Adaptive Output Tracking of Driven Oscillators

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 form𝑧𝜉̇₁̇=𝜙(𝜉,𝑧),𝑧̇_𝜌=⋮𝑧₂,𝑦==𝑧𝑓₁(𝑧,𝜉,𝑢),,(1)where 𝜉=[𝜉₁,𝜉₂,…,𝜉_𝑛−𝜌]^𝑇∈ℝ^𝑛−𝜌 are the state variables of the zero dynamics; 𝑧=[𝑧₁,𝑧₂,…,𝑧_𝜌]^𝑇∈ℝ^𝜌 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 𝑦_𝑟, 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 𝑌_𝑟=[𝑦_𝑟,𝑦̇_𝑟,…,𝑦^{𝑟(𝜌−1)}]^𝑇. In this work, it is assumed that ‖𝑌_𝑟‖≤𝑐,|𝑦^𝑟(𝜌)|≤𝜈₁, with known constant 𝑐>0,𝜈₁>0. Denote 𝜉_𝑟 the “steady-state” response of the tracking zero dynamics, governed by the differential equation 𝜉̇_𝑟=𝜙(𝜉_𝑟,𝑌_𝑟), and 𝜉=𝜉−𝜉_𝑟. It is also assumed that ‖𝜉_𝑟‖≤𝑐_𝜉, where 𝑐_𝜉>0 is a known constant. Denote 𝑒=𝑧−𝑌_𝑟=[𝑒₁,𝑒₂,…,𝑒_𝜌]^𝑇,𝑒_𝑐=[𝑒₁,𝑒₂,…,𝑒_𝜌−1]^𝑇, and 𝑒_𝑠=Λ^𝑇𝑒_𝑐+𝑒_𝜌, where Λ=[𝜆₁,𝜆₂,…,𝜆_𝜌−1] is to be chosen.

In this paper, radial basis function (RBF) presented in [16] was used to approximate a continuous function 𝜓(𝑥)∶ℝ^𝑝→ℝ𝜓(𝑥)=𝑊^∗𝑆(𝑥)+𝜇_𝑙(𝑥),(2)with approximation error 𝜇_𝑙(𝑥), and basis function vector𝑠𝑆_𝑖(𝑥)=[𝑠₁(𝑥),𝑠₂(𝑥),…,𝑠_𝑙(𝑥)]^𝑇,(𝑥)=exp−(𝑥−𝜑^𝑖)^𝑇(𝑥−𝜑_𝑖)𝜎^2𝑖,𝑖=1,2,…,𝑙,(3)where 𝜑_𝑖 is the center of the receptive field, and 𝜎_𝑖 is the width of the Gaussian function. The ideal weight 𝑊^∗ in (2) is defined as𝑊^∗∶=argmin_{𝑊∈Ω𝑤}sup_𝑥∈Ω|||𝑊_𝑇𝑆(𝑥)−𝜓(𝑥)|||,(4)where Ω_𝑤={𝑊|‖𝑊‖≤𝑤_𝑚} with positive constant 𝑤_𝑚 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,𝜉_𝑟,𝑢)/𝜕𝑢 is known, and there exist a positive constant 𝑏₀ and a nonzero continuous function 𝑏₁(𝑧) such that 0<𝑏₀≤|||𝜕𝑓(𝑧,𝜉^𝑟,𝑢)𝜕𝑢|||≤𝑏₁(𝑧).(5)

