164
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 37. NO. 2, FEBRUARY 1990
Left Atrial Pressure Controller Design for an Artificial Heart TADASHI KITAMURA,
Abstract-This paper describes the application of a dynamic compensator technique to the left atrial controller design for use with a portable artificial heart drive system. The compensator is designed using a method in the field of multivariable control. This controller design is based on the physical models of the actuator and blood pump system. The analysis shows that there exists a minimal compensator with a dimension of one. The computer simulation demonstrates the acceptable, robust control performance of the left atrial pressure for a relatively small parameter variation of the vascular system model when all the poles of the closed-loop system are assigned to appropriate values.
INTRODUCTION XPERIENCE with artificial hearts and circulatory assist devices has been accumulated in clinical use as well as in animal experiments mainly using manually controlled drive systems. Efforts have been made to study and develop a variety of controllers for these drive systems. An eventual goal of these efforts is to discover the optimal pump output of volume that the patient’s circulatory system demands. However, many groups have agreed that a major, practical role of these control systems is to maintain atrial pressures within physiological limits (for example, [ 11). Such physiological control of atrial pressures is essential to balance blood volume of the two sides of the circulation especially in the case of total artificial hearts. This is because abnormally high atrial pressures can cause the death of the patient. It is required to find a control system of the left atrial pressure so that i) the controlled variable stays within physiological limits, ii) the closed-loop system of control is asymptotically stable with a desired fast response, iii) these two properties are maintained for physically possible variations of the vascular system, iv) the control system specification is easy to change, v) measurements of the blood pressure for control are stable and noninvasive. Most techniques for controlling the left atrial pressure which have been employed clear i) and ii). However, they do not satisfy iii) and iv) because they are empirical and device-dependent based on classical automatic control theory (for example, [2]). Modem control theory concepts have been applied in controlling atrial pressures for
E
Manuscript received October 20, 1988; revised May 30, 1989. The author is with the Department of Mechanical Systems Engineering, College of Computer Science and Systems Engineering, Kyushu Institute of Technology, Fukuoka 820, Japan. IEEE Log Number 8932199.
MEMBER, IEEE
circulatory assist devices and artificial hearts. McInnis and his co-workers [3], for instance, developed an adaptive control system for a left heart assist device using a selftuning PID-controller to achieve i) through iv). Their technique, however, has difficulty in estimating left atrial pressure for control, since it employs an ARMA-type (ARMA-autoregressive moving average), nonphysical model. All control techniques that have been employed need long-term direct measurements of blood pressures where v) of the above conditions is not guaranteed. A dynamic compensator and pole assignment technique has been applied by Kitamura [8] to design a left atrial pressure controller satisfying all the above conditions for the use with a portable pneumatically-driven left heart assist device (Whang [5]). This controller design is based on physical models of the components of the driver and blood pump system so that these physical models make it possible to estimate the left atrial pressure with acceptable accuracy for on-line control, (Kitamura [6], [7]). The present paper shows the control system design taking into consideration the actual computational time for the online pressure estimation and discusses results of a computer simulation that was conducted to test the use of the designed dynamic compensator.
DESCRIPTION OF
THE
SYSTEM
The schematic diagram of the portable pneumatic driver and blood pump system is shown in Fig. 1. The system consists of a linear actuator with a piston, cylinder, and linear dc-motor, as well as a blood pump, and inlet and outlet prosthetic heart valves. The blood pump has a flexible plastic diaphragm separating blood and air. Reciprocating motion of the piston actuated by the linear motor moves the pump diaphragm back and forth through the air in the closed pneumatic space between the piston and the diaphragm of the blood pump. This reciprocating motion of the pump diaphragm results in the pulsatile blood flow across each valve of the pump. The cylinder pressure and the piston displacement which are located extracorporeal are measured and then used for the estimation of the atrial pressure, (Kitamura, [6], [7]). In this study the inlet of the blood pump is assumed to be connected to the left atrium, and the outlet to the aorta. The first goal of the control of an artificial heart or a circulatory assist device to be discussed in this paper is to
0018-9294/90/0200-0164$01.OO O 1990 IEEE
KITAMURA: DESIGN FOR ARTIFICIAL HEART
Ti
a
Prosthetic Heart Valves
[T:
,,
Diaphragm
I65
where D ( k ) is the sum of Do( r ) over each set of four beats. The estimate of LAP(k) to be used for control at k+lis
Drive L i n e L l n e a r DC-motor
Blood Pump
maintain the left atrial pressure within physiological limits. This goal can be achieved by regulating either the pump rate or pump diaphragm volume displacement since the left atrial pressure inverse-proportionally depends on the pump output volume per beat. The piston amplitude, however, is taken as an actual controlling variable because the piston amplitude and the pump output volume per beat are linearly related to each other as shown below. The beat-by-beat basis discrete time models of the left atrium and of the pneumatic actuator and blood pump are approximately given by [8]
Z(k
(1)
+ 1)
+ Ao{Xm(r +
= ~m(r>
1) - X m ( r ) }
x(k
Ao(xm(r + 1) - x , ( Y ) } .
