Pergamon 0967-0661 (94)00027-1

Contrvl Eag Prac~e, Vo[ 2, No 4,pp 659-664, 1994 Copyright © 1994 Elscvaer Sexenee Ltd

Pnnted m Great Bntam All rights reserved 0967-0661/9457004-000

SELF-ADAPTIVE CONTROL FOR BLOOD PRESSURE

XJ. Qu and Z. Mao Shaanxl Mechamcal and Electr4cal Research Institute, P 0 Box 1, Xtan Yang, 712099 Shaanx~, PRC

Abstract. The blood pressure control technology descnbed m flus paper m based on the stochasttc self-adaptive control theory. The maxtmum hkehhood method and reeurslve LS algorithm are used to establhsh the mathemaucal model of an orgamsmts blood pressure and esumate the model coefficients, respectavely. The methods of weighted control and control rate are adopted m order to hmlt overshoot and rapid change of the control The expermaental results on ammals and m a clinic m&cate that the performance of the controller m excellent. This paper ts mainly concerned wzth the chmcal expertments.

Key words. Blocontrol, orgamsm, parameter estimatton; sclf-adapttve control; self-tuning regulator

1 . I N T R O D U C T I O N

The internal ad jus tment o f b lood pressure is a problem that of ten appears and is difficult to solve. F o r the internal ad jus tment o f a b lood pressure system, the tmbalane¢ f rom various factors makes biosystematac b lood pressure abnormal . Therefore , in a clinical environ- ment , it is necessary to ~dentafy the pressure first, and normahse the pressure using rele- vant mechcataon

After open-heart surgery, pataents often show high arterial blood pressures, which are dangerous to them. In thls case, the pataent should be refused wxth the dmstohc drug, so- chum mtroprusside (SNP), to decrease the blood pressure in blood vessels and then re- duce the posslbdltaes of the other diseases. Sometames, patients, after open-heart sur- gery, show low arterial blood pressure, whlch also causes danger to them. In order to avert the danger, the pataent must be treated with systohc drugs to increase the blood pressure In order to hold the blood pressure at the de- sared level, what should be done is to momtor the blood pressure and then adjust the speed of the drug mfumon. Currently, m chines, manual methods of controlling the pressure are very exhaustang when undertaken for a long tame, and have poor precasion.

In recent years, m a n y papers abou t b lood

pressure cont ro l have been pubhshed H a m m o n d , Klrkendal l and Calfee (1979) presented a b lood pressure controller . Thin control ler was developed for pataents and mainta ined the b lood pressure within + 20 m m H g at 94% o f the t tme history. K a u f m a n , Ro y and X m h e (1984) exper imented with mongrel dogs and the b lood pressure was main ta ined to within + 5 m m H g within 185 seconds.

Thls paper wtll give a self-adaptave control approach whlch ts based on stochastac adaptave control theory. The structure of the controlled blosystem can be estabhshed using the maxlmum hkehhood method, then the model/s parameters are estimated w~th an LS algorithm on-hne. The controlling value (that is the flow speed of drug infusion) may be obtained by a control cost functaon Am- real and clmlcal experiments were imple- mented m order to test whether thls method is useful and rehable Some hmlts are added to the system to ensure the safety of patients.

2. BLOOD PRESSURE MODELLING OF AN ORGANISM

F o r the opt imal cont ro l o f b lood pressure, in- fo rmaUon ab o u t the control led plant should be ob ta ined , m order to const ruct the mathematacal model o f the control led plant.

659

660

From analyses of the dynamic response of blood pressure to drugs, and a bmsystem/s characteristics, ~t ~s known that an orgamsm has six properties (Qu, 1987) These are complexity, t~me-varlatmn, nonlinearity, long delay, strong random disturbance and &fferences between mdlwduals The &ffer- ences between any orgamsms are very great.

