# Adaptive Output Tracking of Driven Oscillators

Authors: Lili Diao; Martin Guay
Journal: Journal of Control Science and Engineering
 (2008)

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## 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.


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## Body

## 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. Section2 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
form(1)ξ˙=ϕ(ξ,z),z˙1=z2,⋮z˙ρ=f(z,ξ,u),y=z1,where ξ=[ξ1,ξ2,…,ξn−ρ]T∈ℝn−ρ are the state variables of the zero dynamics; z=[z1,z2,…,zρ]T∈ℝρ are the state variables of the main dynamics; u∈ℝ and y∈ℝ are the system input and output, respectively.
The mapping f(z,ξ,u) is assumed to be an unknown continuous
function of z and u and is assumed to be globally Lipschitz in ξ.
Given a reference trajectory yr,
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.LetYr=[yr,y˙r,…,yr(ρ−1)]T.
In this work, it is assumed that ∥Yr∥≤c,|yr(ρ)|≤ν¯1,
with known constant c>0,ν¯1>0.
Denote ξr the “steady-state” response of the
tracking zero dynamics, governed by the differential equation ξ˙r=ϕ(ξr,Yr),
and ξ¯=ξ−ξr.
It is also assumed that ∥ξr∥≤cξ,
where cξ>0 is a known constant. Denote e=z−Yr=[e1,e2,…,eρ]T,ec=[e1,e2,…,eρ−1]T,
and es=ΛTec+eρ,
where Λ=[λ1,λ2,…,λρ−1] is to be chosen.In this paper, radial basis function (RBF) presented
in [16] was used to
approximate a continuous function ψ(x):ℝp→ℝ(2)ψ(x)=W*S(x)+μl(x),with approximation error μl(x),
and basis function vector(3)S(x)=[s1(x),s2(x),…,sl(x)]T,si(x)=exp[−(x−φi)T(x−φi)σi2],i=1,2,…,l,where φi is the center of the receptive field, and σi is the width of the Gaussian function. The
ideal weight W* in (2)
is defined as(4)W*:=argminW∈Ωw{supx∈Ω|WTS(x)−ψ(x)|},where Ωw={W|∥W∥≤wm} with positive constant wm to be chosen at the design stage, and Ω is a compact set. Universal approximation
results stated in [16, 17] (and the
references therein) indicate that if l is chosen sufficiently large, then W*TS(x) 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∂f(z,ξr,u)/∂u is known, and there exist a positive constant b¯0 and a nonzero continuous function b¯1(z) such that (5)0<b¯0≤|∂f(z,ξr,u)∂u|≤b¯1(z).Assumption.
The
approximation error satisfies|μl(x(t))|≤μ¯l with unknown constant μ¯l>0 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 trajectoryYr and ξr,
system (1) can be rewritten in terms of the tracking error e and ξ¯=ξ−ξr as follows:(6)ξ¯˙=ϕ(ξ¯+ξr,e+Yr)−ϕ(ξr,Yr),(7)e˙c=Acec+Bceρ,e˙ρ=f(e+Yr,ξ¯+ξr,u)−yr(ρ), where(8)Ac=[01……0001…0⋮⋮⋮⋱⋮00……100……0],Bc=[00⋮01].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 U(ξ¯),
such that the following is satisfied: (9)c1∥ξ¯∥2≤U(ξ¯)≤c2∥ξ¯∥2,U˙(ξ¯)≤−c3∥ξ¯∥2+c4∥ξ¯∥∥e∥,where c1,c2,c3,
and c4 are positive constants, and U˙(ξ¯) is the time derivative of U(ξ¯) along the solution of (1).Consideringe as a “disturbance” to the tracking error
zero dynamics (6), Assumption 3 ensures that ξ¯ dynamics in (6) are input-to-state stable (ISS) with respect
to e.By∂f(z,ξr,u)/∂u≠0,∀z∈ℝρ,∀u∈ℝ (Assumption 2), it follows from the implicit
function theorem [19]
that there exists a continuous function α(z,ξr) such that f(z,ξr,α(z,ξr))=0.
The function f(z,ξ,u) may be re-expressed as(10)f(z,ξ,u)=f(z,ξr,u)+f(z,ξ,u)−f(z,ξr,u)=f(z,ξr,α(z,ξr))+∫01∂f(z,ξr,uλ)∂uλdλ(u−α(z,ξr))+[f(z,ξ,u)−f(z,ξr,u)]=b(z,ξr,u)(u−α(z,ξr))+[f(z,ξ,u)−f(z,ξr,u)],where uλ=λu+(1−λ)α(z,ξr),b(z,ξr,u)=∫01(∂f(z,ξr,uλ)/∂uλ)dλ,
