The analysis and design of observed-based nonlinear control of a heartbeat tracking system is investigated in this paper. Two of Zeeman’s heartbeat models are investigated and modified by adding the control input as a pacemaker, thereby creating the control-affine nonlinear system models that capture the general heartbeat behavior of the human heart. The control objective is to force the output of the heartbeat models to track and generate a synthetic electrocardiogram (ECG) signal based on the actual patient reference data, obtained from the William Beaumont Hospitals, Michigan, and the PhysioNet database. The formulations of the proposed heartbeat tracking control systems consist of two phases: analysis and synthesis. In the analysis phase, nonlinear controls based on input-output feedback linearization are considered. This approach simplifies the difficult task of developing nonlinear controls. In the synthesis phase, observer-based controls are employed, where the unmeasured state variables are estimated for practical implementations. These observer-based nonlinear feedback control schemes may be used as a control strategy in electronic pacemakers. In addition, they could be used in a software-based approach to generate a synthetic ECG signal to assess the effectiveness of diagnostic ECG signal processing devices.
Heartbeat Model; Electrocardiogram; Nonlinear Control of Biological Systems; Input-Output Feedback Linearization; Observer-Based Nonlinear Control Systems1. Introduction
The human heart is a complex yet robust system. One of the most important signals that are being generated during the operation of the human heart is the electrocardiogram (ECG). It is a time-varying signal measuring the potential of the electrical activities in the cardiac tissue. A single cycle of the ECG consists of the contraction and relaxation of the heart, which is the heart pumping action. The ECG can be measured by recording the potential differences between two electrodes placed on the skin at pre-determined positions. The characteristic information extracted from the ECG can be used to assess the state of the cardiac health and potential heart problems [1,2].
The research and development of mathematical models for heartbeats or heart rhythms have been very active in the literature [3-17]. One of the important developments was accomplished by Zeeman [3], where the models captured, at least qualitatively, three essential characteristics of the cardiac dynamics: 1) a stable equilibrium; 2) a threshold for triggering the action potential; and 3) the return to equilibrium. The resulting models are a 2nd- order nonlinear ordinary differential equation (ODE) of the Liénard-type representing the heartbeat dynamics, and a 3rd-order nonlinear ODE that can be applied to the nerve impulse. These models are based on the normal pacemaker generated by the sino-atrial (SA) node [4], which is the dominant pacemaker as compared to the slower one produced by the atrio-ventricular (AV) junction. This slower pacemaker is regarded in [5] as a passive conduit. In [6], the authors modified the 2nd-order ODE heartbeat system in [3] by incorporating an on-off control variable representing the pacemaker mechanism of contraction-relaxation of the heart. Reference [7] modified the 3rd-order nonlinear ODE model in [3] by adding control parameters that affect the frequency of the oscillation to control the heart rate variability and used a neural network to produce the ECG signal.
Another well-known approach to modeling the cardiac induction system is based on the van der Pol (VdP) type oscillators [8]. The idea of this approach stems from the analogy between the properties of relaxation oscillators and those of biological pacemakers. In contrast to [3], a coupled VdP oscillator assumes a more active role for the AV pacemaker, and considers the coupling effect between the SA and the AV pacemakers in the normal electrophysiological dynamics. A study of the synchronization properties of the rhythms produced by the SA and AV nodes using two coupled VdP oscillators was investigated in [9-13]. Furthermore, reference [14] proposed a proportional feedback control algorithm to synchronize the rhythm of the SA and AV pacemakers using the model from [11,12]. Recently, Gois and Savi [15] proposed a heart rhythm model that consists of three coupled modified VdP oscillators with time delay. Numerical simulations were carried out to reproduce the ECG signal that could be altered by changing the coupling parameters. A dynamical model that generates a synthetic ECG signal, based on statistical information such as the mean, standard deviation of the heart rate, and the power spectrum of the RR tachogram was proposed in [16]. The model does not rely on the dynamics of the heart but the ECG statistical information as a priori data to generate the signal. Reference [17] studied the synchronization between the models from [15,16] as a tracking control problem by using the model from [15] as a process plant and the ECG generated by the model from [16] as a reference signal, and proposed a heart rhythm proportional controller with tracking error as one of the control parameters.
