- Email: [email protected]
Iterative Learning Control of a Left Ventricular Assist Device: Nonlinear Model Integration
10th IFAC Symposium on Biological and Medical Systems 10th IFAC Symposium on Biological and Medical Systems 10th IFAC Symposium on Biological and Medical Systems São Paulo, Brazil, September 3-5, 2018 10th IFAC on and Medical Systems 10th IFAC Symposium Symposium on Biological Biological and Medical Systems São Paulo, Brazil, September 3-5, 2018 Available online at www.sciencedirect.com São Paulo, Brazil, September 3-5, 2018 São Paulo, Paulo, Brazil, September September 3-5, 2018 2018 10th IFAC Symposium on Biological and Medical Systems São Brazil, 3-5, São Paulo, Brazil, September 3-5, 2018
ScienceDirect
IFAC PapersOnLine 51-27 (2018) 152–157
Iterative Learning Control of a Left Iterative Learning Control of a Left Iterative Learning Control of a Left Iterative Learning Control of a Left Ventricular Assist Device: Nonlinear Model Ventricular Assist Device: Nonlinear Model Iterative Learning Control of a Left Ventricular Assist Device: Nonlinear Model Ventricular Assist Device: Nonlinear Model Integration Integration Ventricular Assist Device: Nonlinear Model Integration Integration ∗ ∗∗ Integration M. Ketelhut ∗∗∗ S. Stemmler orner ∗∗∗ D. Abel ∗∗∗ ∗ M. Hein ∗∗ D. K¨ ∗ ∗∗
M. Ketelhut D. K¨ o rner M. Ketelhut ∗∗ S. S. Stemmler Stemmler ∗∗ M. M. Hein Hein ∗∗ D. K¨ o rner ∗∗ D. D. Abel Abel ∗∗ M. D. o M. Ketelhut Ketelhut ∗ S. S. Stemmler Stemmler ∗ M. M. Hein Hein ∗∗ D. K¨ K¨ orner rner ∗ D. D. Abel Abel ∗ ∗∗ ∗ M. Ketelhut S. Stemmler M. Hein D. K¨ o rner D. Abel ∗ Institute of Automatic Control, RWTH Aachen University, Aachen, ∗ Institute of Control, RWTH Aachen University, Aachen, ∗ InstituteGermany of Automatic Automatic Control, RWTH Aachen University, Aachen, ∗ Institute of Automatic Control, RWTH Aachen University, Aachen, (e-mail: [email protected]). InstituteGermany of Automatic Control, RWTH Aachen University, Aachen, (e-mail: [email protected]). ∗ ∗∗ Germany (e-mail: [email protected]). Germany (e-mail: [email protected]). for Anesthesiology, University Hospitel Aachen, Aachen, Institute of Automatic Control, RWTH Aachen University, Aachen, ∗∗ Clinic Germany (e-mail: [email protected]). ∗∗ Clinic for Anesthesiology, University Hospitel Aachen, Aachen, ∗∗ for Anesthesiology, University Hospitel Aachen, Aachen, ∗∗ Clinic ClinicGermany for Germany Anesthesiology, University Hospitel Aachen, Aachen, (e-mail: [email protected]) (e-mail: [email protected]). Clinic for Anesthesiology, University Hospitel Aachen, Aachen, Germany (e-mail: [email protected]) ∗∗ (e-mail: [email protected]) Germany (e-mail: [email protected]) Clinic for Germany Anesthesiology, University Hospitel Aachen, Aachen, Germany (e-mail: [email protected]) Germany (e-mail: [email protected]) Abstract: Norm-optimal iterative learning control algorithms use plant models to predict the Abstract: Norm-optimal Norm-optimal iterative iterative learning learning control control algorithms algorithms use use plant plant models models to to predict predict the the Abstract: Abstract: Norm-optimal iterative learning control algorithms use plant models to predict the system behavior. In this paper, we focus on improving the performance in the norm-optimal Abstract: Norm-optimal iterative learning control algorithms use plant models to predict the system behavior. behavior. In In this this paper, paper, we we focus focus on on improving improving the the performance performance in in the the norm-optimal norm-optimal system system In paper, we focus improving the in the norm-optimal iterative learning control left ventricular assist devices. For this Abstract: Norm-optimal iterative control algorithms usepurpose, plant models to predictused the system behavior. In this this of paper, we learning focus on on improving the performance inthe thepreviously norm-optimal iterativebehavior. learning control of left ventricular ventricular assist devices. Forperformance this purpose, the previously used iterative learning control of left assist devices. For this purpose, the previously used iterative learning control of left ventricular assist devices. For this purpose, the previously used simple plant model is replaced by a piecewise linearized version of a nonlinear cardiovascular system behavior. In this paper, we focus on improving the performance in the norm-optimal iterative learning control of left by ventricular assist devices. version For thisof purpose, the previously used simple plant plant model is replaced replaced a piecewise piecewise linearized nonlinear cardiovascular simple model is by a linearized version aaa nonlinear cardiovascular simple plant model is by a linearized version of cardiovascular system model including the left ventricular assist device. Simulations are carried out to study iterative learning control of left ventricular assist devices. For thisof purpose, the previously used simple plant model is replaced replaced by a piecewise piecewise linearized version of a nonlinear nonlinear cardiovascular system model including the left ventricular assist device. Simulations are carried out to study system modelmodel including the leftby ventricular assist device.and Simulations are carried carried out to study study system model including the left ventricular assist device. Simulations are out to the controller response to end-diastolic volume setpoint preload changes. The results show simple plant is replaced a piecewise linearized version of a nonlinear cardiovascular system model including the left ventricular assist device. Simulations are carried out to study the controller controller response response to to end-diastolic end-diastolic volume volume setpoint setpoint and and preload preload changes. changes. The The results results show show the the controller to end-diastolic volume setpoint preload changes. results show minor improvements regarding tracking performance and the rejection of disturbances but system model response including leftthe ventricular assist device.and Simulations are carried out to study the controller response tothe end-diastolic volume setpoint and preload changes. The results show minor improvements regarding the tracking performance and the rejection rejection of The disturbances but minor improvements regarding the tracking performance and the of disturbances but minor improvements regarding the tracking performance and the rejection of disturbances but also an increase of computational effort compared to the previous algorithm. the controller response to end-diastolic volume setpoint and preload changes. The results show minor improvements regarding theeffort tracking performance and the rejection of disturbances but also an increase of computational compared to the previous algorithm. also an increase compared to algorithm. also animprovements increase of of computational computational effort compared to the the previous previous algorithm. minor regarding theeffort tracking performance and the rejection of disturbances but also an increase of computational compared to the previous algorithm. © 2018, IFAC (International Federationeffort of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. also an increase of computational effort compared to the previous algorithm. Keywords: Biomedical system modeling, simulation and visualization, ventricular assist device, Keywords: Biomedical Biomedical system modeling, simulation and and visualization, visualization, ventricular ventricular assist assist device, device, Keywords: system modeling, simulation Keywords: Biomedical system modeling, simulation and visualization, ventricular assist device, norm-optimal iterative learning control Keywords: Biomedical system modeling, simulation and visualization, ventricular assist device, norm-optimal iterative learning control norm-optimal iterative learning control norm-optimal iterative system learningmodeling, control simulation and visualization, ventricular assist device, Keywords: Biomedical norm-optimal iterative learning control norm-optimal iterative learning control 1. INTRODUCTION As stated by Stevens et al. (2018) the system consist1. INTRODUCTION INTRODUCTION As stated by Stevens et al. (2018) the system consist1. As stated by Stevens et al. (2018) the system consist1. INTRODUCTION INTRODUCTION As stated by Stevens et al. (2018) the system consisting of LVAD and cardiovascular system (CVS) contains 1. As stated by and Stevens et al. (2018) the (CVS) system contains consisting of LVAD cardiovascular system ing of LVAD and cardiovascular system (CVS) contains 1. INTRODUCTION ing of LVAD and cardiovascular system (CVS) contains multiple nonlinearities, uncertainties, perturbations and As stated by Stevens et al. (2018) the system consisting of LVAD and cardiovascular system (CVS) contains multiple nonlinearities, uncertainties, perturbations and Terminal congestive heart failure is one of the most commultiple nonlinearities, uncertainties, perturbations and Terminal congestive congestive heart heart failure failure is is one one of of the the most most comcom- multiple nonlinearities, uncertainties, perturbations and constraints. There exist various different control strateing of LVAD and cardiovascular system (CVS) contains Terminal multiple nonlinearities, uncertainties, perturbations and constraints. There exist various different control strateTerminal congestive heart failure is one of the most common causes of death in Europe. In the optimal case, the Terminal congestive heart failure is one of the most comconstraints. There exist various different control stratemon causes of death in Europe. In the optimal case, the constraints. There exist various different control strategies addressing different aims. Reviews of different control multiple nonlinearities, uncertainties, perturbations and mon causes of death in Europe. In the optimal case, the constraints. There exist various different control strategies addressing different aims. Reviews of different control mon causes of death in Europe. In the optimal case, the treatment includes the implantation of a donor heart. As Terminal congestive heart failure is one of the most common causesincludes of death Europe. In the case, the addressing different aims. Reviews of different control treatment theinimplantation implantation of aaoptimal donor heart. heart. As gies gies addressing different aims. Reviews of different control strategies for LVADs are done by AlOmari et al. (2012), constraints. There exist various different control stratetreatment includes the of donor As gies addressing different aims. Reviews of different control for LVADs are done by AlOmari et al. (2012), treatment includes thein implantation of aaoptimal donor heart. As strategies stated by Aissaoui et al. (2017) patients are waiting mon of death Europe. In the case,very the treatment includes the of donor heart. As strategies for LVADs are done by et statedcauses by Aissaoui Aissaoui et al.implantation (2017) patients patients are waiting very strategies foral. LVADs are done by AlOmari AlOmari et al. al. (2012), (2012), Amacher et (2014) and Bozkurt (2016). Iterative learngies addressing different aims. Reviews of different control stated by et al. (2017) are waiting very strategies for LVADs are done by AlOmari et al. (2012), Amacher et al. (2014) and Bozkurt (2016). Iterative learnstated by Aissaoui et al. (2017) patients are waiting very long for a transplant and may die before a donor heart is treatment includes the implantation of a donor heart. As stated et and al. (2017) patients waiting very et al. (2014) and Bozkurt (2016). Iterative learnlong for forbyaa Aissaoui transplant may die die before are a donor donor heart is Amacher Amacher et al. (2014) and Bozkurt (2016). Iterative learning control (ILC) algorithms are open-loop control stratestrategies for LVADs are done by AlOmari et al. (2012), long transplant and may before a heart is Amacher et al. (2014) and Bozkurt (2016). Iterative learning control (ILC) algorithms are open-loop control stratelong for a transplant and may die before a