and the following is assumed:(11)|f(z,ξ,u)−f(z,ξr,u)|≤L1∥ξ¯∥,where L1 is a Lipschitz constant.Approximate the unknown functionα(z) as(12)α(z,ξr)=W∗TS(z,ξr)+μ1(x(t)).Let W^ denote the estimate of W∗.
The parameter estimation error is given by W˜=W^−W∗.Note that the boundedness of|∂f(z,ξr,u)/∂u| (Assumption 2) implies that b(z,ξr,u) is bounded as follows:(13)b0≤b(z,ξr,u)≤b1(z,ξr),where b0 is a positive constant, and b1(z,ξr) is a nonzero continuous function.Using (12), the error dynamics (7) can be written as(14)e˙c=Acec+Bceρ,e˙ρ=b(z,ξr,u)(u−α(z))−yr(ρ)+[f(z,ξ,u)−f(z,ξr,u)].The state feedback design and parameter update law are
given by(15)u=k(z,ξr)es+W^TS(z,ξr),(16)W^˙=γwProj(W^,c(t)es),where k(z,ξr) is the controller gain function, γw is a positive constant, and Proj(⋅) is a projection algorithm.The dynamics ofc(t) are chosen as follows:(17)c˙T(t)=−(b1kt−12BcTPBc)cT(t)−b1ST(z,ξr)=−K(t)cT(t)+B(t),where K(t)=b1kt−(1/2)BcTPBc>0,
and B(t)=−b1ST(z,ξr).
Note that K(t) can always be made negative by a suitable
choice of the gain constant kt.
The matrix P is a positive definite solution of the
Riccati-like equation:(18)PAc+AcTP+2γ1PBcBcTP+12k2AcTAc+Q=0,for some positive definite
symmetric matrix Q and positive γ1>0 and k2>0 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 constantT>0 and kN>0 such that
(19)∫tt+Tc(τ)cT(τ)dτ≥kNIN,where cT(t) is the solution of (17), IN is a N-dimensional identity matrix.The following lemma will be used in the
sequel.Lemma 1.
Consider the differential equation(20)z˙(t)=−ϕ(t)ϕT(t)z(t),where z(t)∈ℝ+→ℝn and ϕ(t)∈ℝ+→ℝn are both column vectors. Assume that there
exist a T>0 and a k>0 such that(21)∫tt+Tϕ(τ)ϕT(τ)dτ≥kI,then the origin of (20) is a
globally exponentially stable equilibrium of the system.The proof of this lemma can be found in [20].Theorem1
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 c(t) is such that
(22)∫tt+Tc(τ)cT(τ)dτ≥kNIN,for positive constants T>0 and kN>0,
where cT(t) is a solution of (17).Given Assumptions1 to 3, all the signals of the
closed-loop system are bounded. The parameter estimation errors W˜ converge exponentially to a small neighborhood
of the origin.The mean square tracking error satisfies(23)1t∫0te12dt≤α0tVs(0)+1k(μ¯l2+ν¯12+wm2),where α0 is a positive constant, and Vs(0) is a positive constant depending on system
initial conditions.Furthermore, the tracking error is such
that(24)∥e(t)∥≤α1e−β1(t−t0)+α2k(μ¯l2+ν¯12+wm2).where α1, α2, β1,
and k 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:(25)e^˙1=e^2+l1ϵ(e1−e^1),e^˙2=e^3+l2ϵ2(e1−e^1),⋮e^˙ρ=lρϵρ(e1−e^1),where [l1,l2,…,lρ]T are the coefficients of a Hurwitz polynomial, ϵ is some small positive constant, and e^i,i=1,…,ρ  are the estimated tracking errors.Following [21], we define the scaled estimation errors:(26)ηi=ei−e^iϵn−i=e˜iϵn−i,1≤i≤n.Using (6) and (25), the dynamics
of the scaled estimation errors are given by(27)ϵη˙=A0η+ϵB0(ydn−f(x,u)),where the matrix A0∈ℝn×n and the vector B0∈ℝn assume the following form:(28)A0=[−α11……0−α201…0⋮⋮⋮⋱⋮−αn−10……1−αn0……0],B0=[00⋮01].We define the Lyapunov function W(η)=(1/2)ηTP0η,
where P0 is the symmetric positive definite matrix
solution of(29)P0A0+A0TP0=−I.Let V2=(1/2)ecTPec+(1/2)es2+(1/2γw)W˜TW˜ and consider the compact sets Ω={e∈ℝn,W˜∈ℝl∣V2≤c1},
where c1>((1/2kd)ν¯12+(1/2kμ)μ¯l2+(2/kw+2/γ)wm2)=c2.
Define the positive constant b such that 0<c2≤b<c1 and the set Ωb={V2(e,W˜)≤b}.
For the estimation errors, we define the set Σ={W(η)≤ρϵ2}.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 Ω.
