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Journal of Control Science and Engineering
Volume 2008, Article ID 641847, 14 pages
http://dx.doi.org/10.1155/2008/641847
Research Article

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 [14], 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 [1012]. 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 forṁ𝜉=𝜙(𝜉,𝑧),̇𝑧1=𝑧2,̇𝑧𝜌=𝑓(𝑧,𝜉,𝑢),𝑦=𝑧1,(1)where 𝜉=[𝜉1,𝜉2,,𝜉𝑛𝜌]𝑇𝑛𝜌 are the state variables of the zero dynamics; 𝑧=[𝑧1,𝑧2,,𝑧𝜌]𝑇𝜌 are the state variables of the main dynamics; 𝑢 and 𝑦 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.

Let 𝑌𝑟=[𝑦𝑟,̇𝑦𝑟,,𝑦𝑟(𝜌1)]𝑇. In this work, it is assumed that 𝑌𝑟𝑐,|𝑦𝑟(𝜌)|𝜈1, with known constant 𝑐>0,𝜈1>0. Denote 𝜉𝑟 the “steady-state” response of the tracking zero dynamics, governed by the differential equation ̇𝜉𝑟=𝜙(𝜉𝑟,𝑌𝑟), and 𝜉=𝜉𝜉𝑟. It is also assumed that 𝜉𝑟𝑐𝜉, where 𝑐𝜉>0 is a known constant. Denote 𝑒=𝑧𝑌𝑟=[𝑒1,𝑒2,,𝑒𝜌]𝑇,𝑒𝑐=[𝑒1,𝑒2,,𝑒𝜌1]𝑇, and 𝑒𝑠=Λ𝑇𝑒𝑐+𝑒𝜌, where Λ=[𝜆1,𝜆2,,𝜆𝜌1] is to be chosen.

In this paper, radial basis function (RBF) presented in [16] was used to approximate a continuous function 𝜓(𝑥)𝑝𝜓(𝑥)=𝑊𝑆(𝑥)+𝜇𝑙(𝑥),(2)with approximation error 𝜇𝑙(𝑥), and basis function vector𝑆(𝑥)=[𝑠1(𝑥),𝑠2(𝑥),,𝑠𝑙(𝑥)]𝑇,𝑠𝑖(𝑥)=exp(𝑥𝜑𝑖)𝑇(𝑥𝜑𝑖)𝜎2𝑖,𝑖=1,2,,𝑙,(3)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 as𝑊=argmin𝑊Ω𝑤sup𝑥Ω|||𝑊𝑇|||𝑆(𝑥)𝜓(𝑥),(4)where Ω𝑤={𝑊|𝑊𝑤𝑚} 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 𝑊𝑇𝑆(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 𝑏0 and a nonzero continuous function 𝑏1(𝑧) such that 0<𝑏0|||𝜕𝑓(𝑧,𝜉𝑟,𝑢)|||𝜕𝑢𝑏1(𝑧).(5)

Assumption. The approximation error satisfies |𝜇𝑙(𝑥(𝑡))|𝜇𝑙 with unknown constant 𝜇𝑙>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 trajectory 𝑌𝑟 and 𝜉𝑟, system (1) can be rewritten in terms of the tracking error 𝑒 and 𝜉=𝜉𝜉𝑟 as follows:̇𝜉=𝜙(𝜉+𝜉𝑟,𝑒+𝑌𝑟)𝜙(𝜉𝑟,𝑌𝑟),(6)̇𝑒𝑐=𝐴𝑐𝑒𝑐+𝐵𝑐𝑒𝜌,̇𝑒𝜌=𝑓(𝑒+𝑌𝑟,𝜉+𝜉𝑟,𝑢)𝑦𝑟(𝜌),(7) where𝐴𝑐=0100010001000,𝐵𝑐=0001.(8)

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: 𝑐1𝜉2𝑈(𝜉)𝑐2𝜉2,̇𝑈(𝜉)𝑐3𝜉2+𝑐4𝜉𝑒,(9)where 𝑐1,𝑐2,𝑐3, and 𝑐4 are positive constants, and ̇𝑈(𝜉) is the time derivative of 𝑈(𝜉) along the solution of (1).

Considering 𝑒 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)/𝜕𝑢0,𝑧𝜌,𝑢 (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 as𝑓(𝑧,𝜉,𝑢)=𝑓(𝑧,𝜉𝑟,𝑢)+𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟,𝑢)=𝑓(𝑧,𝜉𝑟,𝛼(𝑧,𝜉𝑟))+10𝜕𝑓(𝑧,𝜉𝑟,𝑢𝜆)𝜕𝑢𝜆𝑑𝜆(𝑢𝛼(𝑧,𝜉𝑟+))𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟,𝑢)=𝑏(𝑧,𝜉𝑟,𝑢)(𝑢𝛼(𝑧,𝜉𝑟))+𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟,,𝑢)(10)where 𝑢𝜆=𝜆𝑢+(1𝜆)𝛼(𝑧,𝜉𝑟),𝑏(𝑧,𝜉𝑟,𝑢)=10(𝜕𝑓(𝑧,𝜉𝑟,𝑢𝜆)/𝜕𝑢𝜆)𝑑𝜆, and the following is assumed:||𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟||,𝑢)𝐿1𝜉,(11)where 𝐿1 is a Lipschitz constant.

Approximate the unknown function 𝛼(𝑧) as𝛼(𝑧,𝜉𝑟)=𝑊𝑇𝑆(𝑧,𝜉𝑟)+𝜇1(𝑥(𝑡)).(12)Let 𝑊 denote the estimate of 𝑊. The parameter estimation error is given by 𝑊=𝑊𝑊.

Note that the boundedness of |𝜕𝑓(𝑧,𝜉𝑟,𝑢)/𝜕𝑢| (Assumption 2) implies that 𝑏(𝑧,𝜉𝑟,𝑢) is bounded as follows:𝑏0𝑏(𝑧,𝜉𝑟,𝑢)𝑏1(𝑧,𝜉𝑟),(13)where 𝑏0 is a positive constant, and 𝑏1(𝑧,𝜉𝑟) is a nonzero continuous function.