donor heart is available due to their limited ability. In order to extend stated by Aissaoui et al. (2017) patients are waiting very long for a transplant and may die before a donor heart is ing control (ILC) algorithms are open-loop control strateavailable due to their limited ability. In order to extend ing control (ILC) algorithms are open-loop control strategies that only require minor model knowledge, because Amacher et al. (2014) and Bozkurt (2016). Iterative learnavailable due to their limited ability. In order to extend ing control (ILC) algorithms are open-loop control strategies that only require minor model knowledge, because available due to their limited ability. In order to extend the possible waiting time, so called left ventricular assist long for a transplant and may die before a donor heart is available duewaiting to theirtime, limited ability. order to extend that only require minor model knowledge, because the possible possible so called called leftInventricular ventricular assist gies gies that only require minor model knowledge, because they calculate the control signal sequence for the next cycle ing control (ILC) algorithms are open-loop control stratethe waiting time, so left assist gies that onlythe require minor model knowledge, because calculate control signal sequence for the next cycle the possible waiting time, soLVADs called leftInventricular ventricular assist devices (LVADs) are used. forward the blood in available duewaiting to their limited ability. order extend the possible so called left assist they calculate the control signal sequence for the next cycle devices (LVADs) aretime, used. LVADs forward thetoblood blood in they they calculate the control signal sequence for the next cycle based on information from the previous one. As stated by gies that only require minor model knowledge, because devices (LVADs) are used. LVADs forward the in they calculate the control signal sequence for the next cycle based on information from the previous one. As stated by devices (LVADs) are used. LVADs forward the blood in parallel to the beating heart from the left ventricle (LV) the possible waiting time, so called left ventricular assist devices (LVADs) are used. LVADs forward the blood in based based on information from the previous one. As stated by parallel to the beating heart from the left ventricle (LV) on information from the previous one. As stated by Janssens et al. (2013) they execute the same task under they calculate the control signal sequence for the next cycle parallel to the beating heart from the left ventricle (LV) based on information from the previous one. As stated by Janssens et al. (2013) they execute the same task under parallel to the beating heart from the left ventricle (LV) to the aorta (Ao) as shown in Fig. 1. Typically ventricular devices (LVADs) are used. LVADs forward the blood in parallel to the beating heart from the left ventricle (LV) Janssens et al. (2013) they execute the same task under to the aorta (Ao) as shown in Fig. 1. Typically ventricular Janssens et al. (2013) they execute the same task under the same operating conditions and update the system based on information from the previous one. As stated by to the aorta (Ao) as shown in Fig. 1. Typically ventricular Janssens et al. (2013) they execute the same task under the same operating conditions and update the system to the aorta (Ao) as shown in Fig. 1. Typically ventricular assist devices are driven by a motor at constant speed ω(t) parallel to the beating heart from the left ventricle (LV) to the devices aorta (Ao) as shown Fig. 1.atTypically same operating conditions and update the system assist are driven driven by aainmotor motor constant ventricular speed ω(t) ω(t) the the same operating conditions and update the system input iteratively leading to high robustness against model Janssens et al. (2013) they execute the same task under assist devices are by at constant speed the same operating conditions and update the system iteratively leading to high robustness against model assist devices are driven by motor at constant speed ω(t) ω(t) input and thus not adaptive to changes in the hemodynamic to aorta as shown Fig. 1.at Typically ventricular assist devices are driven by motor constant speed input iteratively leading to high robustness against model andthethus thus not(Ao) adaptive to aain changes in the hemodynamic hemodynamic input iteratively leading to high robustness against model uncertainties and slowly changing repetitive disturbances. the same operating conditions and update the system and not adaptive to changes in the input iteratively leading to high robustness against model uncertainties and slowly changing repetitive disturbances. and thus not adaptive to changes in the hemodynamic conditions e.g. an increased blood flow demand during assist devices are driven by a motor at constant speed ω(t) and thus not adaptive to changes in thedemand hemodynamic uncertainties and slowly changing repetitive disturbances. conditions e.g. an increased blood flow during uncertainties and slowly changing repetitive disturbances. Therefore, ILC algorithms are well suited for repetitive input iteratively leading to high robustness against model conditions e.g. an increased blood flow demand during uncertainties and slowly changing repetitive disturbances. Therefore, ILC algorithms are well suited for repetitive conditions e.g.improvement an increased increased blood flow demand during Therefore, ILC algorithms are well suited for repetitive exercise an or decline in heart function. and thusor not adaptive to changes in the hemodynamic conditions e.g. an blood flow demand during exercise or an improvement or decline in heart function. Therefore, ILC algorithms are well suited for repetitive processes such as the cardiovascular system. Walter et al. uncertainties and slowly changing repetitive disturbances. exercise or or an an improvement orblood declineflow in heart heart function. Therefore,such ILCasalgorithms are well system. suited for repetitive the cardiovascular Walter et al. exercise improvement or decline in function. conditions e.g.improvement an increasedor demand during processes exercise or an decline in heart function. processes such as the cardiovascular system. Walter et al. processes such as the cardiovascular system. Walter et al. (2015) use an iterative learning algorithm in combinaTherefore, ILC algorithms are well suited for repetitive processes such as the cardiovascular system. Walter et al. (2015) use an iterative learning algorithm in combinaexercise or an improvement or decline in heart function. (2015) use an iterative learning algorithm in combina(2015) use an iterative learning algorithm in combination with a PID-controller in order to follow a desired processes such as the cardiovascular system. Walter et al. (2015) use an iterative learning algorithm in combination with a PID-controller in order to follow a desired tion with PID-controller in order to follow desired tion with PID-controller in order to follow a desired flow R¨ u schen et al. (2017) the left (2015) useaaa an iterative learning algorithm in a tion trajectory. with PID-controller in order tominimize follow acombinadesired flow trajectory. R¨ u schen et al. (2017) minimize the left flow trajectory. R¨ u schen et al. (2017) left Ao flow R¨ u schen et al. (2017) minimize the left ventricular an iterative learning control tion with astroke PID-controller order tominimize follow a the desired Ao flow trajectory. trajectory. R¨ uwork schenwith et in al. (2017) minimize the left ventricular stroke work with an iterative learning control Ao ventricular stroke work with an iterative learning control Ao ventricular stroke work with an iterative learning control algorithm. In a previous work, a norm-optimal iterative flow trajectory. R¨ u schen et al. (2017) minimize the left Ao ventricular stroke work with an iterative learning control algorithm. In a previous work, a norm-optimal iterative LA algorithm. In a (NOILC) previous work, aiterative norm-optimal iterative Ao LA algorithm. In a previous work, a norm-optimal iterative learning control algorithm for the control of the ventricular stroke work with an learning control algorithm. In a previous work, a norm-optimal iterative LA learning control (NOILC) algorithm for the control of the LA learning control algorithm for the control of the LA learning control (NOILC) algorithm for the control of the end-diastolic (EDV) of the left ventricle has been algorithm. Involume a (NOILC) previous work, a norm-optimal iterative learning control (NOILC) algorithm for the control of the end-diastolic volume (EDV) of the left ventricle has been LA end-diastolic volume (EDV) of the left ventricle has been end-diastolic volume (EDV) of the left ventricle has been designed, Ketelhut et al. (2017). The aim of this paper is learning control (NOILC) algorithm for the control of the end-diastolic volumeet(EDV) of the leftaim ventricle has been designed, Ketelhut al. (2017). The of this paper is designed, Ketelhut et al. (2017). The aim of this paper is designed, Ketelhut et al. (2017). The aim of this paper is to verify if the performance of the NOILC algorithm can end-diastolic volume (EDV) left ventricle has been LV designed, Ketelhut et al. (2017). The aim of this paper is to verify if the performance of the NOILC algorithm can LV to verify if the performance of the NOILC algorithm can LV to verify if the performance of the NOILC algorithm can be improved through further system knowledge. NOILC designed, Ketelhut et al. (2017). The aim of this paper is LV to verify if the performance of the NOILC algorithm can be improved improved through through further further system system knowledge. knowledge. NOILC NOILC LV be be improved through further system knowledge. NOILC algorithms use plant models predict the next cycle’s to if the performance the NOILC algorithm can LV be verify improved through furtherofto system knowledge. NOILC algorithms use plant models to predict the next cycle’s algorithms use plant models to predict the next cycle’s algorithms use plant models to predict the next cycle’s controlled variable sequence. avoid the time-consuming be improved through furtherTo system knowledge. algorithms use plant models to predict the next NOILC cycle’s controlled variable sequence. To avoid the time-consuming controlled variable sequence. To avoid the time-consuming controlled variable sequence. To avoid the time-consuming identification experiments prior to the NOILC procedure, algorithms use plant models to predict the next cycle’s controlled variable sequence. To avoid the time-consuming identification experiments prior to the NOILC procedure, identification experiments prior to the NOILC procedure, identification experiments prior to the NOILC procedure, data-driven NOILC algorithms can be used. The controlled variable sequence. To avoid the time-consuming L V A identification experiments prior to the NOILC procedure, D data-driven NOILC NOILC algorithms algorithms can can be be used. used. The The datadataL V A D data-driven dataL V A data-driven NOILC algorithms can be used. The driven NOILC algorithms by Moore et al. (1989) and D identification experiments prior to the NOILC procedure, L V data-driven NOILC algorithms can beetused. The datadataLVA driven NOILC algorithms by Moore al. (1989) and AD D driven NOILC algorithms by Moore et al. (1989) and driven NOILC algorithms by Moore et al. (1989) and Kim and Zou (2008) estimate the system behavior using data-driven NOILC algorithms can be used. The dataLVAD drivenand NOILC algorithms by the Moore et al. (1989)using and Kim Zou (2008) estimate system behavior Kim and Zou (2008) estimate the system behavior using Fig. 1. Schematic drawing of a LVAD connected to the left Kim and Zou (2008) estimate the system behavior using input/output data obtained during the procedure by driven NOILC algorithms by Moore et al. (1989) anda Fig. 1. Schematic drawing of a LVAD connected to the left Kim and Zou (2008) estimate the system behavior using input/output data obtained during the procedure by a Fig. 1. Schematic drawing of a LVAD connected to the left the procedure by a Fig. Schematic ventricle. input/output data obtained during the procedure by a Kim and Zou data (2008)obtained estimateduring the system behavior using Fig. 1. 1. Schematic drawing drawing of of a a LVAD LVAD connected connected to to the the left left input/output ventricle. input/output data obtained during the procedure by a ventricle. ventricle. Fig. 1. Schematic drawing of a LVAD connected to the left input/output data obtained during the procedure by a ventricle. ventricle. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 152Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 152 Copyright © 2018 responsibility IFAC 152 Peer review© under of International Federation of Automatic Control. Copyright © 152 Copyright © 2018 2018 IFAC IFAC 152 10.1016/j.ifacol.2018.11.651 Copyright © 2018 IFAC 152