This paper considers the analysis and design of observer-based controls of the modified Zeeman’s models in [3] using the input-output feedback linearization technique. A previous investigation without observers was presented in [18]. Observer-based techniques and technology are motivated by the needs of practical implementations: for systems with inaccessible state variables, it will be difficult to implement most control schemes requiring feedback of the inaccessible states without an observer for state estimation. An observer-based control scheme is, in general, more difficult to analyze and design. The importance of the observer-based nonlinear feedback tracking control technique developed in this paper is that it may be used as a control strategy for electronic pacemakers, or a software-based generation of a synthetic ECG signal for assessing the effectiveness of diagnostic ECG signal processing devices.
The paper is organized as follows. In Section 2, the dynamic model and its characteristics are investigatedphase portraits and stability analysis are conducted. Section 3 provides the fundamental of nonlinear feedback linearization control and observer theory. The results of the observer-based nonlinear tracking control systems and the simulation results are demonstrated in Section 4. The conclusions are presented in Section 5.
2. The Heartbeat Models
A cycle of the heartbeat consists of two states: diastole which is the relaxed state, and systole which is the contracted state. The cycle starts when the heart is in the diastolic state. The pacemaker that is located at the top of the right atrium—one of the upper chambers of the heart —triggers an electrochemical wave that spreads slowly over the atrium. This electrochemical wave causes the muscle fibers to contract and push the blood into the ventricles—the lower chambers of the heart. The electrochemical wave then spreads rapidly over the ventricles causing the whole ventricle to contract into the systolic state, and pumping the blood into the lung and the arteries. Immediately following the systolic state, the muscle fibers relax quickly and return the heart to the diastolic state to complete one cycle of the heartbeat [6].
A mathematical model that describes the behavior of the heartbeat was developed in [3], where it was suggested that such a model should contain three basic features:
1) A stable equilibrium state representing diastole;
2) A threshold for triggering the electrochemical wave causing the heart to go into systole; and 3) The return of the heart to the diastolic state.
The resulting models that address the characteristics above are given below.
2.1. The 2<sup>nd</sup>-Order Nonlinear Heartbeat Model
The 2nd-order nonlinear heartbeat model is given by
where represents the length of the muscle fiber, is a variable related to electrochemical activities, and can be measured as the potential across the membrane of the muscle fiber—the ECG signal; the parameter is a small positive constant associated with the fast eigenvalue of the system, is a scalar quantity representing a typical length of muscle fiber in the diastolic state, and represents tension in the muscle fiber.
Figure 1 illustrates the phase portrait of (1) with the initial conditions along the left and right diagonals across the plane. The parameter values used to produce the phase portrait are, T = 1, and The cubic line (dashed curve) represents the steady-state of the first equation in (1). When, the equilibrium point of the system is at the origin. All trajectories initiated above the cubic line, i.e., direct downward toward the origin along the cubic line.
Likewise, all trajectories started below the cubic line, that is, , direct upward toward the origin along the cubic line. All trajectories end up at the limit cycle around the equilibrium point. It is obvious that the equilibrium point is unstable as the vector field inside the limit cycle directs away from the point. This conclusion can be affirmed by analyzing the stability of the equilibrium point using the well-known Lyapunov indirect stability theorem [19]. For this purpose, let A be the constant Jacobian matrix of (1) at the origin, it fol-
lows that
The eigenvalues of A are given by and for, and therefore, the origin is unstable since both eigenvalues are real and positive.
In Figure 1, since the vector field around the segment AB and CD always points toward the cubic line, and away from the cubic line in the BC portion, any point along the cubic line in the AB and CD segments is considered to be stable whereas points along the BC section are unstable. The points B and C are important as they specify the threshold—the second basic feature (ii) of the heartbeat model mentioned earlier. These points can be computed easily by considering the eigenvalues of the matrix A in (2)
The condition that the real part of the eigenvalue is negative is. Therefore, the system is stable if which refers to the section AB, and
which describes the section CD. In other words, the thresholds for switching between the diastolic and the systolic states at point B is, and at point C.