Using (12), the error dynamics (7) can be written aṡ𝑒𝑐=𝐴𝑐𝑒𝑐+𝐵𝑐𝑒𝜌,̇𝑒𝜌=𝑏(𝑧,𝜉𝑟,𝑢)(𝑢𝛼(𝑧))𝑦𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟.,𝑢)(14)

The state feedback design and parameter update law are given by 𝑢=𝑘(𝑧,𝜉𝑟)𝑒𝑠+𝑊𝑇𝑆(𝑧,𝜉𝑟̇),(15)𝑊=𝛾𝑤Proj𝑊,𝑐(𝑡)𝑒𝑠,(16)where 𝑘(𝑧,𝜉𝑟) is the controller gain function, 𝛾𝑤 is a positive constant, and Proj() is a projection algorithm.

The dynamics of 𝑐(𝑡) are chosen as follows:̇𝑐𝑇𝑏(𝑡)=1𝑘𝑡12𝐵𝑇𝑐𝑃𝐵𝑐𝑐𝑇(𝑡)𝑏1𝑆𝑇(𝑧,𝜉𝑟)=𝐾(𝑡)𝑐𝑇(𝑡)+𝐵(𝑡),(17)where 𝐾(𝑡)=𝑏1𝑘𝑡(1/2)𝐵𝑇𝑐𝑃𝐵𝑐>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:𝑃𝐴𝑐+𝐴𝑇𝑐𝑃+2𝛾1𝑃𝐵𝑐𝐵𝑇𝑐1𝑃+2𝑘2𝐴𝑇𝑐𝐴𝑐+𝑄=0,(18)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 constant 𝑇>0 and 𝑘𝑁>0 such that
𝑡𝑡+𝑇𝑐(𝜏)𝑐𝑇(𝜏)𝑑𝜏𝑘𝑁𝐼𝑁,(19)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 ̇𝑧(𝑡)=𝜙(𝑡)𝜙𝑇(𝑡)𝑧(𝑡),(20)where 𝑧(𝑡)+𝑛 and 𝜙(𝑡)+𝑛 are both column vectors. Assume that there exist a 𝑇>0 and a 𝑘>0 such that 𝑡𝑡+𝑇𝜙(𝜏)𝜙𝑇(𝜏)𝑑𝜏𝑘𝐼,(21)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 𝑐(𝑡) is such that
𝑡𝑡+𝑇𝑐(𝜏)𝑐𝑇(𝜏)𝑑𝜏𝑘𝑁𝐼𝑁,(22)for positive constants 𝑇>0 and 𝑘𝑁>0, where 𝑐𝑇(𝑡) is a solution of (17).

Given Assumptions 1 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 satisfies1𝑡𝑡0𝑒21𝛼𝑑𝑡0𝑡𝑉𝑠1(0)+𝑘𝜇2𝑙+𝜈21+𝑤2𝑚,(23)where 𝛼0 is a positive constant, and 𝑉𝑠(0) is a positive constant depending on system initial conditions.

Furthermore, the tracking error is such that𝑒(𝑡)𝛼1𝑒𝛽1(𝑡𝑡0)+𝛼2𝑘𝜇2𝑙+𝜈21+𝑤2𝑚.(24)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:̇̂𝑒1=̂𝑒2+𝑙1𝜖(𝑒1̂𝑒1̇),̂𝑒2=̂𝑒3+𝑙2𝜖2(𝑒1̂𝑒1̇),̂𝑒𝜌=𝑙𝜌𝜖𝜌(𝑒1̂𝑒1),(25)where [𝑙1,𝑙2,,𝑙𝜌]𝑇 are the coefficients of a Hurwitz polynomial, 𝜖 is some small positive constant, and ̂𝑒𝑖,𝑖=1,,𝜌 are the estimated tracking errors.

Following [21], we define the scaled estimation errors:𝜂𝑖=𝑒𝑖̂𝑒𝑖𝜖𝑛𝑖=𝑒𝑖𝜖𝑛𝑖,1𝑖𝑛.(26)Using (6) and (25), the dynamics of the scaled estimation errors are given by𝜖̇𝜂=𝐴0𝜂+𝜖𝐵0𝑦𝑛𝑑𝑓(𝑥,𝑢),(27)where the matrix 𝐴0𝑛×𝑛 and the vector 𝐵0𝑛 assume the following form:𝐴0=𝛼110𝛼2010𝛼𝑛101𝛼𝑛00,𝐵0=0001.(28)We define the Lyapunov function 𝑊(𝜂)=(1/2)𝜂𝑇𝑃0𝜂, where 𝑃0 is the symmetric positive definite matrix solution of𝑃0𝐴0+𝐴𝑇0𝑃0=𝐼.(29)Let 𝑉2=(1/2)𝑒𝑇𝑐𝑃𝑒𝑐+(1/2)𝑒2𝑠+(1/2𝛾𝑤)𝑊𝑇𝑊 and consider the compact sets Ω={𝑒𝑛,𝑊𝑙𝑉2𝑐1}, where 𝑐1>((1/2𝑘𝑑)𝜈21+(1/2𝑘𝜇)𝜇2𝑙+(2/𝑘𝑤+2/𝛾)𝑤2𝑚)=𝑐2. Define the positive constant 𝑏 such that 0<𝑐2𝑏<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 constants𝑆𝑢>max𝑒,𝑊Ω𝑘(𝑥)𝑒𝑠+𝑊𝑇,𝑆𝑆(𝑥)𝛿>max𝑒,𝑊Ω𝑏1(𝑥)𝑆(𝑥)𝑒𝑠.(30)The maximization is performed over all (𝑒,𝑊)Ω,𝑌𝑑𝑌. It is assumed that the reference trajectories 𝑌𝑑=[𝑦𝑑,̇𝑦𝑑,,𝑦𝑑(𝑛1)] are bounded and evolve in a compact subset 𝑌 of 𝑛. 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 functionsΨ𝑠𝑒,𝑌𝑑,𝑊=𝑆𝑢sat𝑘(𝑥)𝑒𝑠+𝑊𝑇𝑆(𝑥)𝑆𝑢,Υ𝑠𝑒,𝑌𝑑,𝑊=𝑆𝛿𝑏sat1(𝑥)𝑆(𝑥)𝑒𝑠𝑆𝛿,(31) wheresat(𝑤)=1,if𝑤1,𝑤,if1<𝑤<1,1,if𝑤1.(32)The adaptive learning state feedback is rewritten aṡ𝑊=𝛾Υ𝑠𝑊,if<𝑤𝑚𝑊,or=𝑤𝑚,𝑊𝑇Υ𝑠0,𝛾Υ𝑠𝑊𝑊+𝛾𝑇Υ𝑠𝑊2𝑊,if=𝑤𝑚,𝑊𝑇Υ𝑠<0,𝑢=Ψ𝑠𝑒,𝑌𝑑,𝑊(33) for 𝑒,𝑊Ω. Having bounded the control and the adaptive learning rate, we pose the output-feedback controller ̇̂𝑒𝑖=̂𝑒𝑖+1+𝛼𝑖𝜖𝑖𝑒1̇,1𝑖𝑛1,(34)̂𝑒𝑛=𝛼𝑛𝜖𝑛𝑒1,̇Υ(35)𝑊=𝛾𝑠𝑊,if<𝑤𝑚𝑊,or=𝑤𝑚,𝑊𝑇Υ𝑠Υ0,𝛾𝑠𝑊𝑊+𝛾𝑇Υ𝑠𝑊2𝑊,if=𝑤𝑚,𝑊𝑇Υ𝑠<0,(36)𝑢=Ψ𝑠̂𝑒,𝑌𝑑,𝑊,(37) where Υ𝑠=Υ𝑠(̂𝑒,𝑌𝑑).