IFAC BMS 2018 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
parameter estimator and a model-free inversion-based approach. However, these algorithms either require a priori system information or a frequency-domain representation of the desired output. Janssens et al. (2013) estimate the system’s impulse response using a linear combination of input/output measurements from the previous iteration, but this method only holds if the system is linear time invariant. Barton and Alleyne (2011) determine the impulse response matrix with the linear time variant multi-input multi-output state space model of the system which shall be controlled. As part of this work, the same approach is applied on the control of LVADs. To do so the nonlinear model of the cardiovascular system is piecewise linearized. Highly nonlinear functions are replaced by differentiable functions and the resulting linear model is discretized. To validate the designed algorithm and test the tracking performance as well as the disturbance rejection, two simulative experiments are conducted. In these experiments EDV setpoint and pulmonary vein pressure are varied at different time instances. The remainder of this paper is structured as follows: Section 2 deals with the former norm-optimal iterative learning control algorithm of the LVAD and Section 3 with the model of the cardiovascular system and its integration in the control algorithm. The simulation results are compared and discussed in Section 4. Finally the conclusions are drawn in Section 5. 2. NORM-OPTIMAL ITERATIVE LEARNING CONTROL Iterative learning control algorithms use information from the previous cycle j as for example the previous control signal sequence uj to calculate the control signal sequence (1) uj+1 = uj + ∆uj+1 for the next cycle j+1. The control signal sequence changes (2) ∆uj+1 = η∆uopt,j+1 of norm-optimal iterative learning control algorithms (NOILC) are determined by a learning factor η times the vector of optimal control signal changes ∆uopt,j+1 . These changes result from the quadratic optimization problem 1 T min fT x + x H x (3) s s s s xs 2 s with ∆uopt,j+1 smin,y xs = . (4) smax,y Apart from the vector of optimal control signal changes ∆uopt,j+1 , the slack variables smin,y and smax,y are also considered in the optimization problem to allow boundary violations, thus avoiding conflicts between the constraints described in the following. The linear term fs and quadratic terms Hs in (3) result from the cost function (5) Jj+1 = Je,j+1 + Jδ,j+1 + Juδ,j+1 + Jmin,y,j+1 + Jmax,y,j+1 incorporating the costs Je,j+1 penalizing the predicted tracking error, Jδ,j+1 ensuring a uniform filling of the ventricle and Juδ,j+1 to avoid severe pump speed changes. To inhibit suction and ventricle dilatation the ventricular volume yj+1 shall be confined within a lower Jmin,y,j+1 153
153
and an upper Jmax,yj+1 boundary. Within this paper ventricular volume VLV (t) serves as controller input and angular velocity w(t) as corresponding output. In cycle j they are described as sequences yj (k) and uj (k) ∈ RNj ×1 with the variable number of samples TC (t) Nj = (6) TS per cycle. Whereas TC (t) is the cycle duration, TS the fixed sample time and k ∈ {0, 1, ..., Nj − 1} the time index. The time index k wraps back to zero at the beginning of a new cycle. The number of samples per cycle Nj determines the dimension of the quadratic optimization problem and thereby also the dimension of the controller output for the next cycle j + 1 necessitating the narrowing or stretching of the controller input yj and output uj . Interpolated or ∗ ∗ extrapolated values are denoted by an asterisk (yj , uj ). To incorporate continuous controller operation throughout the cycles the first controller output in the next cycle uj+1 (0) is set equal to the last one in the previous cycle uj (Nj − 1) leading to the linear equality constraint η ∆uoptj+1 (0) = uj (Nj − 1) − uj (0) . (7) Constrained controller output allows to take pump speed limitations predescribed by the manufacturer into account. The controller output uj+1 of the NOILC algorithm serves as setpoint ω0 (t) for the speed controller of the pump, see Fig. 2. The plant’s actual pump speed up,j+1 is equivalent to ω(t) in Fig. 2. As the next cycle’s controlled variable yj+1 the latter is predicted with an impulse response matrix Gu,j+1 ∈ RN ×N containing the corresponding model of pump dynamics. This yields the linear inequality constraints ∗ Ej+1 Gu,j+1 η ∆uopt,j+1 ≥ Ej+1 (up,min,j+1 − up,j ) (8) ∗
Ej+1 Gu,j+1 η ∆uopt,j+1 ≤ Ej+1 (up,max,j+1 − up,j ) , (9) with the upper up,max,j+1 and lower up,min,j+1 pump speed boundary and the identity matrix Ej+1 , where the upper left diagonal entry is changed to zero to avoid conflict with the equality constraint (7). As stated before, it is favorable to confine the ventricular volume within an upper and a lower boundary to prevent suction and dilatation. The corresponding linear inequality constraints ∗ η Gp,j+1 ∆uoptj+1 ≥ (ymin,j+1 − yj ) − smin,y,j+1 (10) smin,y ≥ 0 ∗
η Gp,j+1 ∆uoptj+1 ≤ (ymax,j+1 − yj ) + smax,y,j+1 . (11) smax,y ≥ 0 are fomulated in terms of the slack variables smin,y and smax,y and the block impulse response matrix Gp,j+1 for the prediction of the next cycle’s controlled variable yj+1 .
A detailed description of the algorithm including the singles components of the cost function of the NOILC algorithm (5) is provided by Ketelhut et al. (2017). 3. MODEL INTEGRATION 3.1 Nonlinear Model of the Cardiovascular System A simplified nonlinear model of the systemic circulation is used for preliminary simulative testing of the designed
IFAC BMS 2018 154 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
Repetitive Optimization Reference EDV
Setpoint ω0 (t) -
NOILC
-
Speed controller
Ventricular volume VLV (t)
LVAD
CVS Angular velocity ω(t)
Buffer
Fig. 2. Block diagram of the cascade control of the LVAD. algorithm. Apart from that, this paper deals with the improvement of the NOILC algorithm described in Section 2 by integrating the nonlinear plant model described in the following. The model aims at imitating the heart functionality based on the model of Leaning et al. (1983) and the systemic circulation as well as the baroreceptor reflex similar to the models of Toy et al. (1985) and Colacino et al. (2007). The differential equations PPV (t) − PLA (t) − QMV (t) V˙ LA (t) = RPV V˙ LV (t) = QMV (t) − QAV (t) − QVAD (t) PLV (t) − PAo (t) − RAV QAV (t) Q˙ AV (t) = LAV 0 PLV (t) > PAo (t) ∨ QAV (t) > 0 otherwise ˙ VAW (t) = QAV (t) − QAo (t) + QVAD (t)
PAo (t) − PSp (t) Q˙ Ao (t) = LAo PSp (t) − PSV (t) V˙ Sp (t) = QAo − ∆RSp (t) + RSp P (t) − PCNS (t) Ao P˙ CNS (t) = τCNS
(12) (13) (14)
(15) (16) (17) (18)
describe the cardiovascular system, where VLA (t), VLV (t), VAW (t) and VSp (t) are the atrial, ventricular, aortic and splanchic volumes and PLA (t), PLV (t), PAo (t) and PSp (t) the corresponding pressures. PPV (t) and RPV represent the pulmonary vein pressure and resistance. The flow through mitral valve, aortic valve, LVAD and aorta are described by QMV (t), QAV (t), QVAD (t) and QAo (t). The constants RAV , LAV and LAo are the resistance and inductance of the aortic valve and the aorta, while RSp is a splanchic resistance parameter, ∆RSp (t) the corresponding time-varying splanchic resistance and PSV (t) the splanchic venous pressure. State one to six either represent a volume or a volume flow in the cardiovascular system and the seventh state the central nervous system’s dynamic properties represented by the pressure PCNS (t) with the time constant τCNS . In order to describe the behavior of the pump the two additional differential equations 154
H(t) + PLV (t) − PAo (t) Q˙ VAD (t) = (19) LVAD ω0 (t) − ω(t) ω(t) ˙ = (20) τVAD with the pump pressure head H(t) and the LVAD inductance LVAD are used. The dynamics of the feedbackcontrolled pump speed ω(t) are approximated as a first order system with the time constant τVAD and the setpoint ω0 (t). For more details regarding model structure and parametrization refer to Ketelhut et al. (2017) and Schr¨ odel et al. (2016). 3.2 Linearization The aim of this paper is to determine if the NOILC’s performance can be improved by the integration of further model knowledge. So far a fully linear model is used for the prediction of the next cycle’s controlled variable sequence, which shall be replaced by the more detailed cardiovascular system model described in Section 3.1. The model is nonlinear, but the aim is to create a piecewise linearized version of it. According to this, the nonlinear switching functions included to simulate heart valves and atrial and ventricular pumping of the heart, have to be replaced by differentiable functions. For this purpose, modified arctangent functions are found well. The flow through the mitral valve for example PLA (t) − PLV (t) PLA (t) > PLV QMV (t) = , (21) RMV 0 otherwise
depends on the pressure difference (22) ∆PMV = PLA (t) − PLV (t) and the mitral valve resistance RMV . Rearrangement of (21) and the introduction of the conditional function α yields to ∆PMV (23) QMV (t) = α RMV with 1 ∆PMV > 0 . (24) α= 0 ∆PMV ≤ 0 The conditional function α can be replaced by the modified arctangent function 1 1 α ˜ = · atan(c · (∆PMV )) + (25) π 2
IFAC BMS 2018 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
Conditional function α Modified arctangent function α ˜
which is compressed by π and shifted to lie in the correct value range. The additional factor c results in a compression along ∆PMV to approximate the conditional function α as shown in Fig. 3.
α α, ˜ c = 10
α, ˜ c=1 α, ˜ c = 100
155
0
D1
···
0 .. . .. . .. . .. .
D2 C2 Bd,1 C3 Ad,2 Bd,1 C B 3 d,2 G = C4 Ad,3 Ad,2 Bd,1 (29) C A B 4 d,3 d,2 .. .. . . N −1 N −1 C A B Ad,q Bd,2 · · · DN N d,q d,1 CN q=2
q=3
1
similar to the impulse response matrix of Rockel (2006) and Barton and Alleyne (2011) for multi input multi output linear time variant systems.
0.5
An overview of the newly developed NOILC algorithm incorporating additional model knowledge is shown in Fig. 4.
0 −0.5
0
4. RESULTS AND DISCUSSION
0.5
∆PMV
4.1 Experimental Setup
Fig. 3. Approximation of the conditional function α by the modified arctangent function α. ˜
Consider the resulting nonlinear time variant cardiovascular system (CVS) model x˙ j (t) = f (xj (t), uj (t), t) , yj (t) = g(xj (t), uj (t), t)
(26)
where xj (t) is the n-dimensional state vector at time t, uj (t) the m-dimensional input vector and yj (t) the pdimensional vector of outputs in cycle j. Performing a first-order Taylor series approximation around the corresponding operating trajectory results in the linear time and iteration variant state space system ∆x˙ j (t) = Ak (t)∆xj (t) + Bk (t)∆uj (t) ∆yj (t) = Ck (t)∆xj (t) + Dk (t)∆uj (t)
(27)
where Ak (t) denotes the n × n Jacobian matrix of f (x(t), u(t), t) with respect to x, whose il-th entry is the partial derivative of the i-th component of f (x(t), u(t), t) with respect to the l-th component of x around the operating point defined by xj (k) and uj (k) as stated by Rockel (2006). The matrices Bk (t), Ck (t) and Dk (t) describe the remaining Jacobian matrices of f (x(t), u(t), t) and g(x(t), u(t), t) with respect to u and x respectively.