(3)
Once U ( r ) is designed, the control law for X , ( r ) is given by this equation. The estimate of LAP is updated every four beats using an on-line maximum likelihood technique, [6], 171: the first beat is used for acquiring data of the piston displacement and the driving air pressure, and the remaining three beats for the estimation of LAP. Noting this estimation scheme, the system model (1) and (2) should be changed for convenience by introducing a new sampling time k = 4r incorporating a set of four original sampling time points, i.e., r, r + 1, r + 2, and r 3. Then, assuming that U ( r ) does not change during interval k , we find from (1) and ( 2 ) by algebraic calculation that
+
LAP(k
V,(k
+ 1) = LAP(k) + 1) =
-
(4/CLA)V,(k)
- ( ~ o / c L A ) u ( ~+) ~ ( k ) , ( 4 ) V m ( k )+ 4U(k) (5)
+ 1) = Z ( k ) - LAP(k) + LAPr
(7)
(9) where
A =
1 -4/CLA
0 0
0
1
0 0
1
0
0 0
0
-1 1
[
0
(2)
=
(6)
+ 1 ) = A x ( k ) + b u ( k ) + r ( k ) + ~ ( k ) , (8)
9
where LAP( r ) is the mean left atrial pressure over rth beat, V, ( r ) is the pump diaphragm volume displacement, CLA is the left atrial compliance, Do( r ) is an input dependent on the aortic pressure and right heart output flow and will be treated as unknown disturbance, X , ( r ) is the piston amplitude which is the controlling variable, A,, is the piston cross-sectional area. Time changes in Do( r ) , as discussed later, are negligible compared to the time variations of the controller. A new control U ( r ) to be designed is introduced by u(r)
LAP(k).
where Z ( k ) is the output of the integrator, and LAPr is the reference value to LAP ( k ) . n Letting x = [LAP, V,, LAP, I ] T , the set of system equations (4)-(7) is given by
+ 1) = LAP(r) - (I/CLA)V,(r) + D o ( r ) ,
~ m ( r
=
According to basic linear control theory, an integrator removes the influence of a slow disturbance like D ( k ) in (4).Then the integrator is described
Pneumatic Actuator
Fig. 1. Schematic diagram of actuator and blood pump system.
LAP(r
+ 1)
L%(k
i
Piston-cylinder
b =
ro
i
CONTROLLER DESIGN From basic modem control theory, it can be easily shown that the system ( A , b , C ) defined by (8) through (1 1) is controllable and observable, and that all the poles of the resultant closed-loop system cannot be arbitrarily assigned by a standard output feedback. Therefore, a compensator should be designed in order for all the four poles of the closed-loop system to be arbitrarily assigned. Instead of using a well-known observer-based controller design technique, a more systematic, general design approach that was proposed by Brasch and Pearson [ 101 and modified by Kimura [9] is employed in the present study. Their design technique provides a one-dimensional com-
166
IEEE TRANSACTIONS ON BIOMEDICAL tNGINEERING. VOL
77. NO 2 . FbBRLIARY I Y Y O
pensator for a given set of five eigenvalues { sI,s2,sj, s4, s5 ] of the closed-loop system for the present system ( A ,
b, C ) : u(k)
=
K,Y(k)
+ K2z(k),
(12)
and the compensator is given by
z(k)
=
Fz(k
- 1)
+ G y ( k - 1)
(13)
where the dimension of the compensator p = 1, K , E R I x 3 , K2 E R I x p , F E R p x p , G E Rpx’ are given by an algorithm in Appendix.
SIMULATION COMPUTER To test the control performance of the designed dynamic compensator, a computer simulation was conducted using the mathematical models of the portable driver and blood pump system connected to a vascular system model. The modeling consists of modules for the components of the system: a linear dc-motor, pneumatic actuator, pump diaphragm elastic behavior, and prosthetic heart valves. Details of the models of the driver and blood pump system are illustrated elsewhere [4]. Fig. 2 shows an electric circuit diagram equivalent to the vascular system model. This model is composed of systemic circulation, left atrium, and bronchial circulation that is a bypass from the aorta to the left atrium. The systemic circulation model is a Windkessel model with two components: aortic compliance C A 0 and total peripheral resistance RL. the bronchial circulation is modeled as a linear resistor R,. The right side of the heart and lungs is lumped as a left atrial compliance, CLA. Variations of values of the two parameters, CLA and R,, are physiologically responsible for immediate changes in the left atrial pressure. This simple vascular model does not impair the purposes of the computer simulation for the numerical test of the controller. The minor interaction between the actuator and the vascular system is expressed in the model (8) and (9) in terms of the disturbance D. The structure of the whole system of the actuator and the blood pump plus the vascular system leaves all the variables of the nonlinear models of the actuator and the blood pump involved only in D.The influence of V,,, on LAP through the whole vascular system which is the major physical meaning of D is negligible compared to the direct response of V,,, to LAP through the left atrium. This is because, as shown later, the controlled LAP settles at its desired value within far less than the major time constant, 150 s, of the transfer function of the vascular system from V,nto the right heart output volume [ 1 I]. Thus, the pulmonary circulation, the major time constant of which is 5 s [ 1 I], can be replaced with a small capacitance, or neglected for the purposes of the present study. The present vascular model is also sufficient to evaluate whether or not the designed controller of the left atrial pressure is effective to balance the blood volume of the two sides of the circulation. This is because the left atrial
Fig. 2. Electrical circuit equivalent to the vascular system model: Q,,, and Q,,,, are the inflow and outflow, respectively, and Pa,, and P , , are the aortic pressure and left atrial pressure, respectively.
pressure is a well-known indication representing to what extent those blood volumes are balanced, and as far as that pressure is controlled within the physiological limits, abnormally high pulmonary pressure resulting in the death of the patient can be prevented. Furthermore, the present model can simulate realistic dynamic variations of the left atrial pressure due to hemodynamic disturbances around the left atrium, as shown in the next chapter. A set of differential equations for these models combined with the dynamic compensator presented in equations (12) and (13) is solved with the time increment equivalent to 0.05 ms using the Runge-Kutta-Gill method. The dynamic compensator is simulated by integer arithmetic assuming the same computational circumstances as in the microprocessor for the on-line control of the actual portable drive system.