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 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 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, ̇𝑒0=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: ̇𝑥1=1𝐶1𝑥2,(38a)̇𝑥21=𝐿1[𝑥1+𝑔(𝑥2)+𝑅(𝑥2+𝑥4)]+𝐴cos(2𝜋𝜔𝑡)(38b)̇𝑥3=1𝐶2𝑥4,(38c)̇𝑥41=𝐿2[𝑥3+𝑓(𝑥4)+𝑅(𝑥2+𝑥4)],(38d) 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:𝑓(𝑥4)=𝑥4+13𝑥34,(𝑥2)=𝑥2214,||𝑥2||<12,𝑥2,𝑥2>12,𝑥2,𝑥21<2,𝑔(𝑥2)=𝑥2+13(𝑥2).(39)

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 13. (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 control̇𝑥21=𝐿1[𝑥1+𝑔(𝑥2)+𝑅(𝑥2+𝑥4)]+𝐴cos(2𝜋𝜔𝑡)+𝑢.(40)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 (parameter 𝐶1) and the coupling strength between the two nodes (parameter 𝑅).

In this example, the unknown nonlinearity 𝛼(𝑧,𝜉𝑟) in (12) is as follows:𝛼(𝑥2,𝑟1,𝑟41)=𝐿1(𝑟1+𝑔+𝑅(𝑥2+𝑟4))+𝐴cos(2𝜋𝜔𝑡).(41)

For the simulation, the number of basis functions is 𝑙=11, with 𝜎2=5,𝜙𝑖=𝑖6,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 until 𝑡=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 𝑊𝑇𝑆(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.

fig1
Figure 1: AV block suppression, tracking performances, and control action.
fig2
Figure 2: Unknown nonlinearities approximation and parameter convergence.

