied. The FI had little problem with this challenge. The estimated background increased as expected and the gain maintained a negative value throughout the run. The increased S N P rate, showing only moderate oscillations, kept the M A P near the target level. The effectiveness of the FI is further shown in Fig. 4 . During the disturbance in this experiment, the input and output oscillations ceased when the FI was substituted for the LSI. Note
TIMMONS er a l . : ADAPTIVE CONTROL ENHANCED BY BACKGROUND ESTIMATION DISTURBANCE
TABLE IV ANIMAL RESULTS,PERIOD 111: DISTURBANCE ONSET(MEAN SD)
(80 u g l m i n P H E N Y L E P H R I N E )
21
MEAN A R T E R I A L P R E S S U R E
Measure
SODIUM NITROPRUSSIDE
Mean Error (mmHg) STD. Dev. of Error WmHg) Input STD.
:I
-
LSI
211
7.30 f 2.88 5.81
pVALUE3
N
6.22 f 4.34
0.1807
9
6.54 f 1.70
0.4082
9
78.55 f 44.89
0.3423
9
FI
LSI
+ 2.13
65.10 f 24.36
TABLE V A N I M ARESULTS, L PERIOD IV: CONTINUED DISTURBANCE (MEAN SDI
+
+--
~
FI
--D
~
Measure
LSI
FI
pVALUE'
N
2.54 f 2.13
0.92 f 2.85
0.3209
9
4.09 f 2.03
3.68 & 1.70
0.5769
9
69.11 f 27.35
41.00 f 26.82
0.0027
9
0 N
Mean Error (mmHg) STD. Dev. of Error (mmHg) Input STD. Dev . (dmin)
ESTIMATED BACKGROUND mD 1a _ E- '
r42 0
'Student's paired t-test (two-sided)
turbance (Table V) where the FI was better (with more than a 95 % confidence level). Adjustments for eleven joint measures TABLE I1 ANIMAL RESULTS, PERIOD 1: STARTUP (MEAN SD)
Measure
LSI
Overshoot ("Hg! Settling Time (Sec)
-7.79 f 8.96
-4.78
88.33 f 53.06
96.25 f 68.73
FI
3.56
using the Bonferroni method did not change this result.
pVALUE'
N
0.2797
12
0.6872
12
TABLE 111, A N I M ARESULTS, L PERIOD 11: PREDISTURBANCE (MEANi SD)
Measure
LSI
FI
pVALUE3
N
MeanError (mmHg) STD. Dev. of Error ("Hg) Input STD. Dev (dmin)
-1.03 f 3.18
-0.58 rt 1.24
0.5691
II
2.65 f 2.53
2.65 f 2.09
0.9873
I1
27.00 f 36.58
18.27 f 14.63
0.4578
I1
that only a minor adjustment in the background was needed to stabilize the system. Quantitative measures for the paired luns between the LSI and the FI are summarized in Tables 11-V. Only results from the student's paired t-tests are shown because the two nonparametric tests (the sign test and the Wilcoxon rank sum test) produced similar results. Statistically significant differences occurred only for input standard deviation during continued dis-
DISCUSSION Previous investigators who have applied ARMAX based adaptive control to physiological systems (for example, [ 11, [ 171, [21]) either did not consider the background level or used one of the techniques that are shown here to lead to instability. Clarke et al. [4] developed the K-incremental ( K - I N C ) controller to compensate for background. It is identical to the FI when G ( q - ' ) is specified as a Kth-order delay ( q - K ) . It can be shown that such a choice for G can double the variance of the noise on the modified output data used for dynamics estimation if e ( t ) and e ( t - K ) in (1) are uncorrelated. Another approach similar to the FI may be found in Sripada and Fisher [16]. Their approach was to subtract an estimated mean from the input and output data for dynamics estimation. This method may be useful for processes with known means or well conditioned data. However, it is not designed to handle situations where the input and output means are unknown or changing, such as at startup and during setpoint changes. Tham et al. [ 181 and Latawiec and Chyra [ 1 11 independently proposed an algorithm very close to the FI; however, they assumed that the filtered version of the output will be equal to the setpoint. While simplifying the equations of the algorithm, this assumption is incorrect for plants with a time-varying background. Nevertheless, Tham et al. [18] reported that the integrative action provided by the algorithm resulted in zero steadystate error, and that their controller was less sensitive to colored noise than the K-INC of [4]. Interestingly, comparisons of their algorithm with the K-INC and the LSI showed almost no performance differences [ 181. It is possible that their simplifying
I E E t T R A N S A C T I O N S O N B I O M E D I C A L FNGINEERIN(;. VOL. 3X. NO 3. M A R C H 1991
278 Y
x DATA POINT ~
x
- DATA POINT
FIXED b = T R U E b
\ '
TRUE
/
TRUE b ESTIMATED LINE
FIXED b
x 0
OLD DATA POlNT NEW DATA POINT
OLDb
\
Fig. 5 . Linear regression thought experiinent: estiiiiated line U hen (a) thc correct Y-axis intercept is specified to the LSI. (b) a poor intercept i \ specified to the LSI, and ( c ) the intercept changes while using the ALSI. Note that the estimated line is correct only I n (a).
