Journal of Control Science and Engineering

Volume 2008, Article ID 641847, 14 pages

http://dx.doi.org/10.1155/2008/641847

## Adaptive Output Tracking of Driven Oscillators

Chemical Engineering Department, Queen's University, Kingston, ON, Canada K7L 3N6

Received 1 August 2007; Revised 13 May 2008; Accepted 1 August 2008

Academic Editor: Jing Sun

Copyright © 2008 Lili Diao and Martin Guay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Heart dynamics are usually unknown and require the application of real-time control technique because of the fatal nature of most cardiac arrhythmias. The problem of controlling the heart dynamics in a real-time manner is formulated as an adaptive learning output-tracking problem. For a class of nonlinear dynamic systems with unknown nonlinearities and nonaffine control input , a Lyapunov-based technique is used to develop a control law. An adaptive learning algorithm is exploited that guarantees the stability of the closed-loop system and convergence of the output tracking error to an adjustable neighborhood of the origin. In addition, good approximation of the unknown nonlinearities is also achieved by incorporating a persistent exciting signal in the parameter update law. The effectiveness of the proposed method is demonstrated by an application to a cardiac conduction system modelled by two coupled driven oscillators.

#### 1. Introduction

Heart dynamics are very complicated by nature, and it is widely known that accurate analytical models are difficult to develop for cardiac dynamics and different types of arrhythmias. In addition, real-time control technique is needed because of the fatal nature of cardiac arrhythmias. As a result, real-time model-independent control techniques are needed to control heart dynamics in the presence of cardiac arrhythmias.

The dynamics of cardiac arrhythmias have been closely related to a variety of bifurcations and chaos phenomena. In the recent years [1–4], the theory of chaos control has made contributions to a mechanistic understanding of cardiac arrhythmias. Without detailed knowledge of the heart dynamics model structure, chaos control technique is able to regulate the abnormal heart rhythm by stabilizing the system around a desirable limit cycle. However, this approach is limited because to find a suitable controller parameter, it has to go through a “learning stage,” which comprises precontrol time-series recording and system dynamics estimation. In addition, the “learning stage” based on previous time series can lead to a poor estimation of system dynamics, because of the evolving nature of biological systems. The available adaptive approach [5] concentrated on linear chaos control, which is limited because the linear approach, can only delay or change the bifurcation location, while nonlinear control is needed in order to modify the stability property. According to the above observation, chaos control cannot serve as a real-time control technique to regulate cardiac arrhythmias.

To overcome this difficulty, we propose to apply some adaptive control technique to control cardiac arrhythmias. Rather than treat the unknown heart dynamics as a “black box,” we try to estimate the unknown dynamics using a neural network (NN) approach. NN techniques have undergone great developments and have been successfully applied in many fields, such as pattern recognition, signal processing, modelling, and system control. The approximating ability of NN has been proven in [6]. Multilayer NN identification and control techniques have been developed and demonstrated through simulation [7, 8], following the popularization of the “backpropagation” algorithm. However, analytical results obtained in [9] show that offline training is needed, because stability can be guaranteed only when the initial network weights are chosen sufficiently close to the ideal weight. To avoid the above difficulties in constructing stable neural systems, Lyapunov stability theory has been applied in developing control structure and deriving network weight updating laws [10–12]. Recently, multilayer NN control has been successfully applied to robotic control [13, 14]. In addition, [15] provides a systematic treatment of common problems in robotic control by introducing the G-Lee operator.

The aforementioned NN approaches are restricted to control-affine nonlinear systems. The problem of adaptively controlling systems with unknown nonaffine input nonlinearities is still open in the literature. Heart dynamics generally fall into the category of nonaffine systems, because of the inadequate knowledge of heart dynamics, and limited understanding of how actuators enter the dynamics.

Another limitation of current approaches is that approximation performance of the unknown nonlinearity and parameter estimation convergence are not discussed. Usually a high-gain control is employed to dominate the approximation error to ensure good tracking performance. However, in reality, it is often desirable to identify the unknown part of the dynamics.

In this work, we focus on an adaptive learning technique that is applicable to unknown nonlinear dynamic plants with a class of nonaffine input uncertainties, that are unknown, but continuous, and satisfy a sector constraint. An external signal, which is designed to be persistent exciting, is imbedded in the parameter update law to ensure good approximation and parameter convergence.