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 𝑒𝑐 subsystem (14):𝑉1=12𝑒𝑇𝑐𝑃𝑒𝑐,(A.1)where 𝑃 is a symmetric positive definite function.
The derivative of the Lyapunov function 𝑉1 is given bẏ𝑉1=12𝑒𝑇𝑐𝑃𝐴𝑐+𝐴𝑇𝑐𝑃𝑒𝑐+𝑒𝑇𝑐𝑃𝐵𝑐𝑒𝜌.(A.2)Choosing Λ=(1/2)𝐵𝑇𝑐𝑃, we have𝑒𝑠=𝑒𝜌+12𝐵𝑇𝑐𝑃𝑒𝑐.(A.3)A candidate Lyapunov function for the error dynamics is𝑉2=𝑉1+12𝜂2𝑠,(A.4)where 𝜂𝑠=𝑒𝑠+𝑐𝑇𝑊(𝑡), and 𝑐𝑇(𝑡) 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 𝐵(𝑡)=𝑏1𝑆(𝑧,𝜉𝑟) for any choice of gain 𝑘𝑡 large enough.
The time derivative of 𝑉2 is given bẏ𝑉2=̇𝑉1+𝜂𝑠̇𝑒𝑠+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊=1(𝑡)2𝑒𝑇𝑐𝑃𝐴𝑐+𝐴𝑇𝑐𝑃𝑒𝑐+𝑒𝑇𝑐𝑃𝐵𝑐𝑒𝑠12𝐵𝑇𝑐𝑃𝑒𝑐+𝜂𝑠̇𝑒𝜌+12𝐵𝑇𝑐𝑃̇𝑒𝑐+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊.(𝑡)(A.5)Substitution of 𝑒𝜌=𝑒𝑠(1/2)𝐵𝑇𝑐𝑃𝑒𝑐 yieldṡ𝑉2=12𝑒𝑇𝑐𝑃𝐴𝑐+𝐴𝑇𝑐𝑃𝑒𝑐12𝑒𝑇𝑐𝑃𝐵𝑐𝐵𝑇𝑐𝑃𝑒𝑐+𝑒𝑇𝑐𝑃𝐵𝑐𝑒𝑠+𝜂𝑠𝑦𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟,𝑢)+𝜂𝑠𝑏(𝑧,𝑢)(𝑢𝛼(𝑧,𝜉𝑟))+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊(𝑡)+𝜂𝑠12𝐵𝑇𝑐𝑃𝐴𝑐𝑒𝑐+12𝐵𝑇𝑐𝑃𝐵𝑐𝑒𝑠14𝐵𝑇𝑐𝑃𝐵𝑐𝐵𝑇𝑐𝑃𝑒𝑐.(A.6)Given that 𝑒𝑠=𝜂𝑠𝑐(𝑡)𝑇𝑊, and12𝑒2𝑠𝜂2𝑠+𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊,(A.7)we havė𝑉212𝑒𝑇𝑐𝑃𝐴𝑐+𝐴𝑇𝑐𝑃𝑒𝑐12𝑒𝑇𝑐𝑃𝐵𝑐𝐵𝑇𝑐𝑃𝑒𝑐+𝑘12𝑒𝑇𝑐𝑃𝐵𝑐𝐵𝑇𝑐𝑃𝑒𝑐+1𝑘1𝜂2𝑠+1𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇𝑊(𝑡)+𝜂𝑠𝑦𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟,𝑢)+𝜂𝑠𝑏(𝑧,𝑢)(𝑢𝛼(𝑧,𝜉𝑟))+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊+1(𝑡)2𝜂𝑠𝐵𝑇𝑐𝑃𝐴𝑐𝑒𝑐+12𝜂2𝑠𝐵𝑇𝑐𝑃𝐵𝑐14𝜂𝑠𝐵𝑇𝑐𝑃𝐵𝑐𝐵𝑇𝑐𝑃𝑒𝑐12𝜂𝑠𝐵𝑇𝑐𝑃𝐵𝑐𝑐𝑇(𝑡)𝑊,(A.8)oṙ𝑉212𝑒𝑇𝑐𝑃𝐴𝑐+𝐴𝑇𝑐𝑃𝑒𝑐+𝛾1𝑒𝑇𝑐𝑃𝐵𝑐𝐵𝑇𝑐𝑃𝑒𝑐+𝜂𝑠𝑦𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟,𝑢)+𝜂𝑠𝑏(𝑧,𝑢)(𝑢𝛼(𝑧,𝜉𝑟))+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊+1(𝑡)𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊𝜂𝑠𝑐𝑇𝑊1(𝑡)2𝐵𝑇𝑐𝑃𝐵𝑐+𝛾2𝜂2𝑠+𝑘24𝑒𝑇𝑐𝐴𝑇𝑐𝐴𝑐𝑒𝑐,(A.9)where 𝑘1,𝑘2, and 𝑘3 are positive constants and𝛾1=12+𝑘12+𝑘38,𝛾2=1𝑘1+14𝑘2𝐵𝑇𝑐𝑃𝑃𝐵𝑐+12𝐵𝑇𝐶𝑃𝐵𝑐+18𝑘3𝐵𝑇𝑐𝑃𝐵𝑐𝐵𝑇𝑐𝑃𝐵𝑐.(A.10)The constants 𝑘1 and 𝑘3 are chosen such that 𝛾1<0. If the matrix 𝑃 is chosen as a positive definite solution of the Riccati-like equation𝑃𝐴𝑐+𝐴𝑇𝑐𝑃+2𝛾1𝑃𝐵𝑐𝐵𝑇𝑐1𝑃+2𝑘2𝐴𝑇𝑐𝐴𝑐+𝑄=0(A.11)for some positive definite symmetric matrix 𝑄, then inequality (A.9) becomeṡ𝑉212𝑒𝑇𝑐𝑄𝑒𝑐+𝛾2𝜂2𝑠+𝜂𝑠𝑏(𝑧,𝑢)(𝑢𝛼(𝑧,𝜉𝑟))+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊+1(𝑡)𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊𝜂𝑠𝑐𝑇𝑊1(𝑡)2𝐵𝑇𝑐𝑃𝐵𝑐+𝜂𝑠𝑦𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)𝑓(𝑧,𝜉𝑟.,𝑢)(A.12)
Substituting (12), inequality (A.12) becomes ̇𝑉212𝑒𝑇𝑐𝑄𝑒𝑐+𝛾2𝜂2𝑠+𝜂𝑠×𝑊𝑏(𝑧,𝑢)𝑢𝑇𝑆(𝑧,𝜉𝑟)𝜇1(𝑡)+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊+1(𝑡)𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊𝜂𝑠𝑐𝑇𝑊1(𝑡)2𝐵𝑇𝑐𝑃𝐵𝑐𝜂𝑠𝑦𝑟(𝜌)+𝜂𝑠𝐿1𝜉+𝜂𝑠𝑊𝑏(𝑧,𝑢)𝑇𝑆(𝑧,𝜉𝑟)12𝑒𝑇𝑐𝑄𝑒𝑐+𝛾2𝜂2𝑠+𝑘𝜇2𝑏(𝑧,𝑢)2𝜂2𝑠+12𝑘𝜇𝜇1(𝑥(𝑡))2+𝑘𝑑2𝜂2𝑠+12𝑘𝑑𝑦𝑟(𝜌)2+𝜂𝑠𝐿1𝜉+𝜂𝑠𝑊𝑏(𝑧,𝑢)𝑢𝑇𝑆(𝑧,𝜉𝑟)+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊+1(𝑡)𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊𝜂𝑠𝑐𝑇𝑊1(𝑡)2𝐵𝑇𝑐𝑃𝐵𝑐+𝜂𝑠𝑊𝑏(𝑧,𝑢)𝑇𝑆(𝑧,𝜉𝑟),(A.13)where 𝑘𝜇 and 𝑘𝑑 are positive constants.