As stated before the aim of this paper is to determine if the performance of the NOILC algorithm described in Section 2 can be improved by further model knowledge. Therefore, the piecewise linearized cardiovascular system model described in Section 3.2 is used to predict the next cycle’s controlled variable sequence yj+1 instead of a first order system model. The response of the designed algorithm to changes in EDV setpoint and preload are shown in the following and compared to the response of the former NOILC algorithm. In both cases the NOILC algorithm calculates as predescribed the control sequence uj+1 for the entire next cycle and outputs it as the setpoint ω0 (t) for the low-level speed controller, see Fig. 2. The quadratic-optimization problem is solved using the quadproq-solver from MATLAB. The nonlinear model of the cardiovascular system described in Section 3.1 is used for the simulative testing. Therefore, a pathological version of the CVS model is implemented in MATLAB/Simulink. For more details regarding the setup and its parametrization refer to Ketelhut et al. (2017) and Schr¨odel et al. (2016). In the two experiments presented hereafter, EDV setpoint (EDV setpoint = 130 − 145 − 115 ml) and pulmonary vein pressure (PPV = 8.75 − 10 − 7.5 mmHg) are varied at different time instances. While the aim of the first experiment is to test the tracking performance, the second experiment represents a naturally occuring type of parameter variation. 4.2 EDV variation
3.3 Impulse Response Matrix In order to calculate the impulse response matrix Gp,j+1 for the prediction of the next cycle’s controlled variable, the linear state space model in (27) is discretized with a fixed step size T . The discrete form ∆xj (k + 1) = Ad,k (k) · ∆xj (k) + Bd,k (k) · ∆uj (k) (28) ∆y j (k) = Ck (k) · ∆xj (k) + Dk (k) · ∆uj (k) yields the impulse response matrix 155
The results in previous work show that it is possible to use a NOILC algorithm in order to regulate the EDV of a pathological ventricle, Ketelhut et al. (2017). The aim of this paper is to determine if these results can be further improved. In this first experiment the EDV setpoint is varied as stated above. To evaluate the effect of the model integration, two NOILC setups are being compared: the NOILC algorithm using the impulse response matrix of first order system and the NOILC algorithm with the impulse response matrix of the piecewise linearized
IFAC BMS 2018 156 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
0 ≤ k < Nj − 1 Output uj (k) Measure yj (k)
k = Nj − 1
Get Nj+1
Interpolate ∗ ∗ uj , y j → uj , yj
Piecewise linearization of CVS model around xj (k), uj (k) and discretization Ad,k , bd,k , Ck , Dk → GP,j+1
Quadratic Problem min Jj+1 (xs ) xs
s.t. As,eq xs = bs,eq As,ineq xs ≤ bs,ineq ∗
Get Qe ,Que and Quδ
uj+1 = uj + η∆uopt,j+1
Fig. 4. Overview of one cycle j of the newly developed NOILC algorithm. time variant plant model. Fig. 5 shows the EDV signal for both setups during the EDV setpoint variation. The corresponding results of the NOILC algorithm with the linearized time variant CVS model are shown in blue while those of the NOILC with the first order system are displayed in red, see Fig. 5. Fig. 5 illustrates that compared to the former NOILC algorithm only minor improvements have been achieved. One reason for this might be the linearization of the nonlinear cardiovascular system model or the approximation of the highly nonlinear switching functions by modified arctangent functions as stated in Section 3.2. The maximum absolute error of the NOILC algorithm with the linearized time variant CVS model is 3.3 ml and the root mean square error (RSME) amounts to 0.8 ml compared to a maximum error of 3.7 ml and a RSME of 1.1 ml for the NOILC algorithm with a first order system model. The main drawback of the newly developed algorithm is the increased computational effort and the resulting rise in computational time. PT1
The aim of the second simulative experiment is to test the response of the newly developed algorithm to changes in pulmonary vein pressure PPV . The pressure may vary, e.g. when a patient stands up from a horizontal position. The resulting system responses of the preload variation experiment are shown in Fig. 6. EDV of the NOILC algorithm with the linearized time variant system model is displayed in blue and the results of the former NOILC algorithm and EDV setpoint are shown in red and black. The results illustrate that the newly developed NOILC algorithm is able to keep the EDV slightly closer to its desired setpoint. After a pulmonary vein pressure change, which acts in this case as disturbance, less iterations are required for adaption. Norm-optimal iterative learning control algorithms are simple feed-forward control algorithms in time domain. Feedback is only incorporated in the iteration domain. Thereby, the maximum absolute error is reduced from 3.6 ml to 2.7 ml and the RSME from 1.4 ml to 1.1 ml. EDV setpoint
LTV
145
EDV (ml)
EDV (ml)
EDV setpoint
4.3 Preload variation
130 115 50
100
130 115 50
150
100
150
ILC Cycle j (-)
PT1
EDV setpoint
LTV
145
PT1
LTV
135 EDV (ml)
EDV (ml)
LTV
145
ILC Cycle j (-) EDV setpoint
PT1
137.5
130 125
130 55
65
55
75
65 ILC Cycle j (-)
ILC Cycle j (-)
Fig. 5. Reference variation results.
Fig. 6. Preload variation results. 156
75
IFAC BMS 2018 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
5. CONCLUSION The described integration of further model knowledge aims at improving the performance in the norm-optimal iterative learning control of LVADs. Compared to the original algorithm described by Ketelhut et al. (2017) a linearized version of the nonlinear time variant CVS model is used for the prediction of the system behavior. Two simulations are done to study the controller response to changes in EDV setpoint and pulmonary vein pressure. The results confirm that the integration of further model knowledge yields a slightly improved controller performance and a reduced number of iterations required for convergence. Alternatively, the NOILC algorithm could be combined with a feedback controller to incorporate the rejection of instantaneous disturbances, as suggested by R¨ uschen et al. (2017). Also the definition of a valid EDV range rather than a certain setpoint might be favorable considering the variations in the hemodynamic conditions. Another problem might be the measurement of the ventricular volume, because conductance catheters are highly invasive. Instead computational models based on routinely obtained hemodynamic conditions could be used to estimate the ventricular volume. In summary, a piecewise linearized version of the nonlinear time variant CVS model is incorporated in the newly designed NOILC algorithm yielding an improved disturbance rejection but also an increase of computational effort. ACKNOWLEDGEMENTS This work was supported by the German Research Foundation (DFG) as project Smart Life Support 2.0 (65/151RO2000/17-1). REFERENCES Aissaoui, N., Morshuis, M., Maoulida, H., Salem, J.E., Lebreton, G., Brunn, M., Chatellier, G., Hag`ege, A., Schoenbrodt, M., Puymirat, E., et al. (2017). Management of end-stage heart failure patients with or without ventricular assist device: an observational comparison of clinical and economic outcomes. European Journal of Cardio-Thoracic Surgery, 53(1), 170–177. AlOmari, A.H.H., Savkin, A.V., Stevens, M., Mason, D.G., Timms, D.L., Salamonsen, R.F., and Lovell, N.H. (2012). Developments in control systems for rotary left ventricular assist devices for heart failure patients: a review. Physiological measurement, 34(1), R1. Amacher, R., Ochsner, G., and Schmid Daners, M. (2014). Synchronized pulsatile speed control of turbodynamic left ventricular assist devices: review and prospects. Artificial organs, 38(10), 867–875. Barton, K.L. and Alleyne, A.G. (2011). A norm optimal approach to time-varying ilc with application to a multi-axis robotic testbed. IEEE Transactions on Control Systems Technology, 19(1), 166–180. doi: 10.1109/TCST.2010.2040476. Bozkurt, S. (2016). Physiologic outcome of varying speed rotary blood pump support algorithms: a review study. Australasian physical & engineering sciences in medicine, 39(1), 13–28. Colacino, F.M., Moscato, F., Piedimonte, F., Arabia, M., and Danieli, G.A. (2007). Left ventricle load impedance 157
157
control by apical VAD can help heart recovery and patient perfusion: a numerical study. ASAIO J., 53(3), 263–277. Janssens, P., Pipeleers, G., and Swevers, J. (2013). A data-driven constrained norm-optimal iterative learning control framework for lti systems. IEEE Transactions on Control Systems Technology, 21(2), 546–551. doi: 10.1109/TCST.2012.2185699. Ketelhut, M., Schr¨odel, F., Stemmler, S., Roseveare, J., Hein, M., Gesenhues, J., Albin, T., and Abel, D. (2017). Iterative learning control of a left ventricular assist device. IFAC-PapersOnLine, 50(1), 6684–6690. 20th IFAC World Congress. Ketelhut, M., Stemmler, S., Gesenhues, J., Hein, M., and Abel, D. (2018). Iterative learning control of ventricular assist devices with variable cycle durations. Manuscript submitted for publication. Kim, K.S. and Zou, Q. (2008). Model-less inversion-based iterative control for output tracking: piezo actuator example. In American Control Conference, 2008, 2710– 2715. IEEE. Leaning, M.S., Pullen, H.E., Carson, E.R., Al-Dahan, M., Rajkumar, N., and Finkelstein, L. (1983). Modelling a complex biological system: the human cardiovascular system — 2. Model validation, reduction and development. Transactions of the Institute of Measurement and Control, 5(2), 87–98. Moore, K.L., Dahleh, M., and Bhattacharyya, S. (1989). Iterative learning for trajectory control. In Decision and Control, 1989., Proceedings of the 28th IEEE Conference on, 860–865. IEEE. Moza, A., Gesenhues, J., Autschbach, R., Abel, D., Rossaint, R., Schmitz-Rode, T., and Goetzenich, A. (2017). Parametrization of an in-silico circulatory simulation by clinical datasets–towards prediction of ventricular function following assist device implantation. Biomedical Engineering/Biomedizinische Technik, 62(2), 123–130. Rockel, H. (2006). Analyse und Synthese parametrischer iterativ lernender Regelungen. VDI-Verlag. R¨ uschen, D., Prochazka, F., Amacher, R., Bergmann, L., Leonhardt, S., and Walter, M. (2017). Minimizing left ventricular stroke work with iterative learning flow profile control of rotary blood pumps. Biomedical Signal Processing and Control, 31, 444–451. Schr¨odel, F., Schindler, D., Claver, A., Hein, M., Ketelhut, M., and Abel, D. (2016). A physiological control strategy for continuous-flow left ventricular assist devices: The power ratio controller. In 2016 European Control Conference (ECC), 2416–2422. Stevens, M.C., Stephens, A., AlOmari, A.H.H., and Moscato, F. (2018). Chapter 20 - physiological control. In S.D. Gregory, M.C. Stevens, and J.F. Fraser (eds.), Mechanical Circulatory and Respiratory Support, 627 – 657. Academic Press. doi: https://doi.org/10.1016/B978-0-12-810491-0.00020-5. Toy, S.M., Melbin, J., and Noordergraaf, A. (1985). Reduced models of arterial systems. IEEE Trans Biomed Eng, 32(2), 174–176. Walter, M., R¨ uschen, D., Prochazka, F., L¨oschke, D., Schwandtner, S., and Leonhardt, S. (2015). Modellbildung und iterativ lernende Regelung eines Herzunterst¨ utzungssystems, 19–24. Technische Universit¨ at Dortmund, Dortmund.