Assumption. The approximation error satisfies |𝜇_𝑙(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 trajectory 𝑌_𝑟 and 𝜉_𝑟, system (1) can be rewritten in terms of the tracking error 𝑒 and 𝜉=𝜉−𝜉_𝑟 as follows:̇‌𝑒𝜉_𝑐̇=𝜙(𝜉+𝜉_𝑟,𝑒+𝑌_𝑟)−𝜙(𝜉_𝑟,𝑌_𝑟),(6)𝑒_𝜌̇=𝐴_𝑐𝑒_𝑐+𝐵_𝑐𝑒_𝜌,=𝑓(𝑒+𝑌_𝑟,𝜉+𝜉_𝑟,𝑢)−𝑦^𝑟(𝜌),(7) where𝐴_𝑐=⎡⎢⎢⎢⎢⎢⎢⎢⎣⋮0000⋮1000………⋮1⋱…………⋮0010⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐵_𝑐=⎡⎢⎢⎢⎢⎢⎢⎢⎣⋮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: 𝑈̇(𝑐₁‖‖𝜉‖‖²≤𝑈(𝜉)≤𝑐₂‖‖𝜉‖‖²,𝜉)≤−𝑐₃‖‖𝜉‖‖²+𝑐₄‖‖𝜉‖‖‖‖𝑒‖‖,(9)where 𝑐₁,𝑐₂,𝑐₃, and 𝑐₄ are positive constants, and 𝑈̇(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,𝜉_𝑟,𝑢)/𝜕𝑢≠0,∀𝑧∈ℝ^𝜌,∀𝑢∈ℝ (Assumption 2), it follows from the implicit function theorem [19] that there exists a continuous function 𝛼(𝑧,𝜉_𝑟) such that 𝑓(𝑧,𝜉_𝑟,𝛼(𝑧,𝜉_𝑟))=0. The function 𝑓(𝑧,𝜉,𝑢) may be re-expressed as𝑓(𝑧,𝜉,𝑢)==𝑓𝑓((𝑧𝑧,,𝜉𝜉_𝑟𝑟,𝑢)+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢),𝛼(𝑧,𝜉_𝑟))+¹⁰𝜕𝑓(𝑧,𝜉_𝑟,𝑢_𝜆)𝜕𝑢_𝜆+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟𝑑𝜆(𝑢−𝛼(𝑧,𝜉_𝑟))=𝑏(𝑧,𝜉_𝑟,𝑢),𝑢)(𝑢−𝛼(𝑧,𝜉_𝑟))+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢),(10)where 𝑢_𝜆=𝜆𝑢+(1−𝜆)𝛼(𝑧,𝜉_𝑟),𝑏(𝑧,𝜉_𝑟,𝑢)=∫¹⁰(𝜕𝑓(𝑧,𝜉_𝑟,𝑢_𝜆)/𝜕𝑢_𝜆)𝑑𝜆, and the following is assumed:||𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢)||≤𝐿₁‖‖𝜉‖‖,(11)where 𝐿₁ is a Lipschitz constant.

Approximate the unknown function 𝛼(𝑧) as𝛼(𝑧,𝜉_𝑟)=𝑊^∗𝑇𝑆(𝑧,𝜉_𝑟)+𝜇₁(𝑥(𝑡)).(12)Let 𝑊 denote the estimate of 𝑊^∗. The parameter estimation error is given by 𝑊=𝑊−𝑊^∗.

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

Using (12), the error dynamics (7) can be written as𝑒𝑒_𝜌̇_𝑐̇=𝐴_𝑐𝑒_𝑐+𝐵_𝑐𝑒_𝜌,=𝑏(𝑧,𝜉_𝑟,𝑢)(𝑢−𝛼(𝑧))−𝑦^𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢).(14)

The state feedback design and parameter update law are given by 𝑢=𝑊𝑘̇(𝑧=,𝜉𝛾_𝑟𝑤)𝑒_𝑠+𝑊^𝑇𝑆(𝑧,𝜉_𝑟),(15)Proj𝑊,𝑐(𝑡)𝑒_𝑠,(16)where 𝑘(𝑧,𝜉_𝑟) is the controller gain function, 𝛾_𝑤 is a positive constant, and Proj(⋅) is a projection algorithm.