We first compute the constants(30)Su>maxe,W˜∈Ω(−k(x)es+W^TS(x)),Sδ>maxe,W˜∈Ω(b1(x)S(x)es).The maximization is performed
over all (e,W˜)∈Ω,Yd∈Y.
It is assumed that the reference trajectories Yd=[yd,y˙d,…,yd(n−1)] are bounded and evolve in a compact subset Y of ℝn.
This ensures that the state x of the system are also bounded on Ω×Y.
The state-feedback and the adaptive learning rate can then bounded on Ω by implementing the functions(31)Ψs(e,Yd,W˜)=Susat(−k(x)es+W^TS(x)Su),ϒs(e,Yd,W˜)=Sδsat(b1(x)S(x)esSδ),  where(32)sat(w)={−1,ifw≤−1,w,if−1<w<1,1,ifw≥1.The adaptive learning state
feedback is rewritten as(33)W^˙={−γϒs,if∥W^∥<wm,or∥W^∥=wm,W^Tϒs≥0,−γϒs+γW^W^Tϒs∥W^∥2,if∥W^∥=wm,W^Tϒs<0,u=Ψs(e,Yd,W˜) for e,W˜∈Ω.
Having bounded the control and the adaptive learning rate, we pose the
output-feedback controller (34)e^˙i=e^i+1+αiϵie˜1,1≤i≤n−1,(35)e^˙n=αnϵne˜1,(36)W^˙={−γϒ^s,if∥W^∥<wm,or∥W^∥=wm,W^Tϒ^s≥0,−γϒ^s+γW^W^Tϒ^s∥W^∥2,if∥W^∥=wm,W^Tϒ^s<0,(37)u=Ψs(e^,Yd,W^),  where ϒ^s=ϒs(e^,Yd).Theorem2 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 S×O there exists 0<ϵ<ϵ3* 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 Theorem2 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,e˙0=e1,
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:
(38a)x˙1=1C1x2,(38b)x˙2=−1L1[x1+g(x2)+R(x2+x4)]+Acos(2πωt)(38c)x˙3=1C2x4,(38d)x˙4=−1L2[x3+f(x4)+R(x2+x4)],
where x2 and x4 describe the action potential of the SA and AV
nodes, and x1 and x3 are the voltage corresponding to x2 and x4,C1,C2,L1,
and L2 are some constant parameters in the model, R is the constant coupling parameter, A and ω 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 g and f are some nonlinear functions of the following
form:(39)f(x4)=−x4+13x43,h(x2)={−x22−14,|x2|<12,−x2,x2>12,x2,x2<−12,g(x2)=−x2+13h(x2).Systems (38a) to (38d) are in the form of system (1),
with z1=x2,ξ=[x1,x3,x4]T,ξr=[r1,r3,r4]T,
where r1,r3,
and r4 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 x2,
and that the resulting system is stable for large values of x4 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]: C1=0.25F,L1=0.05H,C2=0.675F,L2=0.027H,
and R=0.11Ω,
the system exhibits a normal 1:1 rhythm. By setting C1=0.15F,
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 control(40)x˙2=−1L1[x1+g(x2)+R(x2+x4)]+Acos(2πωt)+u.Physically, 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 (parameterC1) and the coupling strength between the two
nodes (parameter R).In this example, the unknown nonlinearityα(z,ξr) in (12) is as follows:(41)α(x2,r1,r4)=1L1(r1+g+R(x2+r4))+Acos(2πωt).For the simulation, the number of basis functions isl=11,
with σ2=5,ϕi=i−6,i=1,…,11,
and wm=4.
The following tuning parameters are used, k4=kμ=kd=kw=γ2=25, γw=20.In the simulation, the controller is not turned on
untilt=5.
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 t=5,
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 α(x2,r1,r4) is the solid line, and the approximation W^TS(x2,r1,r4) is the dotted line. After running the
simulation for 300 seconds, good approximation is achieved along the r1 and r4
direction, and
parameter estimations also converge.AV block suppression, tracking  performances, and
control action.
(a)(b)(c)(d)Unknown nonlinearities approximation and parameter
convergence.
(a)(b)(c)(d)


## 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.



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