XJ QuandZ Mao

Uk_k_ n ' e k _ ! , , e k _ n ]

0r----[al, ,a , b 1 . . . . b , el, . ,c a]

Synthesizing the characteristics of an organ- Ism and the reaht~es of the control approach, the CARMA (Controllable Autoregresslve Moving Average) model (Pandit and Wu, 1983) has been selected for thls study.

Assume that the orgamsm model ~s

Yk +a lYk-1 + + asYk-,,

= b l u _a_ l + . + b , u ~ , _ a _ ,

-bek + c l e k _ l + . . + ¢ . e k _ n (1)

where Yk ~S the value of the average arterial pressure minus the desired value Uk lS the speed of drug infusion, ek ~S an independent Ment~cal &stnbuted sequence w~th zero mean and known varmnce ~r 2. n ~s the order of the model, d is the delay of the model, eh, b, and % z= 1, .,n are the modelts coefficients.

The hkehhood funeUon is

N S lnL = -- (ln27t + 1)

2

N S 1 xs 2 I n [ - ~ Z E2(k,0)l (31

k m m + d + l

where NS is the number of samphng. The maximum ltkehhood es)amation is to esUmate 0. Therefore, parameter 0 was selected to make /nL mayamum.

In order to determine the delay of the model, suppose that the cost function Is

N$ T ~ 2

J(n,d) --" Z (Y s, -- X J,- I O) k ~ , + d + l

3l$

k - s t + d ÷ l

(45

By selecting do, let

J (n ,do ) - - m i n J ( n , d ) d - d O

So do ~s the statable model delay.

The differences between the various organ- rams are represented m that every orgamsm has a different model order, delay and coeffiments from the others. Before control can take place, the model structure of the or- gamsm ~s estabhshed. During control, the model coefficients are esumated on-hne.

The model order may be determined by AIC (Akaike reformation cntenon) (Akatke, 19745 which is

A I C ( n ) ffi N S ( i n 2 ~ + 1)

+ NSIn[.I-~ J(n ,d) + 6n] .N ,~

(5)

The pseudo-random signal ]s used as the input signal, Uk, and then the arterial pressure signal, Yk, of the orgamsm is sampled W~th both input and output data, the controlled body's model may be estabhshed The values of the pseudo-random signal and the sam- plmg t~me affect the accuracy of estimatmn, which can be selected according to the theo- nes and the experiments being undertaken The model ~s a function of the drug being used and the mdwidual.

The model coeffictents are esttmated using mamraum hkehhood esttmauon. EquaUon (15 may be rewntten as

T y~ • X k _ l O ( k - - 15+e k (2)

where T

X k - I ----'[Yk-I' 'Yt-n' U k - d - l ' '

By selecting n 0, let

AIC(n o) ffi r a i n AIC(n)

So n o is the statable model order

\ 3. SELF-ADAPTIVE CONTROL OF

AN ORGANISM'S BLOOD PRESSURE

In order to design a control system with ex- cellent performance sucessfully, using either general feedback control or opumal control, the mathematacal model of the controlled process must be known. But ]t is difficult to determine the mathemaUcal models for many controlled plants and processes, which are tnne--varymg. For these kinds of systems, general control methods prowde good con- trol performance without dLfficulty.Bearmg m mind the characteristics of the orgamsm, a

Control for Blood Pressure 661

sclf-adjustang control approach is chosen to rcahze the contol of blood pressure with drugs. The self-adJusting control approach is suitable for an uncertain model of a control- led plant and a control system in a strong random envtronment. In the engineering con- text, itS algorithm is relatively sL, nple.

Using the theory of self-tuning control and the actual control task, an Lmproved self-tun- mg control approach is given. Suppose that the organism model is Equation (1), its pre- chctaon model Is such that

Yk÷d+l _-- _ ~ l y k - - _ r t = y k _ ~ + 1

+/3oU k + +~ tUk_ t 4- el,+#+ ! 4- ~lek+# q- ...