ScienceDirect
IFAC PapersOnLine 51-27 (2018) 152–157
Iterative Learning Control of a Left Iterative Learning Control of a Left Iterative Learning Control of a Left Iterative Learning Control of a Left Ventricular Assist Device: Nonlinear Model Ventricular Assist Device: Nonlinear Model Iterative Learning Control of a Left Ventricular Assist Device: Nonlinear Model Ventricular Assist Device: Nonlinear Model Integration Integration Ventricular Assist Device: Nonlinear Model Integration Integration ∗ ∗∗ Integration M. Ketelhut ∗∗∗ S. Stemmler orner ∗∗∗ D. Abel ∗∗∗ ∗ M. Hein ∗∗ D. K¨ ∗ ∗∗
M. Ketelhut D. K¨ o rner M. Ketelhut ∗∗ S. S. Stemmler Stemmler ∗∗ M. M. Hein Hein ∗∗ D. K¨ o rner ∗∗ D. D. Abel Abel ∗∗ M. D. o M. Ketelhut Ketelhut ∗ S. S. Stemmler Stemmler ∗ M. M. Hein Hein ∗∗ D. K¨ K¨ orner rner ∗ D. D. Abel Abel ∗ ∗∗ ∗ M. Ketelhut S. Stemmler M. Hein D. K¨ o rner D. Abel ∗ Institute of Automatic Control, RWTH Aachen University, Aachen, ∗ Institute of Control, RWTH Aachen University, Aachen, ∗ InstituteGermany of Automatic Automatic Control, RWTH Aachen University, Aachen, ∗ Institute of Automatic Control, RWTH Aachen University, Aachen, (e-mail: [email protected]). InstituteGermany of Automatic Control, RWTH Aachen University, Aachen, (e-mail: [email protected]). ∗ ∗∗ Germany (e-mail: [email protected]). Germany (e-mail: [email protected]). for Anesthesiology, University Hospitel Aachen, Aachen, Institute of Automatic Control, RWTH Aachen University, Aachen, ∗∗ Clinic Germany (e-mail: [email protected]). ∗∗ Clinic for Anesthesiology, University Hospitel Aachen, Aachen, ∗∗ for Anesthesiology, University Hospitel Aachen, Aachen, ∗∗ Clinic ClinicGermany for Germany Anesthesiology, University Hospitel Aachen, Aachen, (e-mail: [email protected]) (e-mail: [email protected]). Clinic for Anesthesiology, University Hospitel Aachen, Aachen, Germany (e-mail: [email protected]) ∗∗ (e-mail: [email protected]) Germany (e-mail: [email protected]) Clinic for Germany Anesthesiology, University Hospitel Aachen, Aachen, Germany (e-mail: [email protected]) Germany (e-mail: [email protected]) Abstract: Norm-optimal iterative learning control algorithms use plant models to predict the Abstract: Norm-optimal Norm-optimal iterative iterative learning learning control control algorithms algorithms use use plant plant models models to to predict predict the the Abstract: Abstract: Norm-optimal iterative learning control algorithms use plant models to predict the system behavior. In this paper, we focus on improving the performance in the norm-optimal Abstract: Norm-optimal iterative learning control algorithms use plant models to predict the system behavior. behavior. In In this this paper, paper, we we focus focus on on improving improving the the performance performance in in the the norm-optimal norm-optimal system system In paper, we focus improving the in the norm-optimal iterative learning control left ventricular assist devices. For this Abstract: Norm-optimal iterative control algorithms usepurpose, plant models to predictused the system behavior. In this this of paper, we learning focus on on improving the performance inthe thepreviously norm-optimal iterativebehavior. learning control of left ventricular ventricular assist devices. Forperformance this purpose, the previously used iterative learning control of left assist devices. For this purpose, the previously used iterative learning control of left ventricular assist devices. For this purpose, the previously used simple plant model is replaced by a piecewise linearized version of a nonlinear cardiovascular system behavior. In this paper, we focus on improving the performance in the norm-optimal iterative learning control of left by ventricular assist devices. version For thisof purpose, the previously used simple plant plant model is replaced replaced a piecewise piecewise linearized nonlinear cardiovascular simple model is by a linearized version aaa nonlinear cardiovascular simple plant model is by a linearized version of cardiovascular system model including the left ventricular assist device. Simulations are carried out to study iterative learning control of left ventricular assist devices. For thisof purpose, the previously used simple plant model is replaced replaced by a piecewise piecewise linearized version of a nonlinear nonlinear cardiovascular system model including the left ventricular assist device. Simulations are carried out to study system modelmodel including the leftby ventricular assist device.and Simulations are carried carried out to study study system model including the left ventricular assist device. Simulations are out to the controller response to end-diastolic volume setpoint preload changes. The results show simple plant is replaced a piecewise linearized version of a nonlinear cardiovascular system model including the left ventricular assist device. Simulations are carried out to study the controller controller response response to to end-diastolic end-diastolic volume volume setpoint setpoint and and preload preload changes. changes. The The results results show show the the controller to end-diastolic volume setpoint preload changes. results show minor improvements regarding tracking performance and the rejection of disturbances but system model response including leftthe ventricular assist device.and Simulations are carried out to study the controller response tothe end-diastolic volume setpoint and preload changes. The results show minor improvements regarding the tracking performance and the rejection rejection of The disturbances but minor improvements regarding the tracking performance and the of disturbances but minor improvements regarding the tracking performance and the rejection of disturbances but also an increase of computational effort compared to the previous algorithm. the controller response to end-diastolic volume setpoint and preload changes. The results show minor improvements regarding theeffort tracking performance and the rejection of disturbances but also an increase of computational compared to the previous algorithm. also an increase compared to algorithm. also animprovements increase of of computational computational effort compared to the the previous previous algorithm. minor regarding theeffort tracking performance and the rejection of disturbances but also an increase of computational compared to the previous algorithm. © 2018, IFAC (International Federationeffort of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. also an increase of computational effort compared to the previous algorithm. Keywords: Biomedical system modeling, simulation and visualization, ventricular assist device, Keywords: Biomedical Biomedical system modeling, simulation and and visualization, visualization, ventricular ventricular assist assist device, device, Keywords: system modeling, simulation Keywords: Biomedical system modeling, simulation and visualization, ventricular assist device, norm-optimal iterative learning control Keywords: Biomedical system modeling, simulation and visualization, ventricular assist device, norm-optimal iterative learning control norm-optimal iterative learning control norm-optimal iterative system learningmodeling, control simulation and visualization, ventricular assist device, Keywords: Biomedical norm-optimal iterative learning control norm-optimal iterative learning control 1. INTRODUCTION As stated by Stevens et al. (2018) the system consist1. INTRODUCTION INTRODUCTION As stated by Stevens et al. (2018) the system consist1. As stated by Stevens et al. (2018) the system consist1. INTRODUCTION INTRODUCTION As stated by Stevens et al. (2018) the system consisting of LVAD and cardiovascular system (CVS) contains 1. As stated by and Stevens et al. (2018) the (CVS) system contains consisting of LVAD cardiovascular system ing of LVAD and cardiovascular system (CVS) contains 1. INTRODUCTION ing of LVAD and cardiovascular system (CVS) contains multiple nonlinearities, uncertainties, perturbations and As stated by Stevens et al. (2018) the system consisting of LVAD and cardiovascular system (CVS) contains multiple nonlinearities, uncertainties, perturbations and Terminal congestive heart failure is one of the most commultiple nonlinearities, uncertainties, perturbations and Terminal congestive congestive heart heart failure failure is is one one of of the the most most comcom- multiple nonlinearities, uncertainties, perturbations and constraints. There exist various different control strateing of LVAD and cardiovascular system (CVS) contains Terminal multiple nonlinearities, uncertainties, perturbations and constraints. There exist various different control strateTerminal congestive heart failure is one of the most common causes of death in Europe. In the optimal case, the Terminal congestive heart failure is one of the most comconstraints. There exist various different control stratemon causes of death in Europe. In the optimal case, the constraints. There exist various different control strategies addressing different aims. Reviews of different control multiple nonlinearities, uncertainties, perturbations and mon causes of death in Europe. In the optimal case, the constraints. There exist various different control strategies addressing different aims. Reviews of different control mon causes of death in Europe. In the optimal case, the treatment includes the implantation of a donor heart. As Terminal congestive heart failure is one of the most common causesincludes of death Europe. In the case, the addressing different aims. Reviews of different control treatment theinimplantation implantation of aaoptimal donor heart. heart. As gies gies addressing different aims. Reviews of different control strategies for LVADs are done by AlOmari et al. (2012), constraints. There exist various different control stratetreatment includes the of donor As gies addressing different aims. Reviews of different control for LVADs are done by AlOmari et al. (2012), treatment includes thein implantation of aaoptimal donor heart. As strategies stated by Aissaoui et al. (2017) patients are waiting mon of death Europe. In the case,very the treatment includes the of donor heart. As strategies for LVADs are done by et statedcauses by Aissaoui Aissaoui et al.implantation (2017) patients patients are waiting very strategies foral. LVADs are done by AlOmari AlOmari et al. al. (2012), (2012), Amacher et (2014) and Bozkurt (2016). Iterative learngies addressing different aims. Reviews of different control stated by et al. (2017) are waiting very strategies for LVADs are done by AlOmari et al. (2012), Amacher et al. (2014) and Bozkurt (2016). Iterative learnstated by Aissaoui et al. (2017) patients are waiting very long for a transplant and may die before a donor heart is treatment includes the implantation of a donor heart. As stated et and al. (2017) patients waiting very et al. (2014) and Bozkurt (2016). Iterative learnlong for forbyaa Aissaoui transplant may die die before are a donor donor heart is Amacher Amacher et al. (2014) and Bozkurt (2016). Iterative learning control (ILC) algorithms are open-loop control stratestrategies for LVADs are done by AlOmari et al. (2012), long transplant and may before a heart is Amacher et al. (2014) and Bozkurt (2016). Iterative learning control (ILC) algorithms are open-loop control stratelong for a transplant and may die before a donor heart is available due to their limited ability. In order to extend stated by Aissaoui et al. (2017) patients are waiting very long for a transplant and may die before a donor heart is ing control (ILC) algorithms are open-loop control strateavailable due to their limited ability. In order to extend ing control (ILC) algorithms are open-loop control strategies that only require minor model knowledge, because Amacher et al. (2014) and Bozkurt (2016). Iterative learnavailable