Consider the control structure shown in (15), we choose the following controller:𝑊𝑢=𝑇𝑆(𝑧,𝜉𝑟)+𝑒𝑠𝑘41𝑏0𝛾2𝑘𝜇2𝑏1𝑘(𝑧)𝑑21𝑏0𝑘𝑤2𝑏1(𝑧)2𝑏0𝑆(𝑧,𝜉𝑟)𝑇𝑆(𝑧,𝜉𝑟)=𝑊𝑇𝑆(𝑧,𝜉𝑟)+𝜂𝑠𝑘41𝑏0𝛾2𝑘𝜇2𝑏1𝑘(𝑧)𝑑21𝑏0𝑘𝑤2𝑏1(𝑧)2𝑏0𝑆(𝑧,𝜉𝑟)𝑇𝑆(𝑧,𝜉𝑟)𝑐𝑇(𝑡)𝑊𝑘𝑡,(A.14)where 𝑘4>0 is a constant, 𝑏0 and 𝑏1(𝑧) are the lower and upper bound in the inequality (13), and 𝑘𝑡 is given by𝑘𝑡=𝑘41𝑏0𝛾2𝑘𝜇2𝑏1𝑘(𝑧)𝑑21𝑏0𝑘𝑤2𝑏1(𝑧)2𝑏0𝑆(𝑧,𝜉𝑟)𝑇𝑆(𝑧,𝜉𝑟).(A.15)The above control action constitutes filtered tracking error and the approximated nonlinearity.
The weight 𝑊 satisfies 𝑊𝑤𝑚, where the upper bound, 𝑤𝑚, is guaranteed in the design of an adaptive law (16).
Substitution of the controller equation (A.14) in the inequality (A.13) giveṡ𝑉212𝑒𝑇𝑐𝑄𝑒𝑐𝑘4𝑏𝜂2𝑠+𝜂𝑠𝑏𝑐𝑇(𝑡)𝑊𝑘𝑡+̇𝑐𝑇(𝑡)𝑊+𝑐𝑇̇𝑊(𝑡)+𝜂𝑠𝑊𝑏(𝑧,𝑢)𝑇𝑆(𝑧,𝜉𝑟)+𝜂𝑠𝐿1𝜉+1𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊𝜂𝑠𝑐𝑇𝑊1(𝑡)2𝐵𝑇𝑐𝑃𝐵𝑐+12𝑘𝜇𝜇1(𝑥(𝑡))2+12𝑘𝑑𝑦𝑟(𝜌)2𝑘𝑤2𝑏1(𝑧)2𝑏0𝑏𝑆(𝑧,𝜉𝑟)𝑇𝑆(𝑧,𝜉𝑟)𝜂2𝑠,(A.16)because of the inequality (13).
The adaptive law (16) is chosen such that 𝑊𝑤𝑚, and𝑊𝑇𝑐(𝑡)𝑒𝑠+1𝛾𝑤𝑊𝑇̇𝑊0.(A.17)It takes the following form:̇𝛾𝑊=𝑤𝑐(𝑡)𝑒𝑠𝑊,if<𝑤𝑚𝑊,or=𝑤𝑚𝑊,and𝑇𝑐(𝑡)𝑒𝑠𝛾0,𝑤𝑐(𝑡)𝑒𝑠𝛾𝑤𝑊𝑊𝑇𝑐(𝑡)𝑒𝑠𝑊2𝑊,if=𝑤𝑚𝑊and𝑇𝑐(𝑡)𝑒𝑠>0.(A.18)
Substitution of the controller equation (A.14), the parameter update law inequality (A.17), and the 𝑐(𝑡) dynamics (17), in the inequality (A.16), giveṡ𝑉212𝑒𝑇𝑐𝑄𝑒𝑐𝑘4𝑏𝜂2𝑠+𝛾𝑤𝑐𝑇(𝑡)𝑐(𝑡)𝜂2𝑠+𝑘𝑤2(𝑏𝑏1)2𝜂2𝑠𝑆𝑇𝑘𝑆𝑤2𝑏21𝑏0𝑏𝑆𝑇𝑆𝜂2𝑠(𝑏1𝑏)𝑘𝑡+𝛾𝑤𝑐𝑇𝑐(𝑡)𝑐(𝑡)𝑇(𝑡)𝑊𝜂𝑠+𝜂𝑠𝐿1𝜉+12𝑘𝜇𝜇1(𝑡)2+12𝑘𝑑𝑦𝑟(𝜌)2+12𝑘𝑤𝑊𝑇1𝑊+𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊,(A.19)where 𝑏1=𝑏1(𝑧)𝑏=𝑏(𝑧,𝑢), and 𝑆=𝑆(𝑧,𝜉𝑟) due to space limits.
Since (𝑏(𝑧,𝑢)/𝑏1(𝑧)1)21, it follows thaṫ𝑉212𝑒𝑇𝑐𝑄𝑒𝑐𝑘4𝑏𝜂2𝑠+𝛾𝑤𝑐𝑇(𝑡)𝑐(𝑡)𝜂2𝑠(𝑏1𝑏)𝑘𝑡+𝛾𝑤𝑐𝑇𝑐(𝑡)𝑐(𝑡)𝑇(𝑡)𝑊𝜂𝑠+𝜂𝑠𝐿1𝜉+12𝑘𝜇𝜇1(𝑡)2+12𝑘𝑑𝑦𝑟(𝜌)2+12𝑘𝑤𝑊𝑇1𝑊+𝑘1𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊.(A.20)
Let 𝑉=𝑉2𝑊+(1/2)𝑇𝑊, and let 𝑘𝑐=(𝑏1𝑏)𝑘𝑡, we havė1𝑉2𝑒𝑇𝑐𝑄𝑒𝑐𝑘4𝑏𝜂2𝑠𝛾𝑤+𝑘𝑤2(𝑘𝑐+𝛾𝑤𝑐𝑇(𝑡)𝑐(𝑡)𝛾𝑤)2𝑐𝑇(𝑡)𝑐(𝑡)𝜂2𝑠+1𝑘1𝛾𝑤𝑊𝑇𝑐(𝑡)𝑐𝑇1(𝑡)𝑊+2𝑘𝑤𝑊𝑇𝑊+𝜂𝑠𝐿1𝜉+12𝑘𝜇𝜇1(𝑥(𝑡))2+12𝑘𝑑𝑦𝑟(𝜌)2.(A.21)
Let 𝑘4=𝑘4𝑏0(𝛾𝑤+(𝑘𝑤/2)(𝑘𝑐+𝛾𝑤𝑐𝑇(𝑡)𝑐(𝑡)𝛾𝑤)2)𝑐𝑇(𝑡)𝑐(𝑡)>0, inequality (A.21) becomeṡ1𝑉2𝑒𝑇𝑐𝑄𝑒𝑐𝑘4𝜂2𝑠+1𝑘1𝛾𝑤𝑊𝑇𝑐(𝑡)𝑐𝑇𝑊+1(𝑡)𝑘𝑤𝑊𝑇𝑊+𝜂𝑠𝐿1𝜉+12𝑘𝜇𝜇1(𝑥(𝑡))2+12𝑘𝑑𝑦𝑟(𝜌)2.(A.22)
Noting that, by assumption, |𝜇𝑙(𝑥(𝑡))|𝜇𝑙,𝑦𝑟(𝜌)𝜈1, and using the fact that𝑊𝑇𝑊=𝑊𝑊𝑇𝑊𝑊𝑊2|||𝑊+2𝑇𝑊|||+𝑊24𝑤2𝑚,(A.23)we obtaiṅ1𝑉2𝑒𝑇𝑐𝑄𝑒𝑐𝑘4𝜂2𝑠+𝜂𝑠𝐿1𝜉+12𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚+1𝑘1𝛾𝑤𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊.(A.24)