that lowering the cutoff degrades the controller's performance during background changes (probably because the dc bias is removed too slowly), while raising the cutoff increases the controller's susceptibility to noise (probably because too much of the signal is removed from the data, leaving only high frequency noise). In the extreme cases, control approaches that of the ALSI (for low frequency cutoffs) or the K-INC (for high frequency cutoffs). Cutoff frequencies near the inverse of the sampling period generally produced good results for us, although this observation may be application dependent. In the animal experiments, the most important finding was that the LSI could be made to freeze and the ALSI to oscillate under conditions where the FI did not (as demonstrated by Figs. 3 and 4). It was also demonstrated that in cases where both the F1 and the LSI produced stable control, the controller performances, as judged by output measures. were not different for the two identifiers. For the same quality of control, however, the input was smoother when using the FI than when using the LSI. Since the MAPiSNP system is well damped, problems developing in the controller will more likely be seen first in the input rather than in the output, as in Fig. 3.
REFERENCES assumption and choice of filter cutoff limited the performance of their controller. As illustrated by the computer simulations reported in this paper, the standard LSI algorithm must bias the A and B parameter estimates after a change in background in order to eliminate steady state errors. This biasing may cause oscillations in the inputs and/or outputs, and slowly decaying mean errors. If the biasing becomes too large the controller may freeze as in Fig. 3 (left column). This effect is easily demonstrated by a simple thought experiment. Consider the case of estimating the slope m of the line y = t 7 ~ x b where b corresponds to background and in to sensitivity. Applying the LSI to estimate m results in a good estimate when h is known [Fig. 5(a)]: however. an estimate with the wrong sign as in Fig. 3 (left column) may be obtained when h is specified incorrectly [Fig. S(b)l. Indeed, the importance of estimating the background is illustrated in Fig. 4 where with just a small improvement in the background estimate, control becomes stabilized. The ALSI explicitly estimates a low frequency term. However, it may not estimate background correctly since it estimates all of the parameters together and at the same rate even if only the background changes [4]. Referring to Fig. S(c), note that before a disturbance in background, the plant parameters may be well estimated, resulting in satisfactory control. However, after a disturbance new data may be incongruous with old data, so that the ALSI may estimate the wrong background and the wrong sensitivity. Such behavior occurred repeatedly in our simulations and animal trials, as i n Fig. 3 (middle column). In all of our simulations except those with large amplitude zero mean noise ( S N R = 1.5). the controller using the FI performed better than the ones using the LSI or the ALSI. It had faster recovery from offset disturbances, smaller amplitude mean output errors, and smaller overshoots. It is possible that the FI's performance for the large noise cases could be improved by using a lower cutoff frequency and/or a higher order filter for
+
G. From our simulation studies, we have found that increasing the filter order tends to decrease the controller's susceptibility to noise and improves background tracking. We have also found
N. A. Normann. "Adaptive control of blood pressure." IEEE Trans. Biott/ed. 0 7 g . . VOI. BME-30. pp. 168-176. 1983. 121 D. W . Clarke, "Self-tuning control of nonminimum-phase systems." Automuricer. vol. 20. pp. 501-517. 1984. 131 D . W . Clarke and P. J. Gawthrop. "Self-tuning controller." Proc,. I E E .. vol. 122. no, 9. pp. 929-934. 1975. 141 D. W . Clarke, A. J . F. Hodgson. and P . S . Tuffs, "Offset problem and K-incremental predictors in self-tuning control," IEE Proc.. vol. 130, part D. no. 5 . pp. 217-225, 1983. [SI G . F. Franklin and J . D. Powell, Digital Control qf Dynmmic S~sterm. Reading MA: Addison-Wesley. 1980. ch. 3. pp. 6673. [61 G. C. Goodwin and K . W . Sin. Adupfive Filtering. Prediction, mid Conrrol. Englewood Cliffs. N J : Prentice-Hall. 1984, ch. 6. pp. 224-228. 171 A. G . Gilnian er e r / . . Eds. Goodtnuri u r d Gilniari's The Pharmaco/o,qiccrl Busis of 7'herupruric.s. 7th ed.. New York: Macmillan, 1985. (81 H. J. Chizeck. R. W . Jelliffe. and P. W . Cheung. (Eds.). "Special issue on adaptive control and drug delivery." vol. BME-34. IEEE Trans. Biomed. Eng.. pp. 56.5-656. Aug. 1987. 191 P. G. Katona. "On-line control of physiological variables and clinical therapy." CRCRr.1,. Bionrc~/.En?.. vol. 8. pp. 281-310, 1982. I IO] A. J . Koivo. "Microprocessor-ha\ed controller for pharmacodynamical applications. ' ' I E E E Trtrr7.s. Automa. Contr. vol. AC26. no. 5 . pp. 1208-1213. 