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A Global Sliding Mode Controller Design for an Intra-Aorta Pump

Chang, Yu; Gao, Bin

doi: 10.1097/MAT.0b013e3181ede369
Biomedical Engineering

Both the intra-aorta pump system and the circulation system are nonlinear systems with external perturbation and internal uncertainty. Classical control methods are suggested for linear systems. Therefore, a global sliding mode controller (GSMC) is reported in this article. A dynamic disturbance compensator was used to estimate the uncertainty of the controlled intra-aorta pump system for eliminating chattering effect. Simulations were performed to verify the robustness and dynamic characters of the controller. Simulation results demonstrate that the chattering effect of the controller output is eliminated. The settling time of step response of flow rate (5 L/min) is 0.08 seconds without overshot or steady-state error. When the load torque step disturbance increases to 0.4 Nm, the settling time of the controlled system is 0.025 seconds. When the desired flow rate is pulsatile flow, the dynamic response time is 0.08 seconds, and the maximum flow rate error is 0.03 L/min. To verify the dynamic character of the GSMC, an experiment was conducted. Because the feedback frequencies of rotational speed and flow rate in the experiment were slower than the ones in the simulation, the performance of the controller deteriorates. The experiment results illustrate that the settling time of step response of flow rate (5 L/min) is 0.26 seconds, and the flow rate error is 0.1 L/min.

From the Department of Biomechanics, School of Life Science and BioEngineering, Beijing University of Technology, Beijing, People's Republic of China.

Submitted for consideration December 2009; accepted for publication in revised form April 2010.

Reprint Requests: Dr. Yu Chang, School of Life Science and BioEngineering, Beijing University of Technology, Beijing 100124, People's Republic of China. Email: changyu@bjut.edu.cn.

The sliding mode control is widely researched for its excellent robustness and transient characteristic.1 In 1992, Utkin et al.2 first proposed the sliding mode control as a useful method for the nonlinear uncertain system. Thereafter, the sliding mode control became an attractive control method for the nonlinear system. Yan et al.3 introduced a novel sliding mode control strategy, a global sliding mode control in 1998, which eliminated the reaching phase and derived the robustness throughout entire process. In 2001, Choi et al.4 designed a brushless dc motor controller based on the global sliding mode control theory that used a polynomial function to establish the time-variation switching surface and achieved good performance.

Ventricular-assist devices (VADs) have been used as bridge to transplantation for decades,5 and recently, the rotary VADs are being used as permanent treatment for end-stage heart failure patients.6 The rotary VADs rotational speed demands to response to the physiological requirement, therefore researchers have been proposing different control strategies to meet this demand. Giridharan and Skliar7 proposed a mathematical model of the circulatory system and a control strategy in 2003, which maintained the physiological perfusion by maintaining a constant mean pressure head (75 mm Hg) between the left ventricle and aorta. Volkron et al.8 suggested a control strategy in 2006, in which the flow rate was adjusted depending on the heart rate. In 2009, Boston and coworkers9 designed a suction detector to indicate whether suction was present.

An intra-aorta pump was developed autonomously by the artificial heart research group at Beijing University of Technology. It consists of two parts: the blood pump and dynamic system (Figure 1). The blood pump is implanted between radix aorta and aortic arch to avoid damaging the aortic valve. It consists of a rotating impeller with blades to impart kinetic energy to the blood, a diffuser with helically curved blades, a pair of bearings, and housing. The dynamic system, which is placed in vitro, consists of a permanent magnet and a brushless dc motor. The designs have some advantages compared with the conventional VADs, for instance, there is no percutaneous wires leading to infection, and the temperature of the blood pump that is placed in vivo does not rise. However, the special structures of intra-aorta pump also introduce more internal uncertainty and external perturbation into the control system. The dynamic system uses magnetic coupling to transport kinetic energy to blood pump, differently than the conventional VAD system that uses winding to drive impeller. The magnetic force has a nonlinear relationship with respect to the distance between dynamic system and blood pump and the rotational speed. The relationship is too complex to get an accurate model; therefore, the uncertainty of intra-aorta pump control system is much greater than that of conventional VAD system. In addition, there is much external perturbation in the intra-aorta pump system, which is a nonlinear function of the variation of flow rate, peripheral resistance, and magnetic force. Moreover, conventional control algorithms such as the proportion-integration-differentiation (PID) algorithm are designed for linear system and not suitable for the system with great uncertainty. Fortunately, the sliding mode control performs well in the uncertain nonlinear system; the sliding mode control has invariability to the internal uncertainty and external perturbation as long as the trajectory of control system reaches the switching surface.1 In addition, the global sliding mode control algorithm eliminates the reaching phase. That is, the trajectory of global sliding mode controller (GSMC) reaches the switching surface at beginning of start up. Therefore, the GSMC is suitable to control the intra-aorta pump. The dynamic disturbance compensator can estimate uncertainty of the system directly without knowing the model of magnetic force.