This paper is organized as follows. Section 2 presents the proposed adaptive controller. Application of the adaptive output feedback tracking technique to cardiac dynamics control is provided in Section 3. In Section 4, brief conclusions of the paper are given.

#### 2. Adaptive Control Design

We consider single-input/single-output (SISO) controllable nonlinear systems of the formwhere are the state variables of the zero dynamics; are the state variables of the main dynamics; and are the system input and output, respectively. The mapping is assumed to be an unknown continuous function of and and is assumed to be globally Lipschitz in . Given a reference trajectory , the control objective is to design an output feedback controller for system (1), which achieves good tracking performance subject to the unknown nonlinearities in the system.

Let . In this work, it is assumed that , with known constant . Denote the “steady-state” response of the tracking zero dynamics, governed by the differential equation , and . It is also assumed that , where is a known constant. Denote , and , where is to be chosen.

In this paper, radial basis function (RBF) presented in [16] was used to approximate a continuous function with approximation error , and basis function vectorwhere is the center of the receptive field, and is the width of the Gaussian function. The ideal weight in (2) is defined aswhere with positive constant to be chosen at the design stage, and is a compact set. Universal approximation results stated in [16, 17] (and the references therein) indicate that if is chosen sufficiently large, then can approximate any continuous function to any desired accuracy on a compact set, given that the centers are chosen close enough.

A number of assumptions are made for system (1).

Assumption. *The sign of is known, and there exist a positive constant and a nonzero continuous function such that *

Assumption. *
The
approximation error satisfies with unknown constant over a compact set. *

The design task is achieved in two steps: firstly, a state feedback adaptive tracking controller is designed; secondly, a high gain observer is used together with the state feedback controller to yield an output feedback adaptive tracking control law. We propose the following adaptive controller design.

Given the reference trajectory and , system (1) can be rewritten in terms of the tracking error and as follows: where

The proof of stability (convergence of the tracking error) is achieved by considering the tracking dynamics (6) and the error dynamics (7) as an interconnected system. With proper constraint imposed on the interconnected term, the overall stability is guaranteed by the use of small gain theorem [18], together with a proper choice of control and parameter update law. First, we must make the following assumption concerning system (6).

Assumption. * The
tracking dynamics of the system given by (6) are input-to-state stable (ISS).
That is, there exists a positive definite function ,
such that the following is satisfied: where ,
and are positive constants, and is the time derivative of along the solution of (1).*

Considering as a “disturbance” to the tracking error
zero dynamics (6), Assumption 3 ensures that dynamics in (6) are *input-to-state stable (ISS)* with respect
to .

By (Assumption 2), it follows from the implicit function theorem [19] that there exists a continuous function such that . The function may be re-expressed aswhere , and the following is assumed:where is a Lipschitz constant.

Approximate the unknown function asLet denote the estimate of . The parameter estimation error is given by .

Note that the boundedness of (Assumption 2) implies that is bounded as follows:where is a positive constant, and is a nonzero continuous function.

Using (12), the error dynamics (7) can be written as

The state feedback design and parameter update law are given by where is the controller gain function, is a positive constant, and is a projection algorithm.

The dynamics of are chosen as follows:where , and . Note that can always be made negative by a suitable choice of the gain constant . The matrix is a positive definite solution of the Riccati-like equation:for some positive definite symmetric matrix and positive and chosen as part of the design.

In addition to the above assumptions, we must ensure that a certain persistence of excitation condition is met to ensure that the unknown nonlinearity is estimated correctly.

Assumption. *
There exist positive constant and such that **where is the solution of (17), is a -dimensional identity matrix.*

The following lemma will be used in the sequel.

Lemma 1. *Consider the differential equation **where and are both column vectors. Assume that there
exist a and a such that **then the origin of (20) is a
globally exponentially stable equilibrium of the system.*

The proof of this lemma can be found in [20].

Theorem 1 gives the main results for the state feedback controller design.

Theorem 1. *
Consider the nonlinear system (1) in closed
loop with the controller and parameter update law provided in (15) and (16).
Assume that the signal is such that **for positive constants and ,
where is a solution of (17).*

Given Assumptions 1 to 3, all the signals of the closed-loop system are bounded. The parameter estimation errors converge exponentially to a small neighborhood of the origin.