RESULTSAND DISCUSSION Multiple eigenvalues of the closed-loop system are chosen in a trial and errors method as s, = s2 = s3 = s4 = s5 = 0.25 so that the following two conditions are satisfied: i) the eigenvalues provide the fastest nonoscillatory response of the left atrial pressure LAP to a step change in the referential value LAPr, ii) the actuator piston amplitude X,,, as a controlling variable stays between the two limits (0 and 10 c m ) . Then the following nominal physiological values of the vascular model are used: the total peripheral resistance RL = 3 mmHg.s/ml, the resistance of the bronchial circulation RI, = 15 mmHg *s/ml, the left atrial compliance CLA = 30 ml/mmHg, and the aortic compliance C A 0 = 2 ml/mmHg. The pump rate is fixed to 85 beats/min. Fig. 3 shows the response to the step change of LA4Pr from 2 to 8 mmHg at r = 1 with the best poles, converging to LAPr with hundred beats. This step change of LAPr is tested to determine the best poles that satisfy the two above conditions i) and ii). It is, however, unlikely that step changes in LAPr will be produced in in vivo experiments in which LAPr is fixed at a physiological value (e.g., 5 mmHg). To test the robustness of the designed compensator, the vascular parameters, Rh and CLA, changes of which directly influence the left atrial pressure, are suddenly changed with LAPr = 5 mmHg (Fig. 4): CLA from 30
167
KITAMURA: DESIGN FOR ARTIFICIAL HEART
lor
10,-
O L
OL
7 -
beat
140
Fig. 3. Response of mean left atrial pressure LAP to step change of reference LAPr (first panel) and control X,,, piston amplitude (second panel).
mmHg
B
0'
1
lor
LAP 5
Xm
cm
-
--
w
._
-
-
I , - - , . * * X ~ L ,
-
,- %
-
L-,~
-
_ _
- ,r 7
to 20 at A , Rh from 15 to 8 at B , and CLA and Rh back to their nominal values at C . As shown on Fig. 4, (middle row) changes of these parameters controlled by the compensator have little effect on the left atrial pressure (LAP). The computer simulation also shows that LAP stays between 4 and 6 mmHg in every beat, which satisfies the practical requirement of the controller. On the other hand, solid circle curves (upper row, Fig. 4) represent the left atrial pressure performance achieved without the control of the original piston stroke as it existed prior to the change of the parameters. At the cost of the speed of the response a higher order (not less than two-d ) dynamic compensator might be needed to reduce a noisy digital input u ( k ) which would be caused due to the scaling of all variables in the use of integer arithmetic to save computational time on an online processor. Since the use of a higher order dynamic compensator could increase the number of terms consisting of the control U and at the same time decrease the absolute value of each coefficient of these terms, the decrement or increment of each variable in these terms by one in integer would reduce possible jumps of U that cause noise of U . The use of integer arithmetic is unavoidable to compute the control U in our circumstances. It is also shown by the computer simulaAon that although nearly four beats are spent to update LAP, the integer arithmetic
L
r
-
r +20
beats --I
beat
Fig. 5 . Response of mean left atrial pressure LAP to sudden large variation of CLA (first panel) and control X,,, (second panel).
computation for the compensator can be finished during these four beats. A large difference can exist between the actual atrial compliance CLA and the nominal one in clinical circumstances where CLA of a patient is unknown. Fig. 5 shows the response of LAP is oscillatory to the sudden change in CLA from 30 to 10. Therefore, to avoid such a poor control performance, a robust controller with respect to a large variation of CLA needs to be developed. However, existing adaptive control techniques seem not to be able to solve this problem. Estimation of CLA also is difficult using standard on-line parameter estimation techniques such as model reference adaptive control techniques, since the inputs u ( k ) and D ( k ) to the plant may not give a proper excitation for the on-line parameter estimation by these techniques. The present approach to controller design is advantageous over both a classical approach to controlling artificial hearts [2] and ARMA-type model approach [ 3 ] because the present approach is based on the use of the physical models of the drive system and the blood pump which makes it possible to on-line estimate the pump output volume and blood pressures [ 6 ] , [7]. Besides this advantage of the present approach, it is obviously much better than device-oriented control techniques of artificial hearts and circulatory assist devices because it is easier to change the control specifications and has wide applicability to other artificial heart systems of the same type due to the use of the physical models of the present actuator and pump system. A standard observer-based dynamic compensator algorithm is also applicable to the present controller design problem; however, the present approach has been employed because it provides a more general algorithm, and is of value as a new application in the medical engineering field. Both algorithms are simple, once programmed in the computer; however, our approach determines the feedback gains in a single, straightforward step, while the former method requires two different steps to obtain the feedback gains; construction of an observer and determination of the state feedback gains.
168
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 37. NO. 2. FEBRUARY 1990
APPENDIX Compensator Design Algorithm
Step 5: Define vectors qi E RI sively such that
Suppose an n-dimensional control system is given by x(n
+ 1) = Ax(n) + B u ( n )
qo = 4 1
y ( n ) = Cx(n> (A01 where A E R n X " B , E R"", and C E R"'". For the simplicity of compensator realization, find the upper limit p m of the minimum dimension of the compensator following:
(Al)
p m = min ( v - 1, p - 1 )
=
-
,A ' - ' B }
*
p = min [ j : rank I C T , A r c T ,
*
*
= n],
(A2)
and uiE RI " recur-
go, Bi
(A6)
- an+p-i&
-
-
an+p-r+~B~
. . . - an+p-lBr-l,
(i = 1,
qr- ,c,
.. .
* *
('47)
.PI
UO = W n ,
=
ur- ,A
-
(i = 1,
2
(A9)
P).
Step 6: Find K , , K 2 , F , and G such that
by calculating the controllability index v and the observability index p given by v = min [ j : rank ( B , AB,
xm
K1 = - ( W 2 B ) - I
, ( A T ) ' - ' C T ]= n ] .