Figure 1.

Figure 1.

In this study, the controller of the intra-aorta pump was designed based on the GSMC with an indices function. The dynamic disturbance compensator was used to estimate the uncertainty and external perturbation of the intra-aorta pump control system directly without knowing the model of magnetic force. The computer simulations and experiment were performed to verify the robustness and the dynamic characteristics of the GSMC controller.

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Materials and Methods

Modeling of the Intra-Aorta Pump

The dynamic system is described by the voltage equation and torque equation,10 which is denoted by Equations 1 and 2:

CV

CV

CV

CV

where U, R, i and e represent the phase voltage, the phase resistance, the phase current, and the back electromotive forces (BEMFs) of the winding, respectively; L denotes the phase inductance of the winding; J is the rotary inertia of the permanent magnet; T em, F, T L(t) and ω are the electromagnetic torque, damping coefficient, load torque, and rotational speed (in radian) of the motor, respectively. If motor has the trapezoidal BEMF, the T em is proportional to the phase current, and the amplitude of the BEMF is proportional to the rotational speed.10 Therefore, they are denoted by Equations 3 and 4:

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CV

CV

CV

where K e represents the torque constant, and K u denotes the BEMF constant.

The mathematic model of the blood pump is described as a function of the flow rate, pressure head, and rotational speed of the pump, which is denoted by Equation 5:

CV

CV

CV

CV

where Q PO represents the flow rate of the pump (L/min), P P is the pressure head of the pump (mm Hg), ω is the rotational speed (R/s), ωlimit denotes the threshold speed, and g( · ) represents the step function that is denoted by Equation 6. Equation 5 is used as a plant of the flow rate loop whose output is the desired speed of the intra-aorta pump. The rotational speed and its derivative are chosen as the state variables. According to Equations 1–4, the speed loop state equation of the intra-aorta pump is described as Equation 7:

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CV

where d(t) is defined as the uncertainty of the control system. It consists of the variation of the parameters, the load torque and its derivative. The load torque is a function of the flow rate, pressure head, and magnetic force.

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Designing of the GSMC

Based on the stated equation of the intra-aorta pump, the switching function and the sliding mode control law were designed. The initial condition of the control system was assumed as

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CV

, the errors of the speed and its derivative are defined as

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CV

, where the ωref(t) is the desired rotational speed, ω(t) represents the actual rotational speed. The uncertainty of the intra-aorta pump is assumed as it varied more slowly than sampling frequency. The switching function is designed as Equation 8:

CV

CV

CV

CV

where C is the parameter matrix of the switching function, and e 0 denotes the initial errors of the system, βi (i=1, 2) denotes the attenuation parameters ((Rei] >0)), which regulates the time of the system reaching stabilization, and E(t) is the diagonal matrix. The control system is on the switching surface at beginning of the startup, and the s(e, t) is approximate to 0 throughout the entire process.4 The poles of the control system are designed at (−0.6493, 0) and βi are both selected as −0.6493 to ensure the maximum response time <0.1 seconds.