The stable equilibrium point that represents the state of diastole can be determined by changing the value of in the second equation of (1) such that it satisfies the stability condition above. Figure 2 displays the phase portrait of the system with. The equilibrium point is stable at (1.024, −0.0497), and qualifies to be the
diastolic equilibrium state, i.e., satisfies the first feature (i): a stable equilibrium.
In Figure 2, all of the trajectories, regardless of their initial condition, end up at the diastolic equilibrium point. Since the equilibrium point is stable, the system will stay at this point forever unless there is an external excitation that forces the system to a new equilibrium point. In [6], the authors modified the system by adding a control input u(t) as shown below:
where the additional parameter represents a typical fiber length when the heart is in the systolic state, and represents the cardiac pacemaker control mechanism that directs the heart into the diastolic and the systolic states. By setting the cardiac pacemaker control signal u(t) in the form of {0} and {1} (on-off controls), the equilibrium point of the system can be switched between the diastolic and the systolic states. Figure 3 displays the phase portrait with the stable equilibrium point is located at (−1.3804, 1.25).
2.2. The 3<sup>rd</sup>-Order Nonlinear Heartbeat Model
The 3rd-order nonlinear heartbeat model is given by
where refers to the length of the muscle fiber, represents tension in the muscle fiber, is related to electrochemical activities, is a positive constant, and u(t) represents cardiac pacemaker control
signal which directs the heart into the diastolic and the systolic states.
The dynamics of the 3rd-order system are similar to those of the 2nd-order system except that the dynamic of the muscle fiber tension is taken into consideration, that is, the constant T in the 2nd-order system becomes a state variable in the 3rd-order system.
Consider a control-affine single-input single-output (SISO) nonlinear system described by
where is the state vector, are the control and output, respectively; f, g are smooth vector fields in a domain D and h a smooth function in D, where D is an open set in
Given the nonlinear system in (6), our goal is to find a transformation function (diffeomorphism) with that transforms the nonlinear system in the x-coordinates to a linear system in the z-coordinates. One of the most important reasons for finding the transformation is that the powerful linear system theory and methodologies can be applied once a nonlinear system has been linearized.
Differentiating the output with respect to t yields
where and denotes the Lie derivatives of with respect to and, respectively. If, then is not a function of. Continuing successive differentiation ρ times until the input appears explicitly, we obtain
The smallest integer for which u(t) appears is referred to as the relative degree. The nonlinear system in (6) is said to have a well-defined relative degree in a region if,; and,. When the relative degree is equal to the dimension of the nonlinear system, that is, , the system is said to be fully linearizable, whereas it is only partially linearizable if (both heartbeat systems considered in Section 4 below have; hence both are partially linearizable).
From (8), we define
where v(t) is a one-dimensional transformed input created by the feedback linearization process. Equation (9) yields the linearizing feedback control law [19-21]
provided is nonsingular.
To develop an overall representation of the system for the partially linearized case with, the transformation function can be expressed as
where, and are chosen such that is a diffeomorphism in a domain. In other words, the Jacobian matrix associated with is nonsingular, and
for all.
The transformation (11) leads to the normal form [20]
where and are in controllable canonical forms given by, respectively,
Equation (13) represents the external dynamics, while (14) is referred to as the internal dynamics of (6). Setting in (14) for all yields
which represents the zero dynamics for (6). The stability of the zero dynamics in (16) is an important issue in designing a controller. A system whose zero dynamics are asymptotically stable in the domain of interest is called a minimum phase system. The local asymptotic stability of the zero dynamics is, clearly, the necessary and sufficient conditions for the local asymptotic stability of the feedback linearized system described in (13)-(15) [21,22]. In the case that the zero dynamics are unstable in the region of interest, the system is known as a non-minimum phase system. Generally, a system of this type cannot be used for state-feedback control system design because some of the state variables will escape to infinity. In this case, the stabilization of the unstable zero dynamics needs to be considered, if possible.