The dynamics of 𝑐(𝑡) are chosen as follows:𝑐̇^𝑇(𝑡)=−𝑏₁𝑘_𝑡−12𝐵^𝑇𝑐=−𝐾(𝑡)𝑐_𝑇𝑃𝐵_𝑐𝑐^𝑇(𝑡)−𝑏₁𝑆^𝑇(𝑧,𝜉_𝑟)(𝑡)+𝐵(𝑡),(17)where 𝐾(𝑡)=𝑏₁𝑘_𝑡−(1/2)𝐵^𝑇𝑐𝑃𝐵_𝑐>0, and 𝐵(𝑡)=−𝑏₁𝑆^𝑇(𝑧,𝜉_𝑟). Note that 𝐾(𝑡) can always be made negative by a suitable choice of the gain constant 𝑘_𝑡. The matrix 𝑃 is a positive definite solution of the Riccati-like equation:𝑃𝐴_𝑐+𝐴^𝑇𝑐𝑃+2𝛾₁𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃+12𝑘²𝐴^𝑇𝑐𝐴_𝑐+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 𝑘_𝑁>0 such that
^{𝑡𝑡+𝑇}𝑐(𝜏)𝑐^𝑇(𝜏)𝑑𝜏≥𝑘_𝑁𝐼_𝑁,(19)where 𝑐^𝑇(𝑡) is the solution of (17), 𝐼_𝑁 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)𝜙^𝑇(𝑡)𝑧(𝑡),(20)where 𝑧(𝑡)∈ℝ₊→ℝ^𝑛 and 𝜙(𝑡)∈ℝ₊→ℝ^𝑛 are both column vectors. Assume that there exist a 𝑇>0 and a 𝑘>0 such that ^{𝑡𝑡+𝑇}𝜙(𝜏)𝜙^𝑇(𝜏)𝑑𝜏≥𝑘𝐼,(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+𝑇}𝑐(𝜏)𝑐^𝑇(𝜏)𝑑𝜏≥𝑘_𝑁𝐼_𝑁,(22)for positive constants 𝑇>0 and 𝑘_𝑁>0, where 𝑐^𝑇(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 satisfies1𝑡^𝑡0𝑒²¹𝑑𝑡≤𝛼⁰𝑡𝑉^𝑠(0)+1𝑘𝜇^2𝑙+𝜈²¹+𝑤^2𝑚,(23)where 𝛼₀ is a positive constant, and 𝑉_𝑠(0) is a positive constant depending on system initial conditions.

Furthermore, the tracking error is such that‖‖𝑒(𝑡)‖‖≤𝛼₁𝑒^{−𝛽1(𝑡−𝑡0)}+𝛼²√𝑘𝜇^2𝑙+𝜈²¹+𝑤^2𝑚.(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:𝑒₁̇̂=𝑒₂̂+𝑙¹𝜖(𝑒¹𝑒₂̇̂−𝑒₁̂),=𝑒₃̂+𝑙²𝜖²𝑒_𝜌̇̂⋮(𝑒₁−𝑒₁̂),=𝑙^𝜌𝜖^𝜌(𝑒₁−𝑒₁̂),(25)where [𝑙₁,𝑙₂,…,𝑙_𝜌]^𝑇 are the coefficients of a Hurwitz polynomial, 𝜖 is some small positive constant, and 𝑒_𝑖̂,𝑖=1,…,𝜌 are the estimated tracking errors.