The mean square tracking error satisfieswhere is a positive constant, and is a positive constant depending on system initial conditions.

Furthermore, the tracking error is such thatwhere , , , and are some positive constants.

*Proof. *See the appendix.

Since in practice, only a limited number of measurements can be obtained, one needs to build an observer to estimate the unmeasured states and implement the state feedback controller with the estimated states. For the tracking error system (7), a high-gain observer [19] is used, which takes the following form:where are the coefficients of a Hurwitz polynomial, is some small positive constant, and are the estimated tracking errors.

Following [21], we define the scaled estimation errors:Using (6) and (25), the dynamics of the scaled estimation errors are given bywhere the matrix and the vector assume the following form:We define the Lyapunov function , where is the symmetric positive definite matrix solution ofLet and consider the compact sets , where . Define the positive constant such that and the set . For the estimation errors, we define the set .

In order to apply the result of [21], we must verify that the state-feedback equation (15) and the learning rate for parameter estimation defined in (16) are globally bounded. Since they are not, we consider the application of a state feedback over the compact set . We first compute the constantsThe maximization is performed over all . It is assumed that the reference trajectories are bounded and evolve in a compact subset of . This ensures that the state of the system are also bounded on . The state-feedback and the adaptive learning rate can then bounded on by implementing the functions whereThe adaptive learning state feedback is rewritten as for . Having bounded the control and the adaptive learning rate, we pose the output-feedback controller where .

Theorem 2 provides the result for output feedback controller design.

Theorem 2. *
Consider the nonlinear system equation (1). If
Assumptions 1 to 3 are met then the dynamic output feedback controller
equations (35)–(37) guarantee that for any initial conditions of the
closed-loop system starting in there exists such that every trajectory of the closed-loop
system enters a small neighborhood of the origin in finite time, and it
converges exponentially to a small adjustable neighborhood of the origin. *

*Proof. *See the appendix.

The proof of Theorem 2 is included in the appendices for the sake of completeness. The approach is very similar to the one in the state-feedback case, except that a saturation function is used to isolate the peaking phenomenon in the estimated state dynamics, so as not to cause instability in the original state dynamics.

*Remark 1. *In many applications, convergence
of the error dynamics to a small neighbourhood of the origin may prove to be a
significant limitation. One mechanism that is known to reduce the onset of
tracking error offsets is the addition of integral action. In the current
context, it is straightforward to include the integral term, ,
which guarantees convergence of the tracking error.

#### 3. Application to A Driven Oscillator System

In the literature discussing the problem of regulating cardiac arrhythmias via chaos control, different types of cardiac models have been used. Some are merely constructed to describe a particular type of arrhythmias, such as the irregular interbeat model [22]. The others are models for different functions in the heart, such as the “black box” models [4] and empirical model [23] for the AV conduction system, and the mechanistic model that accounts for the action potentials of the ventricular myocardium [22].

One distinct physical mechanism in the heart is the pacemaker, consisting of the sinoatrial (SA) node and the atrioventricular (AV) node. The idea of considering this system mathematically as a system of coupled nonlinear oscillators is traced back to [24]. Since then, a lot of researchers have tried to study the dynamics of the heartbeat based on limit cycle oscillators [25, 26]. The model proposed in [27] describes the overall behavior of SA and AV nodes, captures the essential features of the cardiac conduction system, establishes a correspondence between system parameters and the physiological quantities, and is able to simulate different types of cardiac arrhythmias.

In this section, we apply the proposed adaptive output tracking controller to the four-dimensional coupled driven oscillators model in [27]. The model takes the following form: where and describe the action potential of the SA and AV nodes, and and are the voltage corresponding to and , and are some constant parameters in the model, is the constant coupling parameter, and are the amplitude and frequency of the driven signal, which is used to model ectopic pace-makers in some region of the cardiac tissue, and and are some nonlinear functions of the following form:

Systems (38a) to (38d) are in the form of system (1), with , where , and are equilibrium (invariant) trajectories for the zero dynamics. It is also assumed that the right-hand side of (38b) is unknown. It can be readily shown that this system meets Assumptions 1–3. (Just note that the tracking dynamics are linear in the output variable , and that the resulting system is stable for large values of and unstable in a small region containing the origin).