+Taek+l (6)

where 1, m are the model orders. 0tl....Otm, ~1""~1, and "~1 ..'~d are the prediction model coefficients which are obtained by the aug- mented recurslve LS algorithm with a forget- tang factor of 0.98. The relationships: m ~ n , 1 > n and//0 = bl also hold.

Because the drugJs dispersion and action need some time, the organism has a delay with d+l sampling periods, whlch affects the action of output of the control, Uk, on the organism.

The output Yk+d+~ at moment k+d+l may be predicted using the output Yk at moment k and the previous outputs Yk-~ Yk-2.., and then the relative control may be computed, which compensates the random effects on the output at moment k+d+l , and minimizes the error between the orgamsm~s actual blood pressure and the desired pressure.

The control cost function (Qu, 1987) is

j -_ E{(yk+a+ ! _ y,)2 4- p(k)Ul

+ w(k)Au2k } (7)

the control weighUng function, which avmds very large control signals, makes the system suitable for a non-minimum phase system, and has robust stability. The control weight- mg functaon p(k) (Qu, 1987) consists o f two parts:

p(k) -- Po(k) + p,(k) (9) where

/z0 lYkl t> Y,0,

p o ( k ) = _ ,o ,_tO uk >i u,~o,k <~ k 1 ,

Po e lYkl < Y,0'

U t < Ur~o,k > k l (lo)

When a patmngs situation changes abruptly at moment k 1 during drug infusion, a strong disturbance is generated, which causes the control process to become unstable. In this case, a very great variation o f the control val- ue, which is harmful to the patient, wdl be generated At this tame, P0(k) chooses the value Po automatically, to limit the great va- riation of control value u k. p0(k) can make the system stable at all tn'aes. P0, Ymo and urn0 are determined by the actual conditaon of the patient. In general, P0 is between 0.1 and 0.3, Ym0 Is between 30 and 40 mmHg, and urn0 (determined by experience and the value uk) is greater than Uk. The value of • is chos- en to make e -~-kD approximate to zero af- ter several samphng periods The function, pl('k) (Qu, 1987) is

p l ( k 4- i) ----

I KIPl

pK 2p l

~ <~ -- ym1(raise pressure)

or y>~ym1(lower pressure)

Y >~ Y,~l (raise p re s sure )

or y <<. -- Y m ( I ° w e r p r e s s u r e )

t~ < Y .1 (11)

By mlmmlzmg J with respect to uk, the con- trol variable u k can be obtained:

/~o u k-- , [ a lYk+ . . .

Po + p(k) + w(k) 4- cc,.yk_s,+1 -- ~luk_1 -- ..

- ,Otu k_ I + .v 7] w(k)

4- 2 u t - I ~o + p(k) + wfk)

(8)

where y~ is the desired output track, p(k) is

where i = 1, ,N (N is between 5 and 20). K l < 1 and K2> 1. ~ is the mean o f the output Yk. The funcUon, pt(k), can make the system achieve a excellent performance, especially m the Ume-varymg structure, and hold the level o f blood pressure near the desired value. The values, Pl and Ym~, are selected by experience and experiment. Generally, p~ is between 0 1 and 0.3, and Yml IS between 5 and 10mmHg.

AUk= Uk--Uk_ t IS the control increment, and w(k) is the welghUng function which prevents the rapld change of control. Here w(k) is

662 XJ Qu andZ Mac,

mmdar to ~ ] '

w(k) = ~ K3w Auk <" - Au= (12) ~'K4w Au k >t Au

where K 3 < l , K 4 > I , w is between 0.1 and 0.4, and Aum is between 15 and 35 m l / h r

4. ANIMAL EXPERIMENTS

The diagram of the self-adaptive control sys- tem for blood pressure ~s seen m Fig.1 Firstly, the ammaYs arterml pressure m con- vcrted to a relatwe electric signal by the blood pressure sensor. Then, the electric mgnal ~s amplified by a low-noise amplLqer. Next, the electric signal is sampled with A / D of 16 bits. Finally, the sampled signal (samphng in- terval xs 10 seconds) ]s transfered to the com- puter (IBM P C / X T ) , which processes the data (the sampled signal) using the self-adaptive control method and prowdes an opttmal control vanable to the mfumon pump.