(:), 40
;)
(A31 The minimum dimension of the compensator is equal to the smallest p < p m with which the following algorithm [9] works. K2 = (W2B)-I ( 00 0 0 -* .* *. Step 1: Find W , E R("-')'" , W2 E R r X " ,and W T = [ W r , WT 1 E R" '" so that W I B = 0 and W2B is nonsin1 0 - * * gular. F = I/W-'Y, - Y2, G = VW-IPI - P2 Step 2: Find A' = WAW-I. Step 3: For a given set of eigenvalues A = (s,, * * , where s , + ~ ) , f i n d f = ( -ao, - a l , * , -a n - p - , ) T E R " Xso P that A is, identical to the set of the eigenvalues of + i f T 1' = - [ e n , o, * * * O 1 9 where A and h are given by 0 1 0 ... 0
a
A=(&
en
Q=
Q O)
y 1: : I) (! . . .
0 0 0
Y2 = -
eRPxp,
. * *
E R("+ P ) x
1
where A. = A' - e,a,,, e, = (0, a,, \,.. is nth row of A'. Step 4: Find Bi E RI '"( i = 0 , *
('44)
, l ) T E R " , and
*
*
*
, p ) such that n-1
+ U , + , - , A ~+ - . - + a n - , z ) + C = c&C + QP-,CA + . + &CAp
W,(AP+'
aj-,wj
j= 1
* *
[: : ; : : : : ) ...
-an
PI =
b=
(All)
9
o...
0
3
('45)
where wi is the ith row of W , and I is an n-dimensional unit matrix.
(Ala
(:), (:), P2=
v=(;;).
4P
40
In our circumstances, n is given by
-an+p-l
-an+2
-a,+,
=
(A13 1 4 , r = 1 , and m = 3. Then W
... ... .. .. . . 0 1
.. .. 0
- * .
I69
KITAMURA: DESIGN FOR ARTIFICIAL HEART
The controllability index and observability index are: U = 4, and p = 2, respectively. Therefore, p m = 1, and this means a one-dimensional dynamic compensator exists as a minimum dimensional compensator. Following the algorithm from steps l to 6 above, F, G, KI, and K2 are given by
F = -1
G
0.4(4
-
+ qolCLA) - a4,
= [-901F, -q02F - q12, - q 1 3 1 ,
K2 = -CLA/(6Ao), K1 = [-K2401,
--K2q02,
(A14) (A151 (A16)
01
(A17)
where a4 = 2 / 5 - A, - A,,
+ a 4 ) / C + a2 1 + a4 + u3 + (2 CLA/5)a2
-4(2
4;
=
(:) ( q03
=
0
REFERENCES [ I ] H. Oster, D. Olsen, R . Jarvik, T. Stanley, J. Volder, and W. Kolff, “Survival for 18 days with a Jarvik type artificial heart,” Surgery, vol. 77, pp. 113-1 17, 1975. (21 A. D. Brickman, D. L. Landis, W. 0. Bannon, V. D. Mortimer, E. Nisley, Jr., and W. S . Pierce, “Control system implementation in the air-driven artificial heart,’’ Trans. Amer. Soc. Artif. Organs., vol. 20, pp. 690-694, 1974. 131 B. C. Mclnnis, Z. W. Guo. P. C . Lu, and J. C . Wang. “Adaptive control of left ventricular by-pass assist devices,’’ fEEE Trctns. Autoniut. Contr., vol. AC-30. pp. 322-329, 1985. [4] T. Kitamura, “Design of a portable artificial heart drive system based on efficiency analysis,” ASME J. Biomech. E n g . , vol. 108, no. 4 , pp. 350-354, 1986. [5] D. H. Whang, “Motor control and indirect measurement in a pneumatically driven artificial heart,’’ Master’s thesis, Univ. Houston. University Park, Houston. TX, 1986. [6] T. Kitamura and D. H. Whang, “Indirect measurement technique for a portable artificial heart drive system,” ISA Trans., vol. 25, no. 4, pp. 7-11, 1986. [7] T. Kitamurd and D. R. Gross, “A technique of indirect measurement for artificial hearts,” presented at 8th Annu. IEEE Eng. Med. Bio. Soc. Con$, Dallas-Fort Worth, TX, Nov., 1986, pp. 58-61. [8] T . Kitamura, “Physiological controller design for a portable artificial heart drive system,” presented at 10th World Congr. Internat. Fed. Automat. Contr., July, Munich, 1987. [9] M. Ito and H. Kirnura, and S. Hosoe, “Design theory for linear control systems,” Japan Soc. Inst. Eng., pp. 73-80, 1980. [IO] F . M. Brasch and J. B. Pearson, “Pole placement using dynamic compensators,” IEEE Trans. Automat. Conrr., vol. AC-15, pp. 3443, 1970. [ I I ] D. A. Spyker, “Simulation in the analysis and control of a cardiocirculatory assist device,” Simulation, pp. 196-205, 1970.
Tadashi Kitamura (M’85) received the B.S. degree in mechanical engineering from Waseda University, Tokyo, Japan, in 1973, and the M.S. and Ph.D. degrees in systems control engineering from Kyoto University, Kyoto, Japan, in 1975 and 1981, respectively. He was an invited research fellow of the Alexander von Humboldt Foundation and a Visiting Lecturer of Technische Universitat Berlin, West Germany, from 1982 to 1984. From 1984 to 1986 he was a Visiting Assistant Professor in the Department of Electrical Engineering, University of Houston, University Park, Houston, TX, and was an Assistant Professor of Mechanical Engineering at Rose-Hulman Institute of Technology, Terra Haute, IN, from 1986 to 1987. He joined the Faculty of Computer Science and Systems Engineering at Kyushu Institute of Technology, Iizuka, Japan, 1987, and has been a Professor of Mechanical System Engineering at the same Institute since 1988. His field of interest includes control system design of artificial hearts and circulatory assist devices and intelligent support system design for diagnosis and therapy with artificial organs. Dr. Kitamura is currently a member of JSICE, ISAO, and JSME. Y
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 37. NO. 2, FEBRUARY 1990
Left Atrial Pressure Controller Design for an Artificial Heart TADASHI KITAMURA,
Abstract-This paper describes the application of a dynamic compensator technique to the left atrial controller design for use with a portable artificial heart drive system. The compensator is designed using a method in the field of multivariable control. This controller design is based on the physical models of the actuator and blood pump system. The analysis shows that there exists a minimal compensator with a dimension of one. The computer simulation demonstrates the acceptable, robust control performance of the left atrial pressure for a relatively small parameter variation of the vascular system model when all the poles of the closed-loop system are assigned to appropriate values.