1981. I I ] K . Latawiec and M. Chyra. "Low frequency drift and long run effects in self-tuning control." presented at 5th IFAC Symp. Identification Syst. Parameter Estimation. Damstadt, 1979. pp. 1169-1 178. 121 J . S . Milton and J . 0. Tsokos. Srutr.\ric~ulMethods iri the Biologi c ~ crncl / Hculth Scierices. New York: McGraw Hill, 1983. 131 J . Neter. W . Wasserman. and M. H . Kutncr. AppliedLinrarSruti.sric,ul Modcl.\: Rrgrc>.s,\io,i, A / i t r l ~ . ~ i .of' s Varicrnce, and E-uperirticnrul Drsigm, 2nd ed. Homewood. IL: Richard D. Irwin. 1985. 141 PhJ,.\ic,icrtr.c' Desk Rrft?rotic.c,.14th ed. Oradell, NJ: Medical Economics. 1990. 151 J . B. Slate and L. C . Sheppard. "Automatic control of blood pressure by drug infusion," IEE Proc., vol. 129, part A. no. 9 , pp. 639-645, Dec. 1982. 161 N . R. Sripada and D. G. Fisher. "Improved least squares identification for adaptive controllers." in Proc. 1987 ACC, Minneapolis. MN 1987. pp. 2027-2037. 171 K . S. Stern. H. J . Chizeck. B. K . Walker, P. S. Krishnaprasad, P. J . Dauchot. and P. G. Katona, "The self-tuning controller: [ I \ J . M. Arnsparger. B. C . Mclnni\. J . R. Glover. Jr.. and
TIMMONS et ol. : ADAPTIVE CONTROL ENHANCED BY BACKGROUND ESTIMATION
Comparison with human performance in the control of arterial pressure,” Ann. Biomed. Eng., vol. 13, pp. 341-357, 1985. [18] M. T. Tham, A. J. Morris, and G. A. Montague, “Self-tuning process control: A review of some algorithms and their applications,” in Proc. 1987 ACC, Minneapolis, MN, 1987, pp. 9961001. [19] G . I. Voss, H. J. Chizeck, and P. G. Katona, “Regarding selftuning controllers for nonminimum phase plants,’’ Auromatica, vol. 23, no. 3, pp. 405-408, 1987. [20] G. I. Voss, H. J. Chizeck, and P. G. Katona, “Self-tuning controller for drug delivery systems,’’ Inr. J. Contr., vol. 47, no. 5, pp. 1507-1520, 1988. [21] G. I. Voss, P. G. Katona, and H. J . Chizeck, “Adaptive multivariable drug delivery: Control of arterial pressure and cardiac output in anesthetized dogs,” IEEE Trans. Biomed. Eng., vol. BME-34, pp. 617-623, Aug. 1987. [22] D. R. Westenskow, “Automating patient care with closed-loop control,” M.D. Comput., vol. 3, no. 2, pp. 14-20, 1986.
279
Howard J. Chizeck (S’74-M’79-SM’88) was
born on March 27, 1952 in Columbus, OH. He received the B.S. and M S degrees in systems and control engineering from Case Western Reserve University, Cleveland, OH, i n 1974 and 1976, respectively, and the Sc D degree in electrical engineering and computer science from the Massachusetts Institute of Technology, Cambridge, in 1982. He is an Associate Professor in the Departments of Systems Engineering and Biomedical Engineering, Case Western Reserve University His research interests involve stochastic control theory and the applications of control engineering to biomedical problems Current projects include the design and analysis of controllers for the electrical stimulation of paralyzed muscle, adaptive control of drug delivery, identification and control of cerebral ventricle volume dynamics for hydrocephalus management, algebraic modeling of DNA sequence-geometry relationships and applications of algebraic systems theory to coding Dr. Chizeck is a member of AAAS, Sigma Xi and the Rehabilitation Society of North America (RESNA), the Biomedical Engineering Society, and the International Neural Network Society. He is also a member of the International Federation of Automatic Control (IFAC) Technical Committee on Biomedical Engineering. Peter G. Katona (M’69) was born
William Donald Timmons (S’88) was born in Allentown, PA, on March 25, 1961. He received the B.S. and M.S. degrees in biomedical engineering from Case Western Reserve University, Cleveland, OH, in 1983 and 1989, respectively, where he is currently in the doctoral program. Since 1988, has been the Student Activities Coordinator for the IEEE Engineering in Medicine and Biology Magazine, for which he writes the “Student’s corner.” His research interests include adaptiv e control, modeling and identification, biomedical signal processing, neural nets, and computational neurobiology. Mr. Timmons is a member of the IEEE Engineering in Medicine and Biology Society.
in Budapest, Hungary, in 1937. He received the B S degree from the University of Michigan, Ann Arbor, in 1960, and the M S. and Sc.D degrees from Massachusetts Institute of Technology, Cambridge, in 1962 and 1965, respectively, all in electrical engineering. In 1969 he joined the Department of Biomedical Engineering at Case Western Reserve University, Cleveland, OH, where he is now Professor. He served as chairman of the department during 1980-87 Currently he is on a two-year temporary assignment as director of the Biomedical Engineering and Aiding the Disabled program at the National Science Foundation in Washington, D.C. Dr Katona’s research interest is the natural and artificial control of the cardio-pulmonary system.