3.2. Asymptotic Output Tracking
Let the control objective be steering the output to a desired reference signal. This gives rise to an output tracking control problem. Defining the output tracking error as, the main objective is to force such that as. It follows that
A suitable tracking control law for the transformed input v(t) is given by
where, is the constant gain matrix to be determined such that
is Hurwitz, that is, all of the eigenvalues of lie in the open left-haft complex plane. Combination of (18) and (10) yields the nonlinear tracking control law
3.3. Observer for Normal Form
The design of observer-based nonlinear control system is addressed in this section to provide real-time estimates of the inaccessible dynamical states required for the implementation of control laws. It is well-known that a Luenberger observer for a nonlinear control system based on input-output feedback linearization when exists, since the transformed system in the z-coordinates is in linear controllable canonical form. However, this is not true for the normal form, i.e., when, because the internal dynamics (14) are unobservable by the chosen output [19,20]. Nonetheless, by applying the results of [23,24], we will show that an observer for such systems may be possible. Without loss of generality, we assume that the normal form (13)-(15) has the equilibrium point at the origin. First, we linearize the normal form given by (13) and (14) in the following partitioned form:
where, , , , z is given in (11), and where
Equation (20) is in a standard linear system with being considered as a disturbance vector. If is an observable pair, that is,
and the term is Lipschitz so that there exists a Lipschitz constant such that
for all z in a region, then an observer for (20) can be formulated as
where the gain matrix is determined in such a way that is Hurwitz.
Now, let the estimation error associated with (20) and (25) be defined by. We need to show that converges to zero asymptotically. It follows from (20) and (25) that
Consider a Lyapunov candidate function
where P is a real symmetric positive definite matrix and is the solution of the Lyapunov equation
with Q a positive definite symmetric matrix. It follows that
Since is Lipschitz, so is. Substituting (24) into (29) yields
Since, it follows that
is negative definite, provided, so that the estimation error as.
Finally, with reference to (11), the observer given by (26) can be expressed in the x-coordinates as
4. Application to the Heartbeat Systems
We apply the theoretical results above to develop an observer-based nonlinear tracking control for the heartbeat systems (4) and (5). First, we consider the 2nd-order heartbeat system (4), with as the output measurement (recall that can be measured as the potential across the membrane of the muscle fiber).
Differentiating the output with respect to t yields
where u(t) appears, hence the relative degree is. The diffeomorphism T is given by
where satisfies (12). Equation (34) shows that the original system in (4) is already in a normal form when the output is chosen as. We note that (34) reveals that and are the internal and external dynamics, respectively. Rewriting (4) using (34) yields the normal form
The zero dynamics satisfy
There are three equilibrium points for (38):,. We need to analyze the stability of the zero dynamics. Applying the Lyapunov indirect stability theorem [19] to (38), the Jacobian matrices at the origin and are given by
Since T and are positive constants, it follows that and, hence the equilibrium point at the origin is unstable and the equilibrium points at are asymptotically stable. In other words, regardless of the unstable equilibrium at the origin, the steady-state of the zero dynamics will end up at either the point or depending on the initial condition. As a result, the zero dynamics are asymptotically stable. Therefore, the 2nd-order heartbeat system is a minimum-phase system.
To proceed to the output tracking control design, we let the tracking error be where. Using (18), the transformed input v(t) is given by
where K = 100 is obtained by placing the real pole at s = –100 of the complex plane. Consequently, the linearizing feedback control law according to (19) is given by
The development of an observer is accomplished by rewriting (35)-(37) in the form of (20) as:
where. It follows that in (43) is observable, and the term is locally Lipschitz. Therefore, the observer for (43) is given by (25) where the gain matrix is chosen by placing the observer poles at of the complex plane. Finally, the observer-based tracking control law for the 2nd-order heartbeat system is given by
The simulation of the 2nd-order heartbeat control system (4) with the output and the control law (44) was conducted using MATLAB. Figure 4 displays the tracking result of real discrete ECG data from PhysioNet database [25]. In Figure 4(a), the output converges and tracks the ECG reference signal very well. Figure 4(b) displays the pacemaker signal or the control law described in (44).
Figure 5 demonstrates the result of the observer in the x-coordinates along with the estimation errors. The initial condition of is, and that of the estimated states is (0, 0). Both estimated states converge quickly to the real states, especially. Finally, the multiple pulses ECG signal is illustrated in Figure 6.