Consider the following composite Lyapunov function for the closed-loop system (15), (16), (6), and (7):𝑉𝑐=𝑉+𝛼𝑈,(A.25)where 𝛼 is a positive design parameter. The derivative of the Lyapunov function 𝑉𝑐 is given bẏ𝑉𝑐=̇̇𝑈1𝑉+𝛼2𝑒𝑇𝑐𝑄𝑒𝑐𝑘4𝜂2𝑠+𝑘52𝜂2𝑠+12𝑘5𝐿21𝜉2𝑐3𝛼𝜉2𝑘+Γ62𝑒𝑇𝑐𝑒𝑐𝑘+Γ62𝑒2𝑠+12𝑘6𝑐24𝛼2𝜉2+12𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚+1𝑘1𝛾𝑤𝑊𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊,(A.26)where Γ=(1+𝐵𝑇𝑐𝑃𝑃𝐵𝑐), since, by definition, the trajectory error vector is such that𝑒𝑇𝑒=𝑒𝑇𝑐𝑒𝑐+𝑒2𝑛(1+𝐵𝑇𝑐𝑃𝑃𝐵𝑐)𝑒𝑇𝑐𝑒𝑐+𝑒2𝑠,(A.27)or𝑒𝑇𝑒(1+𝐵𝑇𝑐𝑃𝑃𝐵𝑐)(𝑒𝑇𝑐𝑒𝑐+𝑒2𝑠).(A.28)
Define𝜆𝜆=minmin𝑄2𝜆max𝑃𝑘Γ62,𝑘4𝑘52𝑘Γ62,𝑐3𝐿212𝑘5𝛼𝑐2𝑐24𝛼2𝑘6𝑐2,(A.29)Γ𝑘6+(1/𝑘1)+𝜆<𝛾𝑤, and the gains of the two interconnected system (6) and (7) are such that 𝑐3𝐿21/2𝑘5𝛼𝑐2𝑐24𝛼/2𝑘6𝑐2>0, we havė𝑉𝑐𝜆𝑒𝑐𝑃𝑒𝑐𝜆𝜂2𝑠𝑊𝜆𝛼𝑈𝜆𝑇𝑐(𝑡)𝑐𝑇𝑊+1(𝑡)2𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚𝜆2𝑒𝑐𝑃𝑒𝑐𝜆2𝑒2𝑠𝜆21𝛼𝑈+2𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚,(A.30)oṙ𝑉𝑐𝜆𝑉𝑐+12𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚.(A.31)
It follows that the error vector, 𝑒, variable 𝜂𝑠, and the parameter estimation errors, 𝑊, are bounded. Since the tracking trajectory 𝑦𝑟 and its first 𝜌 derivatives are bounded, it follows that the states 𝑧 and 𝜉 are both bounded.
Integration of inequality (A.31) yields an explicit bound for 𝜂𝑠 given by𝜂𝑠𝛼𝜂𝑒𝜆𝜂(𝑡𝑡0)+1𝑘𝜂𝜇2𝑙+𝜈21+𝑤2𝑚,(A.32)where 𝛼𝜂=2𝑉𝑐(0),𝜆𝜂=𝜆/2, and 𝑘𝜂=min[2𝑘𝑑,2𝑘𝜇,𝑘𝑤/4].
Next, we derive a persistency of excitation condition that guarantees the convergence of the parameter estimates to the ideal weights, 𝑊.
A solution to (17) is given by𝑐𝑇(𝑡)=𝑒𝐾(𝑡)(𝑡𝑡0)𝑐𝑇(𝑡0)+𝑡𝑡0𝑒𝐾(𝑡)(𝑡𝜏)𝐵(𝜏)𝑑𝜏.(A.33)
It is obvious that the differential equatioṅ𝑐𝑇(𝑡)=𝐾(𝑡)𝑐𝑇(𝑡)(A.34)is globally exponentially stable.
The element 𝐵(𝑡) is a bounded function of time. Note that𝐵(𝑡)2=𝑏1(𝑧)2𝑆𝑇𝑆.(A.35)For the particular choice of basis functions proposed in this paper, we have 𝑆𝑁, where 𝑁 is the number of weights used in the approximation. The boundedness of 𝑏1(𝑧) is obvious, since 𝑧 is bounded. Therefore, it follows that the norm of 𝐵(𝑡) is bounded by some positive number 𝐵𝑀, that is,𝐵(𝑡)𝐵𝑀.(A.36)
Using the exponential stability of system (A.34) and the bound on 𝐵(𝑡), an explicit bound for the solution of (17) can be obtained as follows:𝑐𝑇(𝑡)𝐶𝑒𝜆𝑐(𝑡𝑡0)𝐵+𝐶𝑀𝜆𝑐,(A.37)where 𝐶=𝑐𝑇(𝑡0)>0 and 𝜆𝑐>0 is a positive constant.
Next, it is shown that the parameter estimation error 𝑊 converges to a neighborhood of the origin. In the following, it is shown that, under Assumption 4, the unperturbed (i.e., 𝜂𝑠0) dynamics of the parameter estimation errors are exponentially stable.