INTRODUCTION XPERIENCE with artificial hearts and circulatory assist devices has been accumulated in clinical use as well as in animal experiments mainly using manually controlled drive systems. Efforts have been made to study and develop a variety of controllers for these drive systems. An eventual goal of these efforts is to discover the optimal pump output of volume that the patient’s circulatory system demands. However, many groups have agreed that a major, practical role of these control systems is to maintain atrial pressures within physiological limits (for example, [ 11). Such physiological control of atrial pressures is essential to balance blood volume of the two sides of the circulation especially in the case of total artificial hearts. This is because abnormally high atrial pressures can cause the death of the patient. It is required to find a control system of the left atrial pressure so that i) the controlled variable stays within physiological limits, ii) the closed-loop system of control is asymptotically stable with a desired fast response, iii) these two properties are maintained for physically possible variations of the vascular system, iv) the control system specification is easy to change, v) measurements of the blood pressure for control are stable and noninvasive. Most techniques for controlling the left atrial pressure which have been employed clear i) and ii). However, they do not satisfy iii) and iv) because they are empirical and device-dependent based on classical automatic control theory (for example, [2]). Modem control theory concepts have been applied in controlling atrial pressures for
E
Manuscript received October 20, 1988; revised May 30, 1989. The author is with the Department of Mechanical Systems Engineering, College of Computer Science and Systems Engineering, Kyushu Institute of Technology, Fukuoka 820, Japan. IEEE Log Number 8932199.
MEMBER, IEEE
circulatory assist devices and artificial hearts. McInnis and his co-workers [3], for instance, developed an adaptive control system for a left heart assist device using a selftuning PID-controller to achieve i) through iv). Their technique, however, has difficulty in estimating left atrial pressure for control, since it employs an ARMA-type (ARMA-autoregressive moving average), nonphysical model. All control techniques that have been employed need long-term direct measurements of blood pressures where v) of the above conditions is not guaranteed. A dynamic compensator and pole assignment technique has been applied by Kitamura [8] to design a left atrial pressure controller satisfying all the above conditions for the use with a portable pneumatically-driven left heart assist device (Whang [5]). This controller design is based on physical models of the components of the driver and blood pump system so that these physical models make it possible to estimate the left atrial pressure with acceptable accuracy for on-line control, (Kitamura [6], [7]). The present paper shows the control system design taking into consideration the actual computational time for the online pressure estimation and discusses results of a computer simulation that was conducted to test the use of the designed dynamic compensator.
DESCRIPTION OF
THE
SYSTEM
The schematic diagram of the portable pneumatic driver and blood pump system is shown in Fig. 1. The system consists of a linear actuator with a piston, cylinder, and linear dc-motor, as well as a blood pump, and inlet and outlet prosthetic heart valves. The blood pump has a flexible plastic diaphragm separating blood and air. Reciprocating motion of the piston actuated by the linear motor moves the pump diaphragm back and forth through the air in the closed pneumatic space between the piston and the diaphragm of the blood pump. This reciprocating motion of the pump diaphragm results in the pulsatile blood flow across each valve of the pump. The cylinder pressure and the piston displacement which are located extracorporeal are measured and then used for the estimation of the atrial pressure, (Kitamura, [6], [7]). In this study the inlet of the blood pump is assumed to be connected to the left atrium, and the outlet to the aorta. The first goal of the control of an artificial heart or a circulatory assist device to be discussed in this paper is to
0018-9294/90/0200-0164$01.OO O 1990 IEEE
KITAMURA: DESIGN FOR ARTIFICIAL HEART
Ti
a
Prosthetic Heart Valves
[T:
,,
Diaphragm
I65
where D ( k ) is the sum of Do( r ) over each set of four beats. The estimate of LAP(k) to be used for control at k+lis
Drive L i n e L l n e a r DC-motor
Blood Pump
maintain the left atrial pressure within physiological limits. This goal can be achieved by regulating either the pump rate or pump diaphragm volume displacement since the left atrial pressure inverse-proportionally depends on the pump output volume per beat. The piston amplitude, however, is taken as an actual controlling variable because the piston amplitude and the pump output volume per beat are linearly related to each other as shown below. The beat-by-beat basis discrete time models of the left atrium and of the pneumatic actuator and blood pump are approximately given by [8]
Z(k
(1)
+ 1)
+ Ao{Xm(r +
= ~m(r>
1) - X m ( r ) }
x(k
Ao(xm(r + 1) - x , ( Y ) } .