Following [21], we define the scaled estimation errors:𝜂_𝑖=𝑒^𝑖−𝑒_𝑖̂𝜖^𝑛−𝑖=𝑒^𝑖𝜖^𝑛−𝑖,1≤𝑖≤𝑛.(26)Using (6) and (25), the dynamics of the scaled estimation errors are given by𝜖𝜂̇=𝐴₀𝜂+𝜖𝐵₀𝑦^𝑛𝑑−𝑓(𝑥,𝑢),(27)where the matrix 𝐴₀∈ℝ^𝑛×𝑛 and the vector 𝐵₀∈ℝ^𝑛 assume the following form:𝐴₀=⎡⎢⎢⎢⎢⎢⎢⎢⎣−𝛼¹−𝛼₂1……0−𝛼⋮_𝑛−1⋮0⋮1⋱…⋮0−𝛼_𝑛0……10……0⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐵⁰=⎡⎢⎢⎢⎢⎢⎢⎢⎣⋮0001⎤⎥⎥⎥⎥⎥⎥⎥⎦.(28)We define the Lyapunov function 𝑊(𝜂)=(1/2)𝜂^𝑇𝑃₀𝜂, where 𝑃₀ is the symmetric positive definite matrix solution of𝑃₀𝐴₀+𝐴^𝑇0𝑃₀=−𝐼.(29)Let 𝑉₂=(1/2)𝑒^𝑇𝑐𝑃𝑒_𝑐+(1/2)𝑒^2𝑠+(1/2𝛾_𝑤)𝑊^𝑇𝑊 and consider the compact sets Ω={𝑒∈ℝ^𝑛,𝑊∈ℝ^𝑙∣𝑉₂≤𝑐₁}, where 𝑐₁>((1/2𝑘_𝑑)𝜈²¹+(1/2𝑘_𝜇)𝜇^2𝑙+(2/𝑘_𝑤+2/𝛾)𝑤^2𝑚)=𝑐₂. Define the positive constant 𝑏 such that 0<𝑐₂≤𝑏<𝑐₁ and the set Ω_𝑏={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 constants𝑆_𝑢>𝑆max_{𝑒,𝑊∈Ω}−𝑘(𝑥)𝑒^𝑠+𝑊^𝑇𝑆(𝑥)_𝛿,>max_{𝑒,𝑊∈Ω}𝑏₁(𝑥)𝑆(𝑥)𝑒_𝑠.(30)The maximization is performed over all (𝑒,𝑊)∈Ω,𝑌_𝑑∈𝑌. It is assumed that the reference trajectories 𝑌_𝑑=[𝑦_𝑑,𝑦̇_𝑑,…,𝑦^{𝑑(𝑛−1)}] are bounded and evolve in a compact subset 𝑌 of ℝ^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 functionsΨ_𝑠𝑒,𝑌^𝑑,𝑊=𝑆_𝑢sat−𝑘(𝑥)𝑒^𝑠+𝑊^𝑇𝑆(𝑥)Υ_𝑠𝑆_𝑢,𝑒,𝑌^𝑑,𝑊=𝑆_𝛿sat𝑏¹(𝑥)𝑆(𝑥)𝑒_𝑠𝑆_𝛿,(31) wheresat(𝑤)=⎧⎪⎪⎨⎪⎪⎩−𝑤1,1,,iiifff𝑤𝑤−≤≥1<1−.𝑤1,<1,(32)The adaptive learning state feedback is rewritten as𝑊̇=⎧⎪⎪⎪⎨⎪⎪⎪⎩−𝛾Υ^𝑠,if‖‖‖‖𝑊𝑊‖‖‖‖=<𝑤𝑤_𝑚𝑚,or−𝛾Υ_𝑠,𝑊^𝑇Υ_𝑠≥0,+𝛾𝑊𝑊^𝑇Υ_𝑠‖‖𝑊‖‖²𝑢=Ψ_𝑠,if‖‖𝑊‖‖=𝑤_𝑚,𝑊^𝑇Υ_𝑠<0,𝑒,𝑌^𝑑,𝑊(33) for 𝑒,𝑊∈Ω. Having bounded the control and the adaptive learning rate, we pose the output-feedback controller 𝑒_𝑖̇̂=𝑒̂_𝑖+1+𝛼^𝑖𝜖^𝑖𝑒_𝑛̇̂𝑒₁,1≤𝑖≤𝑛−1,(34)=𝛼^𝑛𝜖^𝑛𝑊̇=⎧⎪⎪⎪⎨⎪⎪⎪⎩−𝑒𝛾₁Υ,_𝑠(35),if‖‖‖‖𝑊𝑊‖‖‖‖=<𝑤𝑤_𝑚𝑚,or−𝛾Υ_𝑠,𝑊^𝑇Υ_𝑠≥0,+𝛾𝑊𝑊^𝑇Υ_𝑠‖‖𝑊‖‖²𝑢=Ψ_𝑠,if‖‖𝑊‖‖=𝑤_𝑚,𝑊^𝑇Υ_𝑠<0,(36)𝑒,̂𝑌^𝑑,𝑊,(37) where Υ_𝑠=Υ_𝑠(𝑒,̂𝑌_𝑑).

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<𝜖^∗3 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, 𝑒₀̇=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: 𝑥̇₁=1𝐶₁𝑥̇₂𝑥₂,(38a)=−1𝐿₁𝑥̇₃[𝑥₁+𝑔(𝑥₂)+𝑅(𝑥₂+𝑥₄)]+𝐴cos(2𝜋𝜔𝑡)(38b)=1𝐶₂𝑥̇₄𝑥₄,(38c)=−1𝐿₂[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+13𝑥³⁴ℎ(𝑥₂,)=⎧⎪⎪⎨⎪⎪⎩−𝑥²²−14,||𝑥²||<1−𝑥₂2,,𝑥₂>1𝑥₂2,,𝑥₂<−1𝑔(𝑥₂2,)=−𝑥₂+13ℎ(𝑥²).(39)

Systems (38a) to (38d) are in the form of system (1), with 𝑧₁=𝑥₂,𝜉=[𝑥₁,𝑥₃,𝑥₄]^𝑇,𝜉_𝑟=[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 control𝑥̇₂=−1𝐿₁[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,𝜉_𝑟) in (12) is as follows:𝛼(𝑥₂,𝑟₁,𝑟₄)=1𝐿₁(𝑟₁+𝑔+𝑅(𝑥₂+𝑟₄))+𝐴cos(2𝜋𝜔𝑡).(41)

For the simulation, the number of basis functions is 𝑙=11, with 𝜎²=5,𝜙_𝑖=𝑖−6,𝑖=1,…,11, and 𝑤_𝑚=4. The following tuning parameters are used, 𝑘₄=𝑘_𝜇=𝑘_𝑑=𝑘_𝑤=𝛾₂=25, 𝛾_𝑤=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 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.