The sliding mode control law is designed by Equation 10:

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CV

where u eq is the equivalent control variable, and u dis) denotes the switching control variable. Substituting Equations 8 and 9 into, 7 yield Equation 11:

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CV

where A is the system matrix of Equation 7, B is the control matrix of Equation 7, and C is the parameter matrix of the switch function: C=[83,1]. Obviously, the equivalent control cannot be applied directly because the uncertainty d(t) is unknown. Hence, the dynamic disturbance compensator11 is used to estimate the d(t) on line. Because the uncertainty of the system varied much slowly, d(tT S) is used to replace d(t). Then the equivalent control turns into Equation 12:

CV

CV

where T s represents the sampling period, d(tT S) denotes the perturbation at the previous time step. The d(tT S) is calculated from Equation 13:

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CV

where X(tT S) and U(tT S) are the state variable vector and input vector of the last control interval, which are the known quantity, respectively.

The switching control variable is designed as Equation 14:

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CV

where M is the gain of the switching control, which is a small positive number, and sign( · ) is the signal function. Because the dynamic disturbance compensator is used to eliminate the uncertainty of the control system, the switching function s(e, t)approximates to 0 throughout the entire process. However, the switching control variable generates discrete output (±M), when the switching function does not equate to 0. The discrete output is a key factor for the chattering effect,2 which is a phenomenon of finite-frequency, finite-amplitude oscillations appearing in output of conventional sliding mode control. Hence, the switching control is neglected in this article to eliminate the chattering effect, which is a limitation of the conventional global sliding mode control.

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Results

Simulations of the GSMC

Computer simulations of the GSMC intra-aorta pump were performed to verify the dynamic characteristics and the global robustness of the controller. The main parameters of the intra-aorta pump system were the inductance L = 0.336 mH, the resistance R = 1.4 Ω, the BEMF constant K u = 0.0229 Vs/rad, the torque constant K e = 0.5 Nm/A, the rotary inertia J = 8 × 10−5 kg m2, the deboost F = 0.00001 kg m2/s2, the desired flow rate was 5 L/min, and the control frequency was 100 Hz.

The simulation results of the GSMC for a constant desired flow rate are shown in Figures 2–5. It is seen that the settling time of step response of flow rate (5 L/min) is 0.08 seconds without overshot and steady-state error (Figure 2).The settling time is 0.025 seconds when the load torque step disturbance increases to 0.4 Nm at 0.12 seconds (Figure 3). From Figure 4, the chattering effect of the GSMC output is eliminated, as the dynamic disturbance compensator was used to estimate the uncertainty of the control system. Figure 5 is the phase plane plot of the GSMC where the horizontal coordinate is speed error, and the vertical coordinate is acceleration error. The real line in Figure 5 is the designed switching surface, and the dashed line is trajectory of the acceleration error relative to the speed error of the control system. Figure 5 illustrates that the trajectory of the GSMC reaches the switching surface at the beginning of the start up.

Figure 2.

Figure 2.

Figure 3.

Figure 3.

Figure 4.

Figure 4.

Figure 5.

Figure 5.

A pulsatile flow is added to the control system as the desired flow rate to further verify the dynamic characteristics of the GSMC. The data of pulsatile flow rate are derived from actual patients12 and superimposed on a constant flow rate to generate the desired flow rate, which is calculated by Equation 15:

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CV

where f ref(t) represents the desired flow rate; f cons is the constant flow rate that is set at 4.5 L/min; and f plus(t) is the pulsatile flow; the parameter α denotes the gain of the pulsatile flow rate that is set at 0.05.

The simulation result of the GSMC for a pulsatile flow is illustrated in Figure 6, which shows that the response time of the control system is 0.08 seconds, and the maximum error is 0.03 L/min.

Figure 6.

Figure 6.

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Experiments of the GSMC

A closed mock loop was set up (Figure 7) to verify accuracy of the GSMC. A preload chamber generates the left atrial pressure; a φ20 mm silicone tube mimicking the radix aortae has a length of 100 mm; an intra-aorta pump delivers the test fluid from the radix aortae to the aortic arch directly; a φ20 mm silicon tube represents the aortic arch, which has a length of 100 mm; a flow probe is placed after the aortic arch; two pressure transducers are placed in the middle of the radix aortae and the aortic arch to measure the pressure at the inlet and the outlet of the intra-aorta pump, respectively; by adjusting a clamp valve, flow rate and pressure head can be varied. The test fluid is 33% glycerin water (by volume) to simulate the viscosity of the blood at 37°C.

Figure 7.