due to their limited ability. In order to extend ing control (ILC) algorithms are open-loop control strategies that only require minor model knowledge, because available due to their limited ability. In order to extend the possible waiting time, so called left ventricular assist long for a transplant and may die before a donor heart is available duewaiting to theirtime, limited ability. order to extend that only require minor model knowledge, because the possible possible so called called leftInventricular ventricular assist gies gies that only require minor model knowledge, because they calculate the control signal sequence for the next cycle ing control (ILC) algorithms are open-loop control stratethe waiting time, so left assist gies that onlythe require minor model knowledge, because calculate control signal sequence for the next cycle the possible waiting time, soLVADs called leftInventricular ventricular assist devices (LVADs) are used. forward the blood in available duewaiting to their limited ability. order extend the possible so called left assist they calculate the control signal sequence for the next cycle devices (LVADs) aretime, used. LVADs forward thetoblood blood in they they calculate the control signal sequence for the next cycle based on information from the previous one. As stated by gies that only require minor model knowledge, because devices (LVADs) are used. LVADs forward the in they calculate the control signal sequence for the next cycle based on information from the previous one. As stated by devices (LVADs) are used. LVADs forward the blood in parallel to the beating heart from the left ventricle (LV) the possible waiting time, so called left ventricular assist devices (LVADs) are used. LVADs forward the blood in based based on information from the previous one. As stated by parallel to the beating heart from the left ventricle (LV) on information from the previous one. As stated by Janssens et al. (2013) they execute the same task under they calculate the control signal sequence for the next cycle parallel to the beating heart from the left ventricle (LV) based on information from the previous one. As stated by Janssens et al. (2013) they execute the same task under parallel to the beating heart from the left ventricle (LV) to the aorta (Ao) as shown in Fig. 1. Typically ventricular devices (LVADs) are used. LVADs forward the blood in parallel to the beating heart from the left ventricle (LV) Janssens et al. (2013) they execute the same task under to the aorta (Ao) as shown in Fig. 1. Typically ventricular Janssens et al. (2013) they execute the same task under the same operating conditions and update the system based on information from the previous one. As stated by to the aorta (Ao) as shown in Fig. 1. Typically ventricular Janssens et al. (2013) they execute the same task under the same operating conditions and update the system to the aorta (Ao) as shown in Fig. 1. Typically ventricular assist devices are driven by a motor at constant speed ω(t) parallel to the beating heart from the left ventricle (LV) to the devices aorta (Ao) as shown Fig. 1.atTypically same operating conditions and update the system assist are driven driven by aainmotor motor constant ventricular speed ω(t) ω(t) the the same operating conditions and update the system input iteratively leading to high robustness against model Janssens et al. (2013) they execute the same task under assist devices are by at constant speed the same operating conditions and update the system iteratively leading to high robustness against model assist devices are driven by motor at constant speed ω(t) ω(t) input and thus not adaptive to changes in the hemodynamic to aorta as shown Fig. 1.at Typically ventricular assist devices are driven by motor constant speed input iteratively leading to high robustness against model andthethus thus not(Ao) adaptive to aain changes in the hemodynamic hemodynamic input iteratively leading to high robustness against model uncertainties and slowly changing repetitive disturbances. the same operating conditions and update the system and not adaptive to changes in the input iteratively leading to high robustness against model uncertainties and slowly changing repetitive disturbances. and thus not adaptive to changes in the hemodynamic conditions e.g. an increased blood flow demand during assist devices are driven by a motor at constant speed ω(t) and thus not adaptive to changes in thedemand hemodynamic uncertainties and slowly changing repetitive disturbances. conditions e.g. an increased blood flow during uncertainties and slowly changing repetitive disturbances. Therefore, ILC algorithms are well suited for repetitive input iteratively leading to high robustness against model conditions e.g. an increased blood flow demand during uncertainties and slowly changing repetitive disturbances. Therefore, ILC algorithms are well suited for repetitive conditions e.g.improvement an increased increased blood flow demand during Therefore, ILC algorithms are well suited for repetitive exercise an or decline in heart function. and thusor not adaptive to changes in the hemodynamic conditions e.g. an blood flow demand during exercise or an improvement or decline in heart function. Therefore, ILC algorithms are well suited for repetitive processes such as the cardiovascular system. Walter et al. uncertainties and slowly changing repetitive disturbances. exercise or or an an improvement orblood declineflow in heart heart function. Therefore,such ILCasalgorithms are well system. suited for repetitive the cardiovascular Walter et al. exercise improvement or decline in function. conditions e.g.improvement an increasedor demand during processes exercise or an decline in heart function. processes such as the cardiovascular system. Walter et al. processes such as the cardiovascular system. Walter et al. (2015) use an iterative learning algorithm in combinaTherefore, ILC algorithms are well suited for repetitive processes such as the cardiovascular system. Walter et al. (2015) use an iterative learning algorithm in combinaexercise or an improvement or decline in heart function. (2015) use an iterative learning algorithm in combina(2015) use an iterative learning algorithm in combination with a PID-controller in order to follow a desired processes such as the cardiovascular system. Walter et al. (2015) use an iterative learning algorithm in combination with a PID-controller in order to follow a desired tion with PID-controller in order to follow desired tion with PID-controller in order to follow a desired flow R¨ u schen et al. (2017) the left (2015) useaaa an iterative learning algorithm in a tion trajectory. with PID-controller in order tominimize follow acombinadesired flow trajectory. R¨ u schen et al. (2017) minimize the left flow trajectory. R¨ u schen et al. (2017) left Ao flow R¨ u schen et al. (2017) minimize the left ventricular an iterative learning control tion with astroke PID-controller order tominimize follow a the desired Ao flow trajectory. trajectory. R¨ uwork schenwith et in al. (2017) minimize the left ventricular stroke work with an iterative learning control Ao ventricular stroke work with an iterative learning control Ao ventricular stroke work with an iterative learning control algorithm. In a previous work, a norm-optimal iterative flow trajectory. R¨ u schen et al. (2017) minimize the left Ao ventricular stroke work with an iterative learning control algorithm. In a previous work, a norm-optimal iterative LA algorithm. In a (NOILC) previous work, aiterative norm-optimal iterative Ao LA algorithm. In a previous work, a norm-optimal iterative learning control algorithm for the control of the ventricular stroke work with an learning control algorithm. In a previous work, a norm-optimal iterative LA learning control (NOILC) algorithm for the control of the LA learning control algorithm for the control of the LA learning control (NOILC) algorithm for the control of the end-diastolic (EDV) of the left ventricle has been algorithm. Involume a (NOILC) previous work, a norm-optimal iterative learning control (NOILC) algorithm for the control of the end-diastolic volume (EDV) of the left ventricle has been LA end-diastolic volume (EDV) of the left ventricle has been end-diastolic volume (EDV) of the left ventricle has been designed, Ketelhut et al. (2017). The aim of this paper is learning control (NOILC) algorithm for the control of the end-diastolic volumeet(EDV) of the leftaim ventricle has been designed, Ketelhut al. (2017). The of this paper is designed, Ketelhut et al. (2017). The aim of this paper is designed, Ketelhut et al. (2017). The aim of this paper is to verify if the performance of the NOILC algorithm can end-diastolic volume (EDV) left ventricle has been LV designed, Ketelhut et al. (2017). The aim of this paper is to verify if the performance of the NOILC algorithm can LV to verify if the performance of the NOILC algorithm can LV to verify if the performance of the NOILC algorithm can be improved through further system knowledge. NOILC designed, Ketelhut et al. (2017). The aim of this paper is LV to verify if the performance of the NOILC algorithm can be improved improved through through further further system system knowledge. knowledge. NOILC NOILC LV be be improved through further system knowledge. NOILC algorithms use plant models predict the next cycle’s to if the performance the NOILC algorithm can LV be verify improved through furtherofto system knowledge. NOILC algorithms use plant models to predict the next cycle’s algorithms use plant models to predict the next cycle’s algorithms use plant models to predict the next cycle’s controlled variable sequence. avoid the time-consuming be improved through furtherTo system knowledge. algorithms use plant models to predict the next NOILC cycle’s controlled variable sequence. To avoid the time-consuming controlled variable sequence. To avoid the time-consuming controlled variable sequence. To avoid the time-consuming identification experiments prior to the NOILC procedure, algorithms use plant models to predict the next cycle’s controlled variable sequence. To avoid the time-consuming identification experiments prior to the NOILC procedure, identification experiments prior to the NOILC procedure, identification experiments prior to the NOILC procedure, data-driven NOILC algorithms can be used. The controlled variable sequence. To avoid the time-consuming L V A identification experiments prior to the NOILC procedure, D data-driven NOILC NOILC algorithms algorithms can can be be used. used. The The datadataL V A D data-driven dataL V A data-driven NOILC algorithms can be used. The driven NOILC algorithms by Moore et al. (1989) and D identification experiments prior to the NOILC procedure, L V data-driven NOILC algorithms can beetused. The datadataLVA driven NOILC algorithms by Moore al. (1989) and AD D driven NOILC algorithms by Moore et al. (1989) and driven NOILC algorithms by Moore et al. (1989) and Kim and Zou (2008) estimate the system behavior using data-driven NOILC algorithms can be used. The dataLVAD drivenand NOILC algorithms by the Moore et al. (1989)using and Kim Zou (2008) estimate system behavior Kim and Zou (2008) estimate the system behavior using Fig. 1. Schematic drawing of a LVAD connected to the left Kim and Zou (2008) estimate the system behavior using input/output data obtained during the procedure by driven NOILC algorithms by Moore et al. (1989) anda Fig. 1. Schematic drawing of a LVAD connected to the left Kim and Zou (2008) estimate the system behavior using input/output data obtained during the procedure by a Fig. 1. Schematic drawing of a LVAD connected to the left the procedure by a Fig. Schematic ventricle. input/output data obtained during the procedure by a Kim and Zou data (2008)obtained estimateduring the system behavior using Fig. 1. 1. Schematic drawing drawing of of a a LVAD LVAD connected connected to to the the left left input/output ventricle. input/output data obtained during the procedure by a ventricle. ventricle. Fig. 1. Schematic drawing of a LVAD connected to the left input/output data obtained during the procedure by a ventricle. ventricle. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 152Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 152 Copyright © 2018 responsibility IFAC 152 Peer review© under of International Federation of Automatic Control. Copyright © 152 Copyright © 2018 2018 IFAC IFAC 152 10.1016/j.ifacol.2018.11.651 Copyright © 2018 IFAC 152