Here, only two control curves were chosen, the increasing pressure curve (dcslred pres- sure I00 mmHg) and the decreasing pressure curve (deslred pressure 70 mmHg) whlch arc referred m Fig.2 and Fig 3, rcspectwely

Y(mmHg) 120 0

....... A ......... A .... n lOOO i~ ~ r ~1~ i ~ - . I ) , . . . . ~ r l y r w ,

V EO 0

60 0

40 0 •

20 O' 20 40 60 80

t]me(mln )

Fig.2 Ammal experiment m rinsing pressure

tOO

Y(rnmHg) I00 0

I Disturbance

Computer ~ Amplifier

60 0

40 0

F1g.l. The self-control system for blood pressure

The pump for refusing the drug Is drwen by a step-motor to whlch the control mgnals are sent from the computer through an interface. There m a bubble detector m the pump. If a bubble wlthout color or of a hght color, wlth diameter bigger than I mm 3, passes through the mfusmn tube, the motor stops at once and the pump gwes the alarm using a loud- speaker.

Before executang mfusmn, several sampled values are necessary to model the patmnt (the parameters, n and d). Dunng infusion, the pataent modeYs structure ~s not changed and the modeYs coefficaents are only estimated on-hne

20 0 0 10 20 30 40 50 60 70 80

trine(ram )

Fig.3. Ammal experiment m lowering pres- s u r e .

In the experiments, the pressure-increasing drugs, Isoprotcrenol metaramlnol or dopammc, and the pressure-decreasing drug, SNP were used. Different densmes of the drugs, different deslred blood pressure values, and rap~d varmuons of the annnal models wcrc used. The expenmental results have proven that the controlhng system was very stable, rchablc and safe, and have strongly supported the apphcatmn of the self-adaptwe method for physlologlcal and blome&cal uses.

Seventeen rabb~ts were used for the experi- ments (Qu, 1987). The control results were that the arterial blood pressures were mare- tamed to within + 10 mmHg for 100% o f the Ume and + 5 mmHg for 80% of the ttme

5. CLINICAL E X P E R I M E N T S

The chmcal expenments were undertaken wlth mght paUents m an environment of clini- cal supervlslon and nursing after heart

Control for Blood Pressure 663

Table l The results of the clinical experiments

~ o Age Sex Drug DenIlty EV n d

(m/100ml) (mmHI)

.8 e contr, tune tnne m time in

(hours) + 10mmHg + 5mmHg

l 37 F dopamme 50 70 ! 1 0 9 5 3 100% 91 8%

2 31 F SNP 30 75 2 i - 006 5 0 100% 92 5%

3 33 F dopamme 6 0 / 8 0 75 2 2 0 9 41 66 100% 83 3%

4 47 M SNP 30 g0 2 2 --0 14 4 5 100% 78 1%

5 ! 8 F dopamme 60 / 80 90 / 80 2 2 006 14 100% 77 6%

6 6 M dopamme 100 80 2 3 0 27 24 5 100% 90 4%

metarmmmol 2 5

7 32 F SNP 2 0 / 4 0 7 5 / 6 5 ! ! "-0.9 22g 100% 857%

8 26 F SNP 20 85 / 75 I 3 --0 9 8.2 100% 92 2%

Note that EV means the desired value.