Next, consider the 3rd-order heartbeat system (5) with as the output measurement. Differentiating the output with respect to t yields
The relative degree is. We obtain the transformation function
which also shows that the original system (5) is already in a normal form when the output is. Note that and satisfy (12). The normal form is written as
The zero dynamics are given by
There are two equilibrium points associated with (51): the origin, and = (1,–1). Applying the Lyapunov indirect stability theorem [19] to the latter equilibrium point yields
It follows that where represents the eigenvalue; hence, matrix is Hurwitz. Therefore, the equilibrium point at (1, –1) is asymptotically stable. Next, consider the equilibrium point at the origin
The eigenvalues of are 0 and –2. Since one of the eigenvalues is zero, we cannot draw the stability conclusion by the Lyapunov indirect stability theorem. However, using the application of the center manifold theory [19] to determine the stability of the equilibrium point at the origin by analyzing a reduced-order system—a system whose order is exactly equal to the number of the
eigenvalues of with zero real part, we found that the equilibrium point at the origin is asymptotically stable. This conclusion is illustrated by the phase portrait of the zero dynamics (51) as shown in Figure 7. All trajectories with initial condition converge to the origin. We conclude that the normal form system in (47)-(50) is a minimum-phase system.
We proceed to the output tracking control design. Since the relative degree in this case is the same as in the 2nd-order case, the transformed control law v(t) is of the same form as in (41). Subsequently, the tracking control law is given by
Similar to the 2nd-order case, the normal form (47)-(50) can be expressed in the form of (20) as
(55)
where. It follows that in (55) is an observable pair, and the term is locally Lipschitz. Thus, the observer for (55) is given by (25) where the gain matrix is obtained by placing the observer poles at of the complex plane. Finally, the observer-based tracking control law for the 3rd-order heartbeat system is given by
The simulation result for the ECG tracking is shown in Figure 8(a) with the control pacemaker signal displays in Figure 8(b). The results show an effective output tracking of the discrete ECG data from the William Beaumont Hospitals, Michigan.
Figure 9 displays the real and estimated state of to including their estimation errors. It shows that the estimated states converge quickly to the
real states with asymptotically stable error dynamics. Finally, Figure 10 illustrates the multiple pulses ECG signal created by the 3rd-order heartbeat tracking system.
5. Conclusion
We applied the nonlinear control system theory, based on input-output feedback linearization and observer theory, to a model for the biological heartbeat systems. Two Zeeman models were chosen in this study as they not only describe the heartbeat, but also offer direct biophysical relationship to the dynamic variables. The two models were modified by incorporating a control input into the systems, thereby creating two interesting controlaffine SISO nonlinear systems. We showed that the resulting heartbeat models are minimum-phase systems suitable for the design of output tracking control laws; these control laws were also used to generate synthetic ECG signals. In addition, an observer was applied to estimate the unknown variables in the transformed coordinates. The simulation results show that the observerbased tracking control laws effectively force the outputs of the systems to track the real ECG data from the PhysioNet database (Figure 4), and William Beaumont Hospitals, Michigan (Figure 8), with asymptotic stable tracking error. Other biomedical engineering applications of Zeeman’s models are under consideration.
6. Acknowledgements
The authors wish to acknowledge the support of an Oakland University-Beaumont Hospital multidisciplinary research grant for biomedical engineering research under the Oakland University-William Beaumont School of Medicine. We would also like to thank Dr. Robert Hammond of the William Beaumont Hospitals, Royal Oak, MI, for providing a set of ECG data used in the simulation studies; Dr. Bradley Roth of the Department of Physics, and Dr. Edward Gu of the Department of Electrical and Computer Engineering, both at Oakland University, for their valuable comments and suggestions.
REFERENCESReferencesN. Kannathal, C. M. Lim, U. R. Acharya and P. K. Sadasivan, “Cardiac State Diagnosis Using Adaptive NeuroFuzzy Technique,” Medical Engineering & Physics, Vol. 28, No. 8, 2006, pp. 809-815.
doi:10.1016/j.medengphy.2005.11.011L. Y. Shyu and W. Hu, “Intelligent Hybrid Methods for ECG Classification—A Review,” Journal of Medical and Biological Engineering, Vol. 28, No. 1, 2007, pp. 1-10. E. C. Zeeman, “Differential Equations for the Heartbeat and Nerve Impulse,” Towards a Theoretical Biology, Vol. 4, 1972, pp. 8-67.L. Glass, “Synchronization and Rhythmic Processes in Physiology,” Nature, Vol. 410, 2001, pp. 277-284.