Figure 7.

In this experiment, the desired flow rate is set at 5 L/min, and the control frequency is 100 Hz. The experiment results are illustrated in Figures 8 and 9. The settling times of rotational speed and flow rate are 0.26 seconds, which are greater than the simulation value (0.08 seconds). The fluctuations appear in the rotational speed response and flow rate response. In this experiment, hall sensors and flow probe are used to calculate the rotational speed and flow rate, whose feedback frequency are much slower than the ones in simulation where they are assumed to be derived continuously. Therefore, the performances of the experiment deteriorate, when compared with the simulation results. Because of the accuracy of the flow probe, the static error of the flow rate response is 0.1 L/min that is greater the one shown in simulation result (Figure 2).

Figure 8.

Figure 8.

Figure 9.

Figure 9.

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Discussion

The chattering effect is a shortage of the conventional sliding mode control, which maybe deteriorates the stability of control system. The switching control variable, which is used to overcome the uncertainty of the control system, is a key factor of the chattering effect.2 The dynamic disturbance compensator is used in this control system to estimate the uncertainty of the system. Hence, the uncertainty of the system has become known quantity by using the compensator, and the control algorithm can be designed for a nominal system without uncertainty, for which the switching control variable can be neglected for eliminating the chattering effect. The dynamic disturbance compensator requires that the disturbance change more slowly than sample frequency. In the intra-aorta pump control system, the disturbances are load torque and its derivative that are the functions of the flow rate, pressure head, and magnetic force, which change more slowly than sample frequency. Therefore, the dynamic disturbance compensator can estimate the uncertainty of the intra-aorta pump control system accurately.

In simulation, the flow rate and rotational speed are assumed to be measured continuously. However, they are measured discontinuously in the experiment; therefore, the performance deteriorated.1 The accuracy of the flow probe is low; therefore, the static error of flow rate (0.1 L/min) appears in the experiment. From experiment, it is seen that the performance of the controller depends on the feedback frequencies of the rotational speed and flow rate. Hence, the method, which can estimate these signals in time and accurately, is a key issue to maintain the performance of the controller.

The intra-aorta pump is a novel VAD with a simple structure. It demands the intelligent control strategy to automatically regulate the rotational speed responding to the physiological demands. Because of the complexity of the circulatory system, it is difficult to establish an accurate mathematic model for control.13 The fuzzy control theory does not need the mathematic model of the system; it only uses the experience of the engineers to build up the controller.9,14 Because the circulation system does not need the flow rate loop to have high accuracy, the variation tendency of its output copes with the physiological demands. Hence, the fuzzy controller is suitable for the flow rate control, whose inputs are the physiological signal and electrical signal of the intra-aorta pump and whose output is the desired speed. The global sliding mode control has proved its high accuracy and robustness with respect to various internal and external perturbations1; hence, it is suitable to control the speed loop of the intra-aorta pump. The control strategy, which is based on the fuzzy-controlled flow rate loop and global sliding mode-controlled speed loop, will be further researched in the future.

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Conclusion

The mathematic model of the intra-aorta pump was established. The model consists of the pressure-flow rate function and the speed loop state equation. The output of the pressure-flow rate function is the desired rotational speed that is used as the input of the speed loop. The GSMC of the intra-aorta pump was designed. The dynamic disturbance compensator was used to eliminate the system uncertainty for eliminating the chattering effect of the controller output. Simulations were performed to verify the robustness and the dynamic characteristics of the controller. The simulation results demonstrate that the output of the GSMC has no chattering effect, the settling time of the control system is <0.1 second whether the desired flow rate is constant or pulsatile, and the maximum error is 0.03 L/min. An experiment was conducted to verify the dynamic character of the controller. The results of the experiment illustrate that the settling time and static error are 0.26 seconds and 0.1 L/min, respectively, because the feedback frequencies of rotational speed and flow rate in experiment are slower than the ones in simulation, and the accuracy of the flow probe is low. Because of the limitation of the experimental condition, the pulsatile flow experiment was not conducted.

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Acknowledgment

Supported partly by the Beijing Nova Program (2006B12) and Scientific Research of Beijing Municipal Commission of Education (KM200710005033).

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