IFAC BMS 2018 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
parameter estimator and a model-free inversion-based approach. However, these algorithms either require a priori system information or a frequency-domain representation of the desired output. Janssens et al. (2013) estimate the system’s impulse response using a linear combination of input/output measurements from the previous iteration, but this method only holds if the system is linear time invariant. Barton and Alleyne (2011) determine the impulse response matrix with the linear time variant multi-input multi-output state space model of the system which shall be controlled. As part of this work, the same approach is applied on the control of LVADs. To do so the nonlinear model of the cardiovascular system is piecewise linearized. Highly nonlinear functions are replaced by differentiable functions and the resulting linear model is discretized. To validate the designed algorithm and test the tracking performance as well as the disturbance rejection, two simulative experiments are conducted. In these experiments EDV setpoint and pulmonary vein pressure are varied at different time instances. The remainder of this paper is structured as follows: Section 2 deals with the former norm-optimal iterative learning control algorithm of the LVAD and Section 3 with the model of the cardiovascular system and its integration in the control algorithm. The simulation results are compared and discussed in Section 4. Finally the conclusions are drawn in Section 5. 2. NORM-OPTIMAL ITERATIVE LEARNING CONTROL Iterative learning control algorithms use information from the previous cycle j as for example the previous control signal sequence uj to calculate the control signal sequence (1) uj+1 = uj + ∆uj+1 for the next cycle j+1. The control signal sequence changes (2) ∆uj+1 = η∆uopt,j+1 of norm-optimal iterative learning control algorithms (NOILC) are determined by a learning factor η times the vector of optimal control signal changes ∆uopt,j+1 . These changes result from the quadratic optimization problem 1 T min fT x + x H x (3) s s s s xs 2 s with ∆uopt,j+1 smin,y xs = . (4) smax,y Apart from the vector of optimal control signal changes ∆uopt,j+1 , the slack variables smin,y and smax,y are also considered in the optimization problem to allow boundary violations, thus avoiding conflicts between the constraints described in the following. The linear term fs and quadratic terms Hs in (3) result from the cost function (5) Jj+1 = Je,j+1 + Jδ,j+1 + Juδ,j+1 + Jmin,y,j+1 + Jmax,y,j+1 incorporating the costs Je,j+1 penalizing the predicted tracking error, Jδ,j+1 ensuring a uniform filling of the ventricle and Juδ,j+1 to avoid severe pump speed changes. To inhibit suction and ventricle dilatation the ventricular volume yj+1 shall be confined within a lower Jmin,y,j+1 153
153
and an upper Jmax,yj+1 boundary. Within this paper ventricular volume VLV (t) serves as controller input and angular velocity w(t) as corresponding output. In cycle j they are described as sequences yj (k) and uj (k) ∈ RNj ×1 with the variable number of samples TC (t) Nj = (6) TS per cycle. Whereas TC (t) is the cycle duration, TS the fixed sample time and k ∈ {0, 1, ..., Nj − 1} the time index. The time index k wraps back to zero at the beginning of a new cycle. The number of samples per cycle Nj determines the dimension of the quadratic optimization problem and thereby also the dimension of the controller output for the next cycle j + 1 necessitating the narrowing or stretching of the controller input yj and output uj . Interpolated or ∗ ∗ extrapolated values are denoted by an asterisk (yj , uj ). To incorporate continuous controller operation throughout the cycles the first controller output in the next cycle uj+1 (0) is set equal to the last one in the previous cycle uj (Nj − 1) leading to the linear equality constraint η ∆uoptj+1 (0) = uj (Nj − 1) − uj (0) . (7) Constrained controller output allows to take pump speed limitations predescribed by the manufacturer into account. The controller output uj+1 of the NOILC algorithm serves as setpoint ω0 (t) for the speed controller of the pump, see Fig. 2. The plant’s actual pump speed up,j+1 is equivalent to ω(t) in Fig. 2. As the next cycle’s controlled variable yj+1 the latter is predicted with an impulse response matrix Gu,j+1 ∈ RN ×N containing the corresponding model of pump dynamics. This yields the linear inequality constraints ∗ Ej+1 Gu,j+1 η ∆uopt,j+1 ≥ Ej+1 (up,min,j+1 − up,j ) (8) ∗
Ej+1 Gu,j+1 η ∆uopt,j+1 ≤ Ej+1 (up,max,j+1 − up,j ) , (9) with the upper up,max,j+1 and lower up,min,j+1 pump speed boundary and the identity matrix Ej+1 , where the upper left diagonal entry is changed to zero to avoid conflict with the equality constraint (7). As stated before, it is favorable to confine the ventricular volume within an upper and a lower boundary to prevent suction and dilatation. The corresponding linear inequality constraints ∗ η Gp,j+1 ∆uoptj+1 ≥ (ymin,j+1 − yj ) − smin,y,j+1 (10) smin,y ≥ 0 ∗
η Gp,j+1 ∆uoptj+1 ≤ (ymax,j+1 − yj ) + smax,y,j+1 . (11) smax,y ≥ 0 are fomulated in terms of the slack variables smin,y and smax,y and the block impulse response matrix Gp,j+1 for the prediction of the next cycle’s controlled variable yj+1 .
A detailed description of the algorithm including the singles components of the cost function of the NOILC algorithm (5) is provided by Ketelhut et al. (2017). 3. MODEL INTEGRATION 3.1 Nonlinear Model of the Cardiovascular System A simplified nonlinear model of the systemic circulation is used for preliminary simulative testing of the designed
IFAC BMS 2018 154 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
Repetitive Optimization Reference EDV
Setpoint ω0 (t) -
NOILC
-
Speed controller
Ventricular volume VLV (t)
LVAD
CVS Angular velocity ω(t)
Buffer
Fig. 2. Block diagram of the cascade control of the LVAD. algorithm. Apart from that, this paper deals with the improvement of the NOILC algorithm described in Section 2 by integrating the nonlinear plant model described in the following. The model aims at imitating the heart functionality based on the model of Leaning et al. (1983) and the systemic circulation as well as the baroreceptor reflex similar to the models of Toy et al. (1985) and Colacino et al. (2007). The differential equations PPV (t) − PLA (t) − QMV (t) V˙ LA (t) = RPV V˙ LV (t) = QMV (t) − QAV (t) − QVAD (t) PLV (t) − PAo (t) − RAV QAV (t) Q˙ AV (t) = LAV 0 PLV (t) > PAo (t) ∨ QAV (t) > 0 otherwise ˙ VAW (t) = QAV (t) − QAo (t) + QVAD (t)
PAo (t) − PSp (t) Q˙ Ao (t) = LAo PSp (t) − PSV (t) V˙ Sp (t) = QAo − ∆RSp (t) + RSp P (t) − PCNS (t) Ao P˙ CNS (t) = τCNS
(12) (13) (14)
(15) (16) (17) (18)
describe the cardiovascular system, where VLA (t), VLV (t), VAW (t) and VSp (t) are the atrial, ventricular, aortic and splanchic volumes and PLA (t), PLV (t), PAo (t) and PSp (t) the corresponding pressures. PPV (t) and RPV represent the pulmonary vein pressure and resistance. The flow through mitral valve, aortic valve, LVAD and aorta are described by QMV (t), QAV (t), QVAD (t) and QAo (t). The constants RAV , LAV and LAo are the resistance and inductance of the aortic valve and the aorta, while RSp is a splanchic resistance parameter, ∆RSp (t) the corresponding time-varying splanchic resistance and PSV (t) the splanchic venous pressure. State one to six either represent a volume or a volume flow in the cardiovascular system and the seventh state the central nervous system’s dynamic properties represented by the pressure PCNS (t) with the time constant τCNS . In order to describe the behavior of the pump the two additional differential equations 154
H(t) + PLV (t) − PAo (t) Q˙ VAD (t) = (19) LVAD ω0 (t) − ω(t) ω(t) ˙ = (20) τVAD with the pump pressure head H(t) and the LVAD inductance LVAD are used. The dynamics of the feedbackcontrolled pump speed ω(t) are approximated as a first order system with the time constant τVAD and the setpoint ω0 (t). For more details regarding model structure and parametrization refer to Ketelhut et al. (2017) and Schr¨ odel et al. (2016). 3.2 Linearization The aim of this paper is to determine if the NOILC’s performance can be improved by the integration of further model knowledge. So far a fully linear model is used for the prediction of the next cycle’s controlled variable sequence, which shall be replaced by the more detailed cardiovascular system model described in Section 3.1. The model is nonlinear, but the aim is to create a piecewise linearized version of it. According to this, the nonlinear switching functions included to simulate heart valves and atrial and ventricular pumping of the heart, have to be replaced by differentiable functions. For this purpose, modified arctangent functions are found well. The flow through the mitral valve for example PLA (t) − PLV (t) PLA (t) > PLV QMV (t) = , (21) RMV 0 otherwise
depends on the pressure difference (22) ∆PMV = PLA (t) − PLV (t) and the mitral valve resistance RMV . Rearrangement of (21) and the introduction of the conditional function α yields to ∆PMV (23) QMV (t) = α RMV with 1 ∆PMV > 0 . (24) α= 0 ∆PMV ≤ 0 The conditional function α can be replaced by the modified arctangent function 1 1 α ˜ = · atan(c · (∆PMV )) + (25) π 2
IFAC BMS 2018 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
Conditional function α Modified arctangent function α ˜
which is compressed by π and shifted to lie in the correct value range. The additional factor c results in a compression along ∆PMV to approximate the conditional function α as shown in Fig. 3.
α α, ˜ c = 10
α, ˜ c=1 α, ˜ c = 100
155
0
D1
···
0 .. . .. . .. . .. .
D2 C2 Bd,1 C3 Ad,2 Bd,1 C B 3 d,2 G = C4 Ad,3 Ad,2 Bd,1 (29) C A B 4 d,3 d,2 .. .. . . N −1 N −1 C A B Ad,q Bd,2 · · · DN N d,q d,1 CN q=2
q=3
1
similar to the impulse response matrix of Rockel (2006) and Barton and Alleyne (2011) for multi input multi output linear time variant systems.
0.5
An overview of the newly developed NOILC algorithm incorporating additional model knowledge is shown in Fig. 4.
0 −0.5
0
4. RESULTS AND DISCUSSION
0.5
∆PMV
4.1 Experimental Setup
Fig. 3. Approximation of the conditional function α by the modified arctangent function α. ˜
Consider the resulting nonlinear time variant cardiovascular system (CVS) model x˙ j (t) = f (xj (t), uj (t), t) , yj (t) = g(xj (t), uj (t), t)
(26)
where xj (t) is the n-dimensional state vector at time t, uj (t) the m-dimensional input vector and yj (t) the pdimensional vector of outputs in cycle j. Performing a first-order Taylor series approximation around the corresponding operating trajectory results in the linear time and iteration variant state space system ∆x˙ j (t) = Ak (t)∆xj (t) + Bk (t)∆uj (t) ∆yj (t) = Ck (t)∆xj (t) + Dk (t)∆uj (t)
(27)
where Ak (t) denotes the n × n Jacobian matrix of f (x(t), u(t), t) with respect to x, whose il-th entry is the partial derivative of the i-th component of f (x(t), u(t), t) with respect to the l-th component of x around the operating point defined by xj (k) and uj (k) as stated by Rockel (2006). The matrices Bk (t), Ck (t) and Dk (t) describe the remaining Jacobian matrices of f (x(t), u(t), t) and g(x(t), u(t), t) with respect to u and x respectively.