The electrical action potential can be measured by an transvenous electrode, which is a common part of artificial pacemakers. Given the following parameter values [27]: , and , the system exhibits a normal 1:1 rhythm. By setting , one can generate an arrythmia of 2:1 AV block: for every two beats of the SA node, only one beat of AV node is observed. The control objective is to apply the proposed adaptive output tracking controller to make systems (38a) to (38d) track the normal 1:1 rhythm, in other words, to suppress the AV block arrythmia.

We choose to use a perturbation to the right-hand side of (38b) as the controlPhysically, the control action is an electrical impulse sent to the heart through a transvenous electrode, which enters the system in an affine manner, as shown in (40).

*Remark 2. *Other potential control actuators
are perturbation to the intrinsic frequency of the SA node (parameter ) and the coupling strength between the two
nodes (parameter ).

In this example, the unknown nonlinearity in (12) is as follows:

For the simulation, the number of basis functions is , with , and . The following tuning parameters are used, , .

In the simulation, the controller is not turned on until . Figure 1 shows the simulation results of the SA and AV node rhythm, tracking performances, and the control action. In the subplot of SA and AV node action potential, the dotted line is the SA node action potential, and the solid line is the AV node action potential. It can be seen that before the controller is turned on, the rhythm is 2:1, with two beats of the SA node, only one beat of AV node is observed. After the controller is turned on at , the rhythm is altered to 1:1 within one beat. The two subplots of tracking show that good tracking performances are achieved within a short time for the SA node and AV node, respectively. Figure 2 shows the approximation of the unknown nonlinearity and parameter estimation. The unknown nonlinearity is the solid line, and the approximation is the dotted line. After running the simulation for 300 seconds, good approximation is achieved along the and direction, and parameter estimations also converge.

#### 4. Conclusions

In this research work, an adaptive output tracking controller is developed for a class of nonlinear systems with unknown nonlinearities, in order to address the heart dynamics control problem in a real-time framework. It is proved that the proposed controller is able to make the tracking error converges to a neighborhood of the origin exponentially fast. Simulation results show satisfactory performances that can be achieved when applying this technique to regulate irregular heart dynamics. In addition, good approximation of the unknown nonlinearities is also achieved by incorporating a persistent exciting signal in the parameter update law. The proposed technique is an alternative approach to the control of complex chaotic systems with unknown dynamics.

#### Appendix

*Proof of Theorem 1. *
Consider the
following Lyapunov function candidate for the subsystem (14):where is a symmetric positive definite function.

The derivative of the Lyapunov function is given byChoosing ,
we haveA candidate Lyapunov function for
the error dynamics iswhere ,
and is the state of the filter (17) time-varying
function to ensure persistency of excitation condition. Note that this filter
is ISS with respect to the signal for any choice of gain large enough.

The time derivative of is given bySubstitution of yieldsGiven that ,
andwe haveorwhere ,
and are positive constants andThe constants and are chosen such that .
If the matrix is chosen as a positive definite solution of
the Riccati-like equationfor some positive definite
symmetric matrix ,
then inequality (A.9) becomes

Substituting (12), inequality (A.12) becomes where and are positive constants.

Consider the control structure shown in (15), we choose
the following controller:where is a constant, and are the lower and upper bound in the
inequality (13), and is given byThe above control action
constitutes filtered tracking error and the approximated nonlinearity.

The weight satisfies ,
where the upper bound, ,
is guaranteed in the design of an adaptive law (16).

Substitution of the controller equation (A.14) in the
inequality (A.13) givesbecause of the inequality (13).

The adaptive law (16) is chosen such that ,
andIt takes the following
form:

Substitution of the controller equation (A.14), the
parameter update law inequality (A.17), and the dynamics (17), in the inequality (A.16),
giveswhere ,
and due to space limits.

Since ,
it follows that

Let ,
and let ,
we have

Let ,
inequality (A.21) becomes

Noting that, by assumption, ,
and using the fact thatwe obtain

Consider the following composite Lyapunov function for
the closed-loop system (15), (16), (6), and (7):where is a positive design parameter. The derivative
of the Lyapunov function is given bywhere ,
since, by definition, the trajectory error vector is such thator

Define,
and the gains of the two interconnected system (6) and (7) are such that ,
we haveor

It follows that the error vector, ,
variable ,
and the parameter estimation errors, ,
are bounded. Since the tracking trajectory and its first derivatives are bounded, it follows that the
states and are both bounded.