(3)
Once U ( r ) is designed, the control law for X , ( r ) is given by this equation. The estimate of LAP is updated every four beats using an on-line maximum likelihood technique, [6], 171: the first beat is used for acquiring data of the piston displacement and the driving air pressure, and the remaining three beats for the estimation of LAP. Noting this estimation scheme, the system model (1) and (2) should be changed for convenience by introducing a new sampling time k = 4r incorporating a set of four original sampling time points, i.e., r, r + 1, r + 2, and r 3. Then, assuming that U ( r ) does not change during interval k , we find from (1) and ( 2 ) by algebraic calculation that
+
LAP(k
V,(k
+ 1) = LAP(k) + 1) =
-
(4/CLA)V,(k)
- ( ~ o / c L A ) u ( ~+) ~ ( k ) , ( 4 ) V m ( k )+ 4U(k) (5)
+ 1) = Z ( k ) - LAP(k) + LAPr
(7)
(9) where
A =
1 -4/CLA
0 0
0
1
0 0
1
0
0 0
0
-1 1
[
0
(2)
=
(6)
+ 1 ) = A x ( k ) + b u ( k ) + r ( k ) + ~ ( k ) , (8)
9
where LAP( r ) is the mean left atrial pressure over rth beat, V, ( r ) is the pump diaphragm volume displacement, CLA is the left atrial compliance, Do( r ) is an input dependent on the aortic pressure and right heart output flow and will be treated as unknown disturbance, X , ( r ) is the piston amplitude which is the controlling variable, A,, is the piston cross-sectional area. Time changes in Do( r ) , as discussed later, are negligible compared to the time variations of the controller. A new control U ( r ) to be designed is introduced by u(r)
LAP(k).
where Z ( k ) is the output of the integrator, and LAPr is the reference value to LAP ( k ) . n Letting x = [LAP, V,, LAP, I ] T , the set of system equations (4)-(7) is given by
+ 1) = LAP(r) - (I/CLA)V,(r) + D o ( r ) ,
~ m ( r
=
According to basic linear control theory, an integrator removes the influence of a slow disturbance like D ( k ) in (4).Then the integrator is described
Pneumatic Actuator
Fig. 1. Schematic diagram of actuator and blood pump system.
LAP(r
+ 1)
L%(k
i
Piston-cylinder
b =
ro
i
CONTROLLER DESIGN From basic modem control theory, it can be easily shown that the system ( A , b , C ) defined by (8) through (1 1) is controllable and observable, and that all the poles of the resultant closed-loop system cannot be arbitrarily assigned by a standard output feedback. Therefore, a compensator should be designed in order for all the four poles of the closed-loop system to be arbitrarily assigned. Instead of using a well-known observer-based controller design technique, a more systematic, general design approach that was proposed by Brasch and Pearson [ 101 and modified by Kimura [9] is employed in the present study. Their design technique provides a one-dimensional com-
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77. NO 2 . FbBRLIARY I Y Y O
pensator for a given set of five eigenvalues { sI,s2,sj, s4, s5 ] of the closed-loop system for the present system ( A ,
b, C ) : u(k)
=
K,Y(k)
+ K2z(k),
(12)
and the compensator is given by
z(k)
=
Fz(k
- 1)
+ G y ( k - 1)
(13)
where the dimension of the compensator p = 1, K , E R I x 3 , K2 E R I x p , F E R p x p , G E Rpx’ are given by an algorithm in Appendix.
SIMULATION COMPUTER To test the control performance of the designed dynamic compensator, a computer simulation was conducted using the mathematical models of the portable driver and blood pump system connected to a vascular system model. The modeling consists of modules for the components of the system: a linear dc-motor, pneumatic actuator, pump diaphragm elastic behavior, and prosthetic heart valves. Details of the models of the driver and blood pump system are illustrated elsewhere [4]. Fig. 2 shows an electric circuit diagram equivalent to the vascular system model. This model is composed of systemic circulation, left atrium, and bronchial circulation that is a bypass from the aorta to the left atrium. The systemic circulation model is a Windkessel model with two components: aortic compliance C A 0 and total peripheral resistance RL. the bronchial circulation is modeled as a linear resistor R,. The right side of the heart and lungs is lumped as a left atrial compliance, CLA. Variations of values of the two parameters, CLA and R,, are physiologically responsible for immediate changes in the left atrial pressure. This simple vascular model does not impair the purposes of the computer simulation for the numerical test of the controller. The minor interaction between the actuator and the vascular system is expressed in the model (8) and (9) in terms of the disturbance D. The structure of the whole system of the actuator and the blood pump plus the vascular system leaves all the variables of the nonlinear models of the actuator and the blood pump involved only in D.The influence of V,,, on LAP through the whole vascular system which is the major physical meaning of D is negligible compared to the direct response of V,,, to LAP through the left atrium. This is because, as shown later, the controlled LAP settles at its desired value within far less than the major time constant, 150 s, of the transfer function of the vascular system from V,nto the right heart output volume [ 1 I]. Thus, the pulmonary circulation, the major time constant of which is 5 s [ 1 I], can be replaced with a small capacitance, or neglected for the purposes of the present study. The present vascular model is also sufficient to evaluate whether or not the designed controller of the left atrial pressure is effective to balance the blood volume of the two sides of the circulation. This is because the left atrial
Fig. 2. Electrical circuit equivalent to the vascular system model: Q,,, and Q,,,, are the inflow and outflow, respectively, and Pa,, and P , , are the aortic pressure and left atrial pressure, respectively.
pressure is a well-known indication representing to what extent those blood volumes are balanced, and as far as that pressure is controlled within the physiological limits, abnormally high pulmonary pressure resulting in the death of the patient can be prevented. Furthermore, the present model can simulate realistic dynamic variations of the left atrial pressure due to hemodynamic disturbances around the left atrium, as shown in the next chapter. A set of differential equations for these models combined with the dynamic compensator presented in equations (12) and (13) is solved with the time increment equivalent to 0.05 ms using the Runge-Kutta-Gill method. The dynamic compensator is simulated by integer arithmetic assuming the same computational circumstances as in the microprocessor for the on-line control of the actual portable drive system.