Lemma 1 establishes that the origin of the differential equatioṅ𝑊=𝛾𝑤𝑐(𝑡)𝑐𝑇(𝑡)𝑊(A.38)is an exponentially stable equilibrium. In fact, it follows from the proof of Lemma 1 that the Lyapunov function 𝑉𝑤=(1/2𝛾𝑤)𝑊𝑇𝑊 is such thaṫ𝑉𝑤𝑊=𝑇𝑐(𝑡)𝑐𝑇(𝑡)𝑊𝑐𝑤𝑊2(A.39)for 𝑐𝑤>0 a positive constant.
It follows from the property of the projection algorithm that the rate of change of 𝑉𝑤 along the trajectories of (A.18) is given bẏ𝑉𝑤𝑊𝑇𝑐(𝑡)𝑐𝑇𝑊(𝑡)𝑊𝑇𝑐(𝑡)𝜂𝑠.(A.40)Completing the squares, we geṫ𝑉𝑤12𝑊𝑇𝑐(𝑡)𝑐𝑇1(𝑡)𝑊+2𝜂2𝑠.(A.41)Substitution of (A.32) yieldṡ𝑉𝑤𝑐𝑤𝛾𝑤𝑉𝑤+𝛼2𝜂𝑒2𝜆𝜂(𝑡𝑡0)+1𝑘𝜂𝜇2𝑙+𝜈21+𝑤2𝑚.(A.42)By integrating, we get𝑉𝑤𝑉max𝑤(𝑡0|||𝛼),2𝜂𝑐𝑤𝛾𝑤2𝜆𝜂|||,1𝑐𝑤𝛾𝑤𝑘𝜂𝜇2𝑙+𝜈21+𝑤2𝑚𝑐×expmin𝑤𝛾𝑤,2𝜆𝜂𝑡𝑡0+1𝑐𝑤𝛾𝑤𝑘𝜂𝜇2𝑙+𝜈21+𝑤2𝑚.(A.43)Consequently, the parameter estimation error is guaranteed to decay exponentially as𝑊𝛼𝑤𝑒𝜆𝑤(𝑡𝑡0)+2𝑘𝜂𝑐𝑤𝜇2𝑙+𝜈21+𝑤2𝑚1/2.(A.44)
Taking the limit as 𝑡 confirms that the estimation error converges to a small adjustable neighborhood of the origin given bylim𝑡𝑊2𝑘𝜂𝑐𝑤𝜇2𝑙+𝜈21+𝑤2𝑚1/2.(A.45)
Under the assumption that the persistency of excitation condition is fulfilled, we have demonstrated that the parameter estimation error and the redefined state variable, 𝜂𝑠, 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 aṡ𝑉𝑐12𝜆min𝑃𝑒𝑇𝑐𝑒𝑐𝜆2𝑒2𝑠𝜆21𝛼𝑈+2𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚,(A.46)where 𝜆min{𝑄} is the smallest eigenvalue of 𝑄. Integration of inequality (A.46) gives𝑉𝑐(𝑡)𝑉𝑐(0)𝑡012𝜆min𝑃𝑒𝑐(𝜎)𝑇𝑒𝑐𝜆(𝜎)+2𝑒𝑠(𝜎)2+𝜆2𝛼𝑈(𝜎)2+𝑡𝑑𝜎212𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚.(A.47)This result implies that the following inequality holds:1𝑡𝑡0𝑒1(𝜎)2𝑑𝜎2𝑉𝑐(0)𝑡𝜆min𝑃+1𝑘𝜇2𝑙+𝜈21+𝑤2𝑚,(A.48)where 𝑘=(1/𝜆min{𝑄})min{𝑘𝜇,𝑘𝑑,𝑘𝑤/8}. Hence, the boundedness of the mean square error stated in inequality (23) is achieved with 𝛼0=2/𝜆min{𝑃} as required.
Integration of inequality (A.31) gives𝑉𝑐(𝑡)𝑉𝑐(0)𝑒𝜆(𝑡𝑡0)+12𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚×𝑡𝑡0𝑒𝜆(𝑡𝜏)𝑑𝜏𝑉𝑐(0)𝑒𝜆(𝑡𝑡0)+12𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚.(A.49)
Given (A.7), we have12𝑝𝑐𝑒2𝑠𝜂2𝑠+𝑊2,(A.50)where 𝑝𝑐=sup𝑡>0𝑐(𝑡)2. With the positive constant𝑝𝑚𝜆=minmin𝑃,12𝑝𝑐,(A.51)inequality (A.49) simplifies to12𝑒𝑇𝑐𝑒𝑐+12𝑒2𝑠𝑉𝑐(0)𝑝𝑚𝑒𝜆(𝑡𝑡0)+1𝑝𝑚12𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚.(A.52)
It follows from inequality (A.52) that𝑒𝑇𝑉𝑒Γ𝑐(0)𝑝𝑚𝑒𝜆(𝑡𝑡0)1+Γ𝑝𝑚12𝑘𝑑𝜈21+12𝑘𝜇𝜇2𝑙+1𝑘𝑤4𝑤2𝑚,(A.53)where Γ=2(1+𝐵𝑇𝑐𝑃𝑃𝐵𝑐). As a result, the adaptive learning tracking control guarantees that the tracking error, 𝑒, fulfills the following inequality:𝑒𝛼1𝑒𝛽1(𝑡𝑡0)+𝛼21𝑘𝜇𝑙+𝜈1+𝑤𝑚,(A.54)where 𝛼1=Γ𝑉(0)𝑝𝑚1/2,𝛽1𝛼=𝜆,2=Γ𝑝𝑚1/2,𝑘=min2𝑘𝑑,2𝑘𝜇,𝑘𝑤4.(A.55)This completes the proof.