Integration of inequality (A.31) yields an explicit
bound for given bywhere ,
and .

Next, we derive a persistency of excitation condition
that guarantees the convergence of the parameter estimates to the ideal
weights, .

A solution to (17) is given by

It is obvious that the differential
equationis globally exponentially
stable.

The element is a bounded function of time. Note
thatFor the particular choice of
basis functions proposed in this paper, we have ,
where is the number of weights used in the
approximation. The boundedness of is obvious, since is bounded. Therefore, it follows that the
norm of is bounded by some positive number ,
that is,

Using the exponential stability of system (A.34) and the
bound on ,
an explicit bound for the solution of (17) can be obtained as
follows:where and is a positive constant.

Next, it is shown that the parameter estimation error converges to a neighborhood of the origin. In
the following, it is shown that, under Assumption 4, the unperturbed (i.e., ) dynamics of the parameter estimation errors
are exponentially stable.

Lemma 1 establishes that the origin of the
differential equationis an exponentially stable
equilibrium. In fact, it follows from the proof of Lemma 1 that the Lyapunov
function is such thatfor a positive constant.

It follows from the property of the projection
algorithm that the rate of change of along the trajectories of (A.18) is given
byCompleting the squares, we
getSubstitution of (A.32)
yieldsBy integrating, we
getConsequently, the parameter
estimation error is guaranteed to decay exponentially as

Taking the limit as confirms that the estimation error converges
to a small adjustable neighborhood of the origin given by

Under the assumption that the persistency of
excitation condition is fulfilled, we have demonstrated that the parameter
estimation error and the redefined state variable, ,
converge exponentially fast to an adjustable neighborhood of the origin. The
size of the neighborhood can be changed by increasing the size of the
controller gain and by reducing the size of the approximation error.

Inequality (23) can be deduced from inequality (A.30) as
follows. Inequality (A.30) can be written aswhere is the smallest eigenvalue of .
Integration of inequality (A.46) givesThis result implies that the
following inequality holds:where .
Hence, the boundedness of the mean square error stated in inequality (23) is
achieved with as required.

Integration of inequality (A.31) gives

Given (A.7), we havewhere .
With the positive constantinequality (A.49) simplifies to

It follows from inequality (A.52) thatwhere .
As a result, the adaptive learning tracking control guarantees that the
tracking error, ,
fulfills the following inequality:where This completes the proof.

*Proof of Theorem 2. *
We consider the
Lyapunov function .
We first note that ,
where and write the closed-loop system as follows:By Lipschitz continuity of the
projection algorithm [28], it follows that for a sufficiently small the inequalityholds on for some positive nonzero constant and all .

Similarly, it follows from the continuous
differentiability of and that the followingholds on for some positive nonzero constant and all .

Using (A.31) and (A.42), the derivative of the Lyapunov
function along the trajectories of the closed-loop
system is such thatwhere is a positive constant arising from the
constant terms in (A.31) and (A.42). In the light of the Lipschitz inequalities
(A.57) and (A.58), we obtainwhere ,
and is the maximum of on .
Similarly, the derivative of along the trajectories of the closed-loop
system is given bySince , , ,
and ,
we obtainThen, it follows that on as long as .
Similarly, on if .
As a result, we show that the set is positively invariant for all .

Consider initial conditions ,
and where is a compact subset and is a compact subset of .
It follows that ,
where is a positive constant which is related to the
size of the compact sets and .

Note that is bounded on .
That is, .
By construction, the adaptive learning rate is such that .
As a result, the closed-loop trajectories of the system starting in are such thatTherefore, there is a time at which the closed-loop trajectory escapes the set .

From (A.62), it follows thatif .
Therefore, we have thatorfor .
As a result, we can find an and a such that .
Moreover, it is always possible to pick small enough such that .
As a result, picking ensures that the trajectories of the
closed-loop process starting in enter the compact set in finite time .

On the set ,
it is shown thatHence, we havewhen and, therefore, the trajectories of the system
must be such that .
Define ,
if we pick such thatthen there is a finite time such that for all Since is of order in the set ,
it follows that there exist an and a finite time such that .
Therefore, taking ensures that there is a finite time such that and .

Finally, we can establish the exponential converge of
the trajectories of the closed-loop system to a small adjustable neighbourhood
of the origin. To do this, we consider the Lyapunov function