doi:10.1038/35065745M. Vassalle, “The Relationship among Cardiac Pacemakers: Overdrive Suppression,” Circulation Research, Vol. 41, No. 3, 1977, pp. 269-277.
doi:10.1161/01.RES.41.3.269D. S. Jones and B. D. Sleeman, “Differential Equations and Mathematical Biology,” Chapman & Hall/CRC, London, 2003.N. Jafarnia-Dabanloo, D. C. McLernon, H. Zhang, A. Ayatollahi and V. Johari-Majd, “A Modified Zeeman Model for Producing HRV Signals and Its Application to ECG Signal Generation,” Journal of Theoretical Biology, Vol. 244, No. 2, 2007, pp. 180-189.
doi:10.1016/j.jtbi.2006.08.005B. van der Pole and J. van der Mark, “The Heart Beat Considered as a Relaxation-Oscillation and an Electrical Model of the Heart,” Philosophical Magazine Series 7, Vol. 6, No. 38, 1928, pp. 763-775. B. J. West, A. L. Goldberger, G. Rovner and V. Bhargava, “Nonlinear Dynamics of the Heartbeat, the AV Junction: Passive Conduit or Active Oscillator?” Physica D: Nonlinear Phenomena, Vol. 17, No. 2, 1985, pp. 198-206.
doi:10.1016/0167-2789(85)90004-1C. R. Katholi, F. Urthaler, J. Macy Jr. and T. N. James, “A Mathematical Model of Automaticity in the Sinus Node and AV Junction Based on Weakly Coupled Relaxation Oscillators,” Computers and Biomedical Research, Vol. 10, No. 6, 1977, pp. 529-543.
doi:10.1016/0010-4809(77)90011-8M. G. Signorini, S. Cerutti and D. D. Bernardo, “Simulation of Heartbeat Dynamics: A Nonlinear Model,” International Journal of Bifurcation and Chaos, Vol. 8, No. 8, 1998, pp. 1725-1731. doi:10.1142/S0218127498001418D. D. Bernardo, M. G. Signorini and S. Cerutti, “A Model of Two Nonlinear Coupled Oscillators for the Study of Heartbeat Dynamics,” International Journal of Bifurcation and Chaos, Vol. 8, No. 10, 1998, pp. 1975-1985.
doi:10.1142/S0218127498001637A. M. dos Santos, S. R. Lopes and R. L. Viana, “Rhythm Synchronization and Chaotic Modulation of Coupled van der Pol Oscillators in a Model for the Heartbeat,” Physica A: Statistical Mechanics and Its Applications, Vol. 338, No. 3-4, 2004, pp. 335-355.
doi:10.1016/j.physa.2004.02.058M. E. Brandt, G. Wang and H. T. Shih, “Feedback Control of a Nonlinear Dual-Oscillator Heartbeat Model,” Bifurcation Control, Vol. 293, 2003, pp. 715-718.S. R. F. S. M. Gois and M. A. Savi, “An Analysis of Heart Rhythm Dynamics Using a Three-Coupled Oscillator Model,” Chaos, Solitons and Fractals, Vol. 41, No. 15, 2009, pp. 2553-2565.
doi:10.1016/j.chaos.2008.09.040P. E. McSharry, G. D. Clifford, L. Tarassenko and L. A. Smith, “A Dynamical Model for Generating Synthetic Electrocardiogram Signals,” IEEE Transactions on Biomedical Engineering, Vol. 50, No. 3, 2003, pp. 289-294.