As stated before the aim of this paper is to determine if the performance of the NOILC algorithm described in Section 2 can be improved by further model knowledge. Therefore, the piecewise linearized cardiovascular system model described in Section 3.2 is used to predict the next cycle’s controlled variable sequence yj+1 instead of a first order system model. The response of the designed algorithm to changes in EDV setpoint and preload are shown in the following and compared to the response of the former NOILC algorithm. In both cases the NOILC algorithm calculates as predescribed the control sequence uj+1 for the entire next cycle and outputs it as the setpoint ω0 (t) for the low-level speed controller, see Fig. 2. The quadratic-optimization problem is solved using the quadproq-solver from MATLAB. The nonlinear model of the cardiovascular system described in Section 3.1 is used for the simulative testing. Therefore, a pathological version of the CVS model is implemented in MATLAB/Simulink. For more details regarding the setup and its parametrization refer to Ketelhut et al. (2017) and Schr¨odel et al. (2016). In the two experiments presented hereafter, EDV setpoint (EDV setpoint = 130 − 145 − 115 ml) and pulmonary vein pressure (PPV = 8.75 − 10 − 7.5 mmHg) are varied at different time instances. While the aim of the first experiment is to test the tracking performance, the second experiment represents a naturally occuring type of parameter variation. 4.2 EDV variation
3.3 Impulse Response Matrix In order to calculate the impulse response matrix Gp,j+1 for the prediction of the next cycle’s controlled variable, the linear state space model in (27) is discretized with a fixed step size T . The discrete form ∆xj (k + 1) = Ad,k (k) · ∆xj (k) + Bd,k (k) · ∆uj (k) (28) ∆y j (k) = Ck (k) · ∆xj (k) + Dk (k) · ∆uj (k) yields the impulse response matrix 155
The results in previous work show that it is possible to use a NOILC algorithm in order to regulate the EDV of a pathological ventricle, Ketelhut et al. (2017). The aim of this paper is to determine if these results can be further improved. In this first experiment the EDV setpoint is varied as stated above. To evaluate the effect of the model integration, two NOILC setups are being compared: the NOILC algorithm using the impulse response matrix of first order system and the NOILC algorithm with the impulse response matrix of the piecewise linearized
IFAC BMS 2018 156 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
0 ≤ k < Nj − 1 Output uj (k) Measure yj (k)
k = Nj − 1
Get Nj+1
Interpolate ∗ ∗ uj , y j → uj , yj
Piecewise linearization of CVS model around xj (k), uj (k) and discretization Ad,k , bd,k , Ck , Dk → GP,j+1
Quadratic Problem min Jj+1 (xs ) xs
s.t. As,eq xs = bs,eq As,ineq xs ≤ bs,ineq ∗
Get Qe ,Que and Quδ
uj+1 = uj + η∆uopt,j+1
Fig. 4. Overview of one cycle j of the newly developed NOILC algorithm. time variant plant model. Fig. 5 shows the EDV signal for both setups during the EDV setpoint variation. The corresponding results of the NOILC algorithm with the linearized time variant CVS model are shown in blue while those of the NOILC with the first order system are displayed in red, see Fig. 5. Fig. 5 illustrates that compared to the former NOILC algorithm only minor improvements have been achieved. One reason for this might be the linearization of the nonlinear cardiovascular system model or the approximation of the highly nonlinear switching functions by modified arctangent functions as stated in Section 3.2. The maximum absolute error of the NOILC algorithm with the linearized time variant CVS model is 3.3 ml and the root mean square error (RSME) amounts to 0.8 ml compared to a maximum error of 3.7 ml and a RSME of 1.1 ml for the NOILC algorithm with a first order system model. The main drawback of the newly developed algorithm is the increased computational effort and the resulting rise in computational time. PT1
The aim of the second simulative experiment is to test the response of the newly developed algorithm to changes in pulmonary vein pressure PPV . The pressure may vary, e.g. when a patient stands up from a horizontal position. The resulting system responses of the preload variation experiment are shown in Fig. 6. EDV of the NOILC algorithm with the linearized time variant system model is displayed in blue and the results of the former NOILC algorithm and EDV setpoint are shown in red and black. The results illustrate that the newly developed NOILC algorithm is able to keep the EDV slightly closer to its desired setpoint. After a pulmonary vein pressure change, which acts in this case as disturbance, less iterations are required for adaption. Norm-optimal iterative learning control algorithms are simple feed-forward control algorithms in time domain. Feedback is only incorporated in the iteration domain. Thereby, the maximum absolute error is reduced from 3.6 ml to 2.7 ml and the RSME from 1.4 ml to 1.1 ml. EDV setpoint
LTV
145
EDV (ml)
EDV (ml)
EDV setpoint
4.3 Preload variation
130 115 50
100
130 115 50
150
100
150
ILC Cycle j (-)
PT1
EDV setpoint
LTV
145
PT1
LTV
135 EDV (ml)
EDV (ml)
LTV
145
ILC Cycle j (-) EDV setpoint
PT1
137.5
130 125
130 55
65
55
75
65 ILC Cycle j (-)
ILC Cycle j (-)
Fig. 5. Reference variation results.
Fig. 6. Preload variation results. 156
75
IFAC BMS 2018 São Paulo, Brazil, September 3-5, 2018
M. Ketelhut et al. / IFAC PapersOnLine 51-27 (2018) 152–157
5. CONCLUSION The described integration of further model knowledge aims at improving the performance in the norm-optimal iterative learning control of LVADs. Compared to the original algorithm described by Ketelhut et al. (2017) a linearized version of the nonlinear time variant CVS model is used for the prediction of the system behavior. Two simulations are done to study the controller response to changes in EDV setpoint and pulmonary vein pressure. The results confirm that the integration of further model knowledge yields a slightly improved controller performance and a reduced number of iterations required for convergence. Alternatively, the NOILC algorithm could be combined with a feedback controller to incorporate the rejection of instantaneous disturbances, as suggested by R¨ uschen et al. (2017). Also the definition of a valid EDV range rather than a certain setpoint might be favorable considering the variations in the hemodynamic conditions. Another problem might be the measurement of the ventricular volume, because conductance catheters are highly invasive. Instead computational models based on routinely obtained hemodynamic conditions could be used to estimate the ventricular volume. In summary, a piecewise linearized version of the nonlinear time variant CVS model is incorporated in the newly designed NOILC algorithm yielding an improved disturbance rejection but also an increase of computational effort. ACKNOWLEDGEMENTS This work was supported by the German Research Foundation (DFG) as project Smart Life Support 2.0 (65/151RO2000/17-1). REFERENCES Aissaoui, N., Morshuis, M., Maoulida, H., Salem, J.E., Lebreton, G., Brunn, M., Chatellier, G., Hag`ege, A., Schoenbrodt, M., Puymirat, E., et al. (2017). Management of end-stage heart failure patients with or without ventricular assist device: an observational comparison of clinical and economic outcomes. European Journal of Cardio-Thoracic Surgery, 53(1), 170–177. AlOmari, A.H.H., Savkin, A.V., Stevens, M., Mason, D.G., Timms, D.L., Salamonsen, R.F., and Lovell, N.H. (2012). Developments in control systems for rotary left ventricular assist devices for heart failure patients: a review. Physiological measurement, 34(1), R1. Amacher, R., Ochsner, G., and Schmid Daners, M. (2014). Synchronized pulsatile speed control of turbodynamic left ventricular assist devices: review and prospects. Artificial organs, 38(10), 867–875. Barton, K.L. and Alleyne, A.G. (2011). A norm optimal approach to time-varying ilc with application to a multi-axis robotic testbed. IEEE Transactions on Control Systems Technology, 19(1), 166–180. doi: 10.1109/TCST.2010.2040476. Bozkurt, S. (2016). Physiologic outcome of varying speed rotary blood pump support algorithms: a review study. Australasian physical & engineering sciences in medicine, 39(1), 13–28. Colacino, F.M., Moscato, F., Piedimonte, F., Arabia, M., and Danieli, G.A. (2007). Left ventricle load impedance 157
157
control by apical VAD can help heart recovery and patient perfusion: a numerical study. ASAIO J., 53(3), 263–277. Janssens, P., Pipeleers, G., and Swevers, J. (2013). A data-driven constrained norm-optimal iterative learning control framework for lti systems. IEEE Transactions on Control Systems Technology, 21(2), 546–551. doi: 10.1109/TCST.2012.2185699. Ketelhut, M., Schr¨odel, F., Stemmler, S., Roseveare, J., Hein, M., Gesenhues, J., Albin, T., and Abel, D. (2017). Iterative learning control of a left ventricular assist device. IFAC-PapersOnLine, 50(1), 6684–6690. 20th IFAC World Congress. Ketelhut, M., Stemmler, S., Gesenhues, J., Hein, M., and Abel, D. (2018). Iterative learning control of ventricular assist devices with variable cycle durations. Manuscript submitted for publication. Kim, K.S. and Zou, Q. (2008). Model-less inversion-based iterative control for output tracking: piezo actuator example. In American Control Conference, 2008, 2710– 2715. IEEE. Leaning, M.S., Pullen, H.E., Carson, E.R., Al-Dahan, M., Rajkumar, N., and Finkelstein, L. (1983). Modelling a complex biological system: the human cardiovascular system — 2. Model validation, reduction and development. Transactions of the Institute of Measurement and Control, 5(2), 87–98. Moore, K.L., Dahleh, M., and Bhattacharyya, S. (1989). Iterative learning for trajectory control. In Decision and Control, 1989., Proceedings of the 28th IEEE Conference on, 860–865. IEEE. Moza, A., Gesenhues, J., Autschbach, R., Abel, D., Rossaint, R., Schmitz-Rode, T., and Goetzenich, A. (2017). Parametrization of an in-silico circulatory simulation by clinical datasets–towards prediction of ventricular function following assist device implantation. Biomedical Engineering/Biomedizinische Technik, 62(2), 123–130. Rockel, H. (2006). Analyse und Synthese parametrischer iterativ lernender Regelungen. VDI-Verlag. R¨ uschen, D., Prochazka, F., Amacher, R., Bergmann, L., Leonhardt, S., and Walter, M. (2017). Minimizing left ventricular stroke work with iterative learning flow profile control of rotary blood pumps. Biomedical Signal Processing and Control, 31, 444–451. Schr¨odel, F., Schindler, D., Claver, A., Hein, M., Ketelhut, M., and Abel, D. (2016). A physiological control strategy for continuous-flow left ventricular assist devices: The power ratio controller. In 2016 European Control Conference (ECC), 2416–2422. Stevens, M.C., Stephens, A., AlOmari, A.H.H., and Moscato, F. (2018). Chapter 20 - physiological control. In S.D. Gregory, M.C. Stevens, and J.F. Fraser (eds.), Mechanical Circulatory and Respiratory Support, 627 – 657. Academic Press. doi: https://doi.org/10.1016/B978-0-12-810491-0.00020-5. Toy, S.M., Melbin, J., and Noordergraaf, A. (1985). Reduced models of arterial systems. IEEE Trans Biomed Eng, 32(2), 174–176. Walter, M., R¨ uschen, D., Prochazka, F., L¨oschke, D., Schwandtner, S., and Leonhardt, S. (2015). Modellbildung und iterativ lernende Regelung eines Herzunterst¨ utzungssystems, 19–24. Technische Universit¨ at Dortmund, Dortmund.