RESULTSAND DISCUSSION Multiple eigenvalues of the closed-loop system are chosen in a trial and errors method as s, = s2 = s3 = s4 = s5 = 0.25 so that the following two conditions are satisfied: i) the eigenvalues provide the fastest nonoscillatory response of the left atrial pressure LAP to a step change in the referential value LAPr, ii) the actuator piston amplitude X,,, as a controlling variable stays between the two limits (0 and 10 c m ) . Then the following nominal physiological values of the vascular model are used: the total peripheral resistance RL = 3 mmHg.s/ml, the resistance of the bronchial circulation RI, = 15 mmHg *s/ml, the left atrial compliance CLA = 30 ml/mmHg, and the aortic compliance C A 0 = 2 ml/mmHg. The pump rate is fixed to 85 beats/min. Fig. 3 shows the response to the step change of LA4Pr from 2 to 8 mmHg at r = 1 with the best poles, converging to LAPr with hundred beats. This step change of LAPr is tested to determine the best poles that satisfy the two above conditions i) and ii). It is, however, unlikely that step changes in LAPr will be produced in in vivo experiments in which LAPr is fixed at a physiological value (e.g., 5 mmHg). To test the robustness of the designed compensator, the vascular parameters, Rh and CLA, changes of which directly influence the left atrial pressure, are suddenly changed with LAPr = 5 mmHg (Fig. 4): CLA from 30
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KITAMURA: DESIGN FOR ARTIFICIAL HEART
lor
10,-
O L
OL
7 -
beat
140
Fig. 3. Response of mean left atrial pressure LAP to step change of reference LAPr (first panel) and control X,,, piston amplitude (second panel).
mmHg
B
0'
1
lor
LAP 5
Xm
cm
-
--
w
._
-
-
I , - - , . * * X ~ L ,
-
,- %
-
L-,~
-
_ _
- ,r 7
to 20 at A , Rh from 15 to 8 at B , and CLA and Rh back to their nominal values at C . As shown on Fig. 4, (middle row) changes of these parameters controlled by the compensator have little effect on the left atrial pressure (LAP). The computer simulation also shows that LAP stays between 4 and 6 mmHg in every beat, which satisfies the practical requirement of the controller. On the other hand, solid circle curves (upper row, Fig. 4) represent the left atrial pressure performance achieved without the control of the original piston stroke as it existed prior to the change of the parameters. At the cost of the speed of the response a higher order (not less than two-d ) dynamic compensator might be needed to reduce a noisy digital input u ( k ) which would be caused due to the scaling of all variables in the use of integer arithmetic to save computational time on an online processor. Since the use of a higher order dynamic compensator could increase the number of terms consisting of the control U and at the same time decrease the absolute value of each coefficient of these terms, the decrement or increment of each variable in these terms by one in integer would reduce possible jumps of U that cause noise of U . The use of integer arithmetic is unavoidable to compute the control U in our circumstances. It is also shown by the computer simulaAon that although nearly four beats are spent to update LAP, the integer arithmetic
L
r
-
r +20
beats --I
beat
Fig. 5 . Response of mean left atrial pressure LAP to sudden large variation of CLA (first panel) and control X,,, (second panel).
computation for the compensator can be finished during these four beats. A large difference can exist between the actual atrial compliance CLA and the nominal one in clinical circumstances where CLA of a patient is unknown. Fig. 5 shows the response of LAP is oscillatory to the sudden change in CLA from 30 to 10. Therefore, to avoid such a poor control performance, a robust controller with respect to a large variation of CLA needs to be developed. However, existing adaptive control techniques seem not to be able to solve this problem. Estimation of CLA also is difficult using standard on-line parameter estimation techniques such as model reference adaptive control techniques, since the inputs u ( k ) and D ( k ) to the plant may not give a proper excitation for the on-line parameter estimation by these techniques. The present approach to controller design is advantageous over both a classical approach to controlling artificial hearts [2] and ARMA-type model approach [ 3 ] because the present approach is based on the use of the physical models of the drive system and the blood pump which makes it possible to on-line estimate the pump output volume and blood pressures [ 6 ] , [7]. Besides this advantage of the present approach, it is obviously much better than device-oriented control techniques of artificial hearts and circulatory assist devices because it is easier to change the control specifications and has wide applicability to other artificial heart systems of the same type due to the use of the physical models of the present actuator and pump system. A standard observer-based dynamic compensator algorithm is also applicable to the present controller design problem; however, the present approach has been employed because it provides a more general algorithm, and is of value as a new application in the medical engineering field. Both algorithms are simple, once programmed in the computer; however, our approach determines the feedback gains in a single, straightforward step, while the former method requires two different steps to obtain the feedback gains; construction of an observer and determination of the state feedback gains.
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APPENDIX Compensator Design Algorithm
Step 5: Define vectors qi E RI sively such that
Suppose an n-dimensional control system is given by x(n
+ 1) = Ax(n) + B u ( n )
qo = 4 1
y ( n ) = Cx(n> (A01 where A E R n X " B , E R"", and C E R"'". For the simplicity of compensator realization, find the upper limit p m of the minimum dimension of the compensator following:
(Al)
p m = min ( v - 1, p - 1 )
=
-
,A ' - ' B }
*
p = min [ j : rank I C T , A r c T ,
*
*
= n],
(A2)
and uiE RI " recur-
go, Bi
(A6)
- an+p-i&
-
-
an+p-r+~B~
. . . - an+p-lBr-l,
(i = 1,
qr- ,c,
.. .
* *
('47)
.PI
UO = W n ,
=
ur- ,A
-
(i = 1,
2
(A9)
P).
Step 6: Find K , , K 2 , F , and G such that
by calculating the controllability index v and the observability index p given by v = min [ j : rank ( B , AB,
xm
K1 = - ( W 2 B ) - I
, ( A T ) ' - ' C T ]= n ] .