operations. The sum of the patients / experi- mental t J m e history is 126 hours. The control accuracy is within _+ I0 mmHg at 100% of the time history. The information about the chmcal expernnents zs given m Table I. The chmca] experimental curve of patient No.2 is represented as Fzg.4 (the desired level of rins- ing pressure is 75 mmHg). Figure 5 shows the curve of paUent, No.6, who was infused with the mixed drugs, dopamine and metaraminol (the desired level of lowering pressure is 80 mmHg). Patient No.3 was injected with dopamine of denmty 60rag/100ml. After some tzme, the paUent was rejected with the same drug with denslty of 80rag/100ml, ac- cording to the doctor's suggestion. By chang- mg the drug/s density, the control resuR in Fig.6 is perfect. Figure 7 shows the blood pressure curve of patient No.7 related to the vanalnons of the density and the desired value. Firstly, the paUent was refused with SNP of denmty 20rag/100ml; the desired value of hzs blood pressure was 75 mmHg. Then, the drugts denslty was increased to 40

r a g / 1 0 0 m l and the desired value was low- ered t o 65 mmHg after three hours. When the density and / or the desired value of each pa- tient are changed in experiments 3, 5, 7 and 8, the model structures are not changed and on- ly the model/s coefficients are identified on-line. The other experimental curves are to be seen in (Qu, 1987).

After the heart operation, the patient was transfused with blood, infused with some other drugs, and nursed. These processes brought about disturbances to the blood pressure control model. Although these chs- turbances affected the control performance, the whole experimental process was very sta- ble and reliable.

6. C O N C L U S I O N

In thin paper, self-adapUve control theory is chosen to estabilish the blood pressure model. This method ]s self-adaptive, that is, the

Y ( m m H g ) 1oo.

80 ̧

60

40

I 250 2O

o 50 z0o 150 200 300 t i m e ( r a i n )

Y(mmHg) 90,

80' ~ . . . . . ...h .*,~.j t* .~L r ~ ~q-,, .r.1,1 ,'q~pq

70

60

50

4o!

301 I

t i m e

W

DO 1600

Fig 4 The blood pressure of patient No.2. Fig.5. The blood pressure of patient No.6.

6 6 4 X J QuandZ M a o

Y(mmHg) 90

80

70

60

50

n l l - ' T ] f , i~llqltllUpu, j , r l l J i T e q, I I

4O

30 0 DO 1500 2000 2 5 0 0 hme(rmn )

Fsg.6 The blood pressure ofpatsent No.3.

Y ( r n m H g ) 10o

8,0 .h .J

z v , [ , - r ~q i

40

20 t o 2oo 400 ooo 0oo Iooo 12oo 5400

tlme(rnln )

Fsg.7. The blood pressure of patsent No.7

model/s coefficsents are 5denUfied on-line. When pataent~s condstion changes, the con- trolling actmn can be regulated by on-line es- umatmn to track the dynamic change m the organism and make the blood pressure stable for a long tune. In this method, both the con- trol and the control rate are weighted, to hmit great and rapid change m control actions.

Smee many successful ammal experiments have been reported, this paper mainly stresses the chmcal experiments. From the results of the clinical experiments, this self-adaptzve control method zs successful

The ammal and chmcal experiments have giv- en strong support to the apphcation of the techmque for the modelhng and controlling of organzsms. What wdl be done next, zs to measure the heart rate, increase the reliabzlity and safety of the controller, and reduce the system errors.

REFERENCES

Akazke, H. (1974). A new look at the statisti- cal model ldentzfication. IEEE Trans. Auto. Contr.,Vol. AC-19, pp.716-723.

Hammond, J.J, W.M. Kirkendall, and R V. Calfee. (1979). Hypertensive cnsm man- aged by computer controlled mfumon of sodmm mtroprusmde" A model for the closed loop admzmstrauon of short acung vasoacUve drugs. Computers and Biomedical Research, 12, pp 97-108

Kaufman, H , R. Roy, and X.H. Xu (1984) Model reference adapUve control of drug lnfusmn rate. Automatica,Vol.20, No.2, pp.205-209.

Pandzt, S.M. and S.M. Wu (1983). T~me Se- rws and System Analyszs with Applications. John Wiley Sons, New York

Qu, XJ. (1987). The Patient's Blood pressure control with automatic control theory. The thems of Master of Scmnce. The Beijing Institute of Technology.