doi:10.1109/TBME.2003.808805M. J. Lopez, A. Consegliere, J. Lorenzo and L. Garcia, “Computer Simulation and Method for Heart Rhythm Control Based on ECG Signal Reference Tracking,” WSEAS Transactions on Systems, Vol. 9, No. 3, 2010, pp. 263-272.W. Thanom and R. N. K. Loh, “Nonlinear Control of Heartbeat Models,” Journal on Systemics, Cybernetics and Informatics, Vol. 9, No. 1, 2011, pp. 21-27.H. K. Khalil, “Nonlinear Systems,” 3rd Edition, Prentice Hall, Upper Saddle River, 2002.A. Isidori, “Nonlinear Control Systems,” Springer-Verlag, New York, 1995.M. A. Henson and D. E. Seborg, “Nonlinear Process Control,” Prentice Hall, Upper Saddle River, 1997.C. I. Byrnes and A. Isidori, “Asymptotic Stabilization of Minimum Phase Nonlinear Systems,” IEEE Transactions on Automatic Control, Vol. 36, No. 10, 1991, pp. 1122-1137. doi:10.1109/9.90226G. Ciccarella, M. Dallamora and A. Germani, “A Luenberger-Like Observer for Nonlinear Systems,” International Journal of Control, Vol. 57, No. 3, 1993, pp. 537556. doi:10.1080/00207179308934406N. H. Jo and J. H. Seo, “Input Output Linearization Approach to State Observer Design for Nonlinear System,” IEEE Transactions on Automatic Control, Vol. 45, No. 12, 2000, pp. 2388-2393. doi:10.1109/9.895580A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. Ch. Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C. K. Peng and H. E. Stanley, “PhysioBank, PhysioToolkit, and PhysioNet,” Circulation, Vol. 101, No. 23, 2000, pp. e215-e220.
doi:10.1161/01.CIR.101.23.e215
tracking error as one of the control parameters.
represents the length of the muscle fiber, 
is a variable related to electrochemical activities, and can be measured as the potential across the membrane of the muscle fiber—the ECG signal; the parameter 
is a small positive constant associated with the fast eigenvalue of the system, 
is a scalar quantity representing a typical length of muscle fiber in the diastolic state, and 
represents tension in the muscle fiber.
plane. The parameter values used to produce the phase portrait are
, T = 1, and 
The cubic line (dashed curve) represents the steady-state of the first equation in (1). When
, the equilibrium point of the system is at the origin. All trajectories initiated above the cubic line, i.e., 
direct downward toward the origin along the cubic line.
, direct upward toward the origin along the cubic line. All trajectories end up at the limit cycle around the equilibrium point. It is obvious that the equilibrium point is unstable as the vector field inside the limit cycle directs away from the point. This conclusion can be affirmed by analyzing the stability of the equilibrium point using the well-known Lyapunov indirect stability theorem [
and 
for
, and 
therefore, the origin is unstable since both eigenvalues are real and positive.
. Therefore, the system is stable if 
which refers to the section AB, and
which describes the section CD. In other words, the thresholds for switching between the diastolic and the systolic states at point B is
, and 
at point C.
in the second equation of (1) such that it satisfies the stability condition above. 
. The equilibrium point is stable at (1.024, −0.0497), and qualifies to be the
represents a typical fiber length when the heart is in the systolic state, and 
represents the cardiac pacemaker control mechanism that directs the heart into the diastolic and the systolic states. By setting the cardiac pacemaker control signal u(t) in the form of {0} and {1} (on-off controls), the equilibrium point of the system can be switched between the diastolic and the systolic states. 
the stable equilibrium point is located at (−1.3804, 1.25).
refers to the length of the muscle fiber, 
represents tension in the muscle fiber, 
is related to electrochemical activities, 
is a positive constant, and u(t) represents cardiac pacemaker control
in the 3rd-order system.
is the state vector, 
are the control and output, respectively; f, g are smooth vector fields in a domain D and h a smooth function in D, where D is an open set in 
with 
that transforms the nonlinear system in the x-coordinates to a linear system in the z-coordinates. One of the most important reasons for finding the transformation is that the powerful linear system theory and methodologies can be applied once a nonlinear system has been linearized.
with respect to t yields
and 
denotes the Lie derivatives of 
with respect to 
and
, respectively. If
, then 
is not a function of
. Continuing successive differentiation ρ times until the input 
appears explicitly, we obtain
for which u(t) appears is referred to as the relative degree. The nonlinear system in (6) is said to have a well-defined relative degree 
in a region 
if
,
; and
,
. When the relative degree is equal to the dimension of the nonlinear system, that is, 
, the system is said to be fully linearizable, whereas it is only partially linearizable if 
(both heartbeat systems considered in Section 4 below have
; hence both are partially linearizable).
is nonsingular.