(:), 40
;)
(A31 The minimum dimension of the compensator is equal to the smallest p < p m with which the following algorithm [9] works. K2 = (W2B)-I ( 00 0 0 -* .* *. Step 1: Find W , E R("-')'" , W2 E R r X " ,and W T = [ W r , WT 1 E R" '" so that W I B = 0 and W2B is nonsin1 0 - * * gular. F = I/W-'Y, - Y2, G = VW-IPI - P2 Step 2: Find A' = WAW-I. Step 3: For a given set of eigenvalues A = (s,, * * , where s , + ~ ) , f i n d f = ( -ao, - a l , * , -a n - p - , ) T E R " Xso P that A is, identical to the set of the eigenvalues of + i f T 1' = - [ e n , o, * * * O 1 9 where A and h are given by 0 1 0 ... 0
a
A=(&
en
Q=
Q O)
y 1: : I) (! . . .
0 0 0
Y2 = -
eRPxp,
. * *
E R("+ P ) x
1
where A. = A' - e,a,,, e, = (0, a,, \,.. is nth row of A'. Step 4: Find Bi E RI '"( i = 0 , *
('44)
, l ) T E R " , and
*
*
*
, p ) such that n-1
+ U , + , - , A ~+ - . - + a n - , z ) + C = c&C + QP-,CA + . + &CAp
W,(AP+'
aj-,wj
j= 1
* *
[: : ; : : : : ) ...
-an
PI =
b=
(All)
9
o...
0
3
('45)
where wi is the ith row of W , and I is an n-dimensional unit matrix.
(Ala
(:), (:), P2=
v=(;;).
4P
40
In our circumstances, n is given by
-an+p-l
-an+2
-a,+,
=
(A13 1 4 , r = 1 , and m = 3. Then W
... ... .. .. . . 0 1
.. .. 0
- * .
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KITAMURA: DESIGN FOR ARTIFICIAL HEART
The controllability index and observability index are: U = 4, and p = 2, respectively. Therefore, p m = 1, and this means a one-dimensional dynamic compensator exists as a minimum dimensional compensator. Following the algorithm from steps l to 6 above, F, G, KI, and K2 are given by
F = -1
G
0.4(4
-
+ qolCLA) - a4,
= [-901F, -q02F - q12, - q 1 3 1 ,
K2 = -CLA/(6Ao), K1 = [-K2401,
--K2q02,
(A14) (A151 (A16)
01
(A17)
where a4 = 2 / 5 - A, - A,,
+ a 4 ) / C + a2 1 + a4 + u3 + (2 CLA/5)a2
-4(2
4;
=
(:) ( q03
=
0
REFERENCES [ I ] H. Oster, D. Olsen, R . Jarvik, T. Stanley, J. Volder, and W. Kolff, “Survival for 18 days with a Jarvik type artificial heart,” Surgery, vol. 77, pp. 113-1 17, 1975. (21 A. D. Brickman, D. L. Landis, W. 0. Bannon, V. D. Mortimer, E. Nisley, Jr., and W. S . Pierce, “Control system implementation in the air-driven artificial heart,’’ Trans. Amer. Soc. Artif. Organs., vol. 20, pp. 690-694, 1974. 131 B. C. Mclnnis, Z. W. Guo. P. C . Lu, and J. C . Wang. “Adaptive control of left ventricular by-pass assist devices,’’ fEEE Trctns. Autoniut. Contr., vol. AC-30. pp. 322-329, 1985. [4] T. Kitamura, “Design of a portable artificial heart drive system based on efficiency analysis,” ASME J. Biomech. E n g . , vol. 108, no. 4 , pp. 350-354, 1986. [5] D. H. Whang, “Motor control and indirect measurement in a pneumatically driven artificial heart,’’ Master’s thesis, Univ. Houston. University Park, Houston. TX, 1986. [6] T. Kitamura and D. H. Whang, “Indirect measurement technique for a portable artificial heart drive system,” ISA Trans., vol. 25, no. 4, pp. 7-11, 1986. [7] T. Kitamurd and D. R. Gross, “A technique of indirect measurement for artificial hearts,” presented at 8th Annu. IEEE Eng. Med. Bio. Soc. Con$, Dallas-Fort Worth, TX, Nov., 1986, pp. 58-61. [8] T . Kitamura, “Physiological controller design for a portable artificial heart drive system,” presented at 10th World Congr. Internat. Fed. Automat. Contr., July, Munich, 1987. [9] M. Ito and H. Kirnura, and S. Hosoe, “Design theory for linear control systems,” Japan Soc. Inst. Eng., pp. 73-80, 1980. [IO] F . M. Brasch and J. B. Pearson, “Pole placement using dynamic compensators,” IEEE Trans. Automat. Conrr., vol. AC-15, pp. 3443, 1970. [ I I ] D. A. Spyker, “Simulation in the analysis and control of a cardiocirculatory assist device,” Simulation, pp. 196-205, 1970.
Tadashi Kitamura (M’85) received the B.S. degree in mechanical engineering from Waseda University, Tokyo, Japan, in 1973, and the M.S. and Ph.D. degrees in systems control engineering from Kyoto University, Kyoto, Japan, in 1975 and 1981, respectively. He was an invited research fellow of the Alexander von Humboldt Foundation and a Visiting Lecturer of Technische Universitat Berlin, West Germany, from 1982 to 1984. From 1984 to 1986 he was a Visiting Assistant Professor in the Department of Electrical Engineering, University of Houston, University Park, Houston, TX, and was an Assistant Professor of Mechanical Engineering at Rose-Hulman Institute of Technology, Terra Haute, IN, from 1986 to 1987. He joined the Faculty of Computer Science and Systems Engineering at Kyushu Institute of Technology, Iizuka, Japan, 1987, and has been a Professor of Mechanical System Engineering at the same Institute since 1988. His field of interest includes control system design of artificial hearts and circulatory assist devices and intelligent support system design for diagnosis and therapy with artificial organs. Dr. Kitamura is currently a member of JSICE, ISAO, and JSME. Y