Proof of Theorem 2. We consider the Lyapunov function 𝑉2=(1/2)𝑒𝑇𝑐𝑃𝑒𝑐+(1/2)𝑒2𝑠+(1/2𝛾𝑤)𝑊𝑇𝑊. We first note that ̂𝑒=𝑒𝐷(𝜖)𝜂, where 𝐷(𝜖)=diag[𝜖𝑛1,,1] and write the closed-loop system as follows:̇𝑒=𝐴0𝑒+𝐵0𝑓(𝑥,Ψ𝑠(𝑒𝐷(𝜖)𝜂,𝑊,𝑌𝑑)𝑦𝑑(𝑛),̇Υ𝑊=𝛾𝑠𝑊,if<𝑤𝑚𝑊,or=𝑤𝑚,𝑊𝑇Υ𝑠Υ0,𝛾𝑠𝑊𝑊+𝛾𝑇Υ𝑠𝑊2𝑊,if=𝑤𝑚,𝑊𝑇Υ𝑠>0,𝜖̇𝜂=𝐴0𝜂+𝜖𝐵0𝑓𝑥,Ψ𝑠𝑒𝐷(𝜖)𝜂,𝑊,𝑌𝑑𝑦𝑑(𝑛).(A.56)By Lipschitz continuity of the projection algorithm [28], it follows that for a sufficiently small 𝜖 the inequalityΥProj𝑠(𝑒𝐷(𝜖)𝜂,𝑌𝑑𝑊Υ),Proj𝑠(𝑒,𝑌𝑑𝑊),𝛾𝐿1𝜂(A.57)holds on Ω for some positive nonzero constant 𝐿1 and all 0<𝜖<𝜖1.
Similarly, it follows from the continuous differentiability of 𝑏1(𝑥) and 𝑆(𝑥) that the followingΨ𝑠𝑒𝐷(𝜖)𝜂,𝑌𝑑,𝑊Ψ𝑠𝑒,𝑌𝑑,𝑊𝐿2𝜂(A.58)holds on Ω for some positive nonzero constant 𝐿2 and all 0<𝜖𝜖2.
Using (A.31) and (A.42), the derivative of the Lyapunov function 𝑉2 along the trajectories of the closed-loop system is such thaṫ𝑉2𝑘5𝑉2+𝑐2+𝑒𝑠×Ψ𝑏(𝑥,𝑢)𝑠𝑒𝐷(𝜖)𝜂,𝑌𝑑,𝑊Ψ𝑠𝑒,𝑌𝑑,𝑊+1𝛾𝑊𝑇ΥProj𝑠𝑒𝐷(𝜖)𝜂,𝑌𝑑,𝑊ΥProj𝑠(𝑒,𝑌𝑑𝑊,),(A.59)where 𝑘5=𝜆/2,𝑐2 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 obtaiṅ𝑉2𝑘5𝑉2+𝑐2+𝐿2𝐿3𝑏𝜂+2𝑤𝑚𝐿1𝜂=𝑘5𝑉2+𝑐2+𝑘7𝜂,(A.60)where 𝑏=max𝑒Ω,𝑌𝑑𝑌𝑏1(𝑒,𝑌𝑑), and 𝐿3 is the maximum of 𝑒𝑠 on Ω. Similarly, the derivative of 𝑊(𝜂) along the trajectories of the closed-loop system is given by𝜖̇𝑊𝜂𝑇𝜂+2𝜖𝜂𝑇𝑃0𝐵0×Ψ𝑏(𝑥,𝑢)𝑠𝑒𝐷(𝜖)𝜂,𝑌𝑑,𝑊𝑊𝑆𝑒+𝑌𝑑𝜇𝑙𝑥(𝑡)𝑦𝑑(𝑛).(A.61)Since |Ψ𝑠|𝑆𝑢, 𝑊𝑤𝑚, 𝑆(𝑒+𝑌𝑑)𝑆, and 𝜇𝑙(𝑥(𝑡))𝜇𝑙, we obtain𝜖̇𝑊𝜂𝑇𝜂𝜆𝜂+2𝜖max(𝑃0)𝑏𝑆𝑢+2𝑤𝑚𝑆+𝜇𝑙+𝜈1=𝜂𝑇𝜂𝜆𝜂+2𝜖max(𝑃0)𝑘6.(A.62)Then, it follows that ̇𝑉20 on {𝑉2=𝑐1}×{𝑊(𝜂)𝜌𝜖2} as long as 𝜖(𝑘5𝑐1𝑐2)/𝑘7𝜌/𝜆min(𝑃0)=𝜖3<min{𝜖1,𝜖2}. Similarly, ̇𝑊0 on {𝑉2𝑐1}×{𝑊(𝜂)=𝜌𝜖2} if 𝜌4𝜆max(𝑃0)3𝑘26. As a result, we show that the set {𝑉2𝑐1}×{𝑊(𝜂)𝜌𝜖2} is positively invariant for all 0<𝜖𝜖3.
Consider initial conditions {𝑒(0),𝑊(0)}𝑆, and {̂𝑒(0)}𝑂 where 𝑆 is a compact subset Ω𝑏Ω and 𝑂 is a compact subset of 𝑛. It follows that 𝜂(0)𝑘4/𝜖(𝑛1), where 𝑘4 is a positive constant which is related to the size of the compact sets 𝑆 and 𝑂.
Note that 𝑓(𝑒,𝑌𝑑,𝑊,𝜇)=[𝑒2,𝑒3,,𝑏(𝑥,𝑢)(Ψ𝑠()𝑊𝑇𝑆(𝑒+𝑌𝑑)𝜇𝑙(𝑥(𝑡))]𝑇 is bounded on Ω. That is, 𝑓(𝑥,Ψ𝑠())𝑘5. By construction, the adaptive learning rate is such that Proj(Υ𝑠(),𝑊)𝑘6. As a result, the closed-loop trajectories of the system starting in 𝑆 are such that𝑒(𝑡)𝑒(0)𝑘5𝑡,𝑊(𝑡)𝑊(0)𝑘6𝑡.(A.63)Therefore, there is a time 𝑇𝑒 at which the closed-loop trajectory (𝑒(𝑡),𝑊(𝑡)) escapes the set Ω.
From (A.62), it follows thaṫ1𝑊𝜂2𝜖𝑇𝜂,(A.64)if 𝑊𝜌𝜖2. Therefore, we have thaṫ1𝑊2𝜖𝜆max(𝑃0)𝑊,(A.65)or𝑊(𝑡)=𝑒1/2𝜖𝜆max(𝑃0)𝑡𝑊(0),(A.66)for 𝑊𝜌𝜖2. As a result, we can find an 𝜖<𝜖3 and a 𝑇(𝜖)>0 such that 𝑊(𝑡)𝜌𝜖2,𝑡𝑇(𝜖). Moreover, it is always possible to pick 𝜖3 small enough such that 𝑇(𝜖3)𝑇0. As a result, picking 𝜖1=min{𝜖1,𝜖2,𝜖3} ensures that the trajectories of the closed-loop process starting in 𝑆×𝑂 enter the compact set Λ={𝑉2𝑐1}×{𝑊𝜌𝜖2} in finite time 𝑇(𝜖)0<𝜖𝜖1.
On the set Λ, it is shown thaṫ𝑉2𝑘1𝑉2+𝑐2+𝑘3𝜌𝜆min𝜖.(A.67)Hence, we havė𝑉2𝑘12𝑉2,(A.68)when {𝑉22𝑐2+2𝑘3(𝜌/𝜆min)𝜖} and, therefore, the trajectories of the system must be such that lim𝑡𝑉2(𝑡)2𝑐2𝑘+23𝜖. Define 𝜒=[𝑒,𝑊], if we pick 𝜖 such that𝑉22𝑐2+𝑘3𝜖𝜒𝜇2,(A.69)then there is a finite time 𝑇(𝜇) such that for all 0<𝜖𝜖4𝜒𝜇2,𝑡𝑇(𝜇).(A.70)Since 𝜂 is of order 𝜖 in the set Λ, it follows that there exist an 𝜖=𝜖5 and a finite time 𝑇(𝜇) such that 𝜂𝜇/2𝑡𝑇(𝜇). Therefore, taking 𝜖2=min{𝜖4,𝜖5} ensures that there is a finite time 𝑇2=max{𝑇,𝑇} such that 𝜒𝜇/2 and ̂𝑒𝜇/2𝑡𝑇2.
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 function𝑉=𝑉2+𝑊(𝜂).(A.71