, the transformation function 
can be expressed as
, 
and 
are chosen such that 
is a diffeomorphism in a domain
. In other words, the Jacobian matrix associated with 
is nonsingular, and
.
and 
are in controllable canonical forms given by, respectively,
in (14) for all 
yields
to a desired reference signal
. This gives rise to an output tracking control problem. Defining the output tracking error as
, the main objective is to force 
such that 
as
. It follows that
, 
is the constant gain matrix to be determined such that
is Hurwitz, that is, all of the eigenvalues of 
lie in the open left-haft complex plane. Combination of (18) and (10) yields the nonlinear tracking control law
exists, since the transformed system in the z-coordinates is in linear controllable canonical form. However, this is not true for the normal form, i.e., when
, because the internal dynamics (14) are unobservable by the chosen output 
[19,20]. Nonetheless, by applying the results of [23,24], we will show that an observer for such systems may be possible. Without loss of generality, we assume that the normal form (13)-(15) has the equilibrium point at the origin. First, we linearize the normal form given by (13) and (14) in the following partitioned form:
, 
, 
, 
, z is given in (11), and where
being considered as a disturbance vector. If 
is an observable pair, that is,
is Lipschitz so that there exists a Lipschitz constant 
such that
, then an observer for (20) can be formulated as
is determined in such a way that 
is Hurwitz.
. We need to show that 
converges to zero asymptotically. It follows from (20) and (25) that
is Lipschitz, so is
. Substituting (24) into (29) yields
, it follows that
, so that the estimation error 
as
.
as the output measurement (recall that 
can be measured as the potential across the membrane of the muscle fiber).
. The diffeomorphism T is given by
satisfies (12). Equation (34) shows that the original system in (4) is already in a normal form when the output is chosen as
. We note that (34) reveals that 
and 
are the internal and external dynamics, respectively. Rewriting (4) using (34) yields the normal form
,
. We need to analyze the stability of the zero dynamics. Applying the Lyapunov indirect stability theorem [
are given by
are positive constants, it follows that 
and
, hence the equilibrium point at the origin is unstable and the equilibrium points at 
are asymptotically stable. In other words, regardless of the unstable equilibrium at the origin, the steady-state of the zero dynamics will end up at either the point 
or 
depending on the initial condition. As a result, the zero dynamics are asymptotically stable. Therefore, the 2nd-order heartbeat system is a minimum-phase system.
where
. Using (18), the transformed input v(t) is given by
. It follows that 
in (43) is observable, and the term 
is locally Lipschitz. Therefore, the observer for (43) is given by (25) where the gain matrix 
is chosen by placing the observer poles at 
of the complex plane. Finally, the observer-based tracking control law for the 2nd-order heartbeat system is given by
and the control law (44) was conducted using MATLAB. 
converges and tracks the ECG reference signal very well. 
is
, and that of the estimated states 
is (0, 0). Both estimated states converge quickly to the real states, especially
. Finally, the multiple pulses ECG signal is illustrated in 
as the output measurement. Differentiating the output with respect to t yields
. We obtain the transformation function
. Note that 
and 
satisfy (12). The normal form is written as
= (1,–1). Applying the Lyapunov indirect stability theorem [
where 
represents the 
eigenvalue; hence, matrix 
is Hurwitz. Therefore, the equilibrium point at (1, –1) is asymptotically stable. Next, consider the equilibrium point at the origin
are 0 and –2. Since one of the eigenvalues is zero, we cannot draw the stability conclusion by the Lyapunov indirect stability theorem. However, using the application of the center manifold theory [
with zero real part, we found that the equilibrium point at the origin is asymptotically stable. This conclusion is illustrated by the phase portrait of the zero dynamics (51) as shown in 
converge to the origin. We conclude that the normal form system in (47)-(50) is a minimum-phase system.
(55)
. It follows that 
in (55) is an observable pair, and the term 
is locally Lipschitz. Thus, the observer for (55) is given by (25) where the gain matrix 
is obtained by placing the observer poles at 
of the complex plane. Finally, the observer-based tracking control law for the 3rd-order heartbeat system is given by
to 
including their estimation errors. It shows that the estimated states converge quickly to the