(a)

(b)

(c)

(d)

Figure 1 

AV block suppression, tracking performances, and control action.

(a)

(b)

(c)

(d)

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 𝑒_𝑐 subsystem (14):𝑉₁=12𝑒^𝑇𝑐𝑃𝑒_𝑐,(A.1)where 𝑃 is a symmetric positive definite function.
The derivative of the Lyapunov function 𝑉₁ is given by𝑉₁̇=12𝑒^𝑇𝑐𝑃𝐴^𝑐+𝐴^𝑇𝑐𝑃𝑒_𝑐+𝑒^𝑇𝑐𝑃𝐵_𝑐𝑒_𝜌.(A.2)Choosing Λ=(1/2)𝐵^𝑇𝑐𝑃, we have𝑒_𝑠=𝑒_𝜌+12𝐵^𝑇𝑐𝑃𝑒_𝑐.(A.3)A candidate Lyapunov function for the error dynamics is𝑉₂=𝑉₁+12𝜂^2𝑠,(A.4)where 𝜂_𝑠=𝑒_𝑠+𝑐^𝑇(𝑡)𝑊, and 𝑐^𝑇(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,𝜉_𝑟) for any choice of gain 𝑘_𝑡 large enough.
The time derivative of 𝑉₂ is given by𝑉₂̇=𝑉₁̇+𝜂_𝑠=1𝑒^𝑠̇+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇2𝑒^𝑇𝑐𝑃𝐴^𝑐+𝐴^𝑇𝑐𝑃𝑒_𝑐+𝑒^𝑇𝑐𝑃𝐵_𝑐𝑒^𝑠−12𝐵^𝑇𝑐+𝜂_𝑠𝑃𝑒_𝑐𝑒^𝜌̇+12𝐵^𝑇𝑐𝑃𝑒_𝑐̇+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇.(A.5)Substitution of 𝑒_𝜌=𝑒_𝑠−(1/2)𝐵^𝑇𝑐𝑃𝑒_𝑐 yields𝑉₂̇=12𝑒^𝑇𝑐𝑃𝐴^𝑐+𝐴^𝑇𝑐𝑃𝑒_𝑐−12𝑒^𝑇𝑐𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃𝑒_𝑐++𝜂𝑒^𝑇𝑐_𝑠𝑃𝐵_𝑐𝑒_𝑠+𝜂_𝑠−𝑦^𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢)+𝜂_𝑠𝑏(𝑧,𝑢)(𝑢−𝛼(𝑧,𝜉^𝑟))+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇21𝐵^𝑇𝑐𝑃𝐴_𝑐𝑒_𝑐+12𝐵^𝑇𝑐𝑃𝐵_𝑐𝑒_𝑠−14𝐵^𝑇𝑐𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃𝑒_𝑐.(A.6)Given that 𝑒_𝑠=𝜂_𝑠−𝑐(𝑡)^𝑇𝑊, and12𝑒^2𝑠≤𝜂^2𝑠+𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊,(A.7)we have𝑉₂̇≤12𝑒^𝑇𝑐𝑃𝐴^𝑐+𝐴^𝑇𝑐𝑃𝑒_𝑐−12𝑒^𝑇𝑐𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃𝑒_𝑐+𝑘¹2𝑒^𝑇𝑐𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃𝑒_𝑐+1𝑘₁𝜂^2𝑠+1𝑘₁+𝜂_𝑠𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊+𝜂_𝑠−𝑦^𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢)+1𝑏(𝑧,𝑢)(𝑢−𝛼(𝑧,𝜉^𝑟))+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇2𝜂^𝑠𝐵^𝑇𝑐𝑃𝐴_𝑐𝑒_𝑐+12𝜂^2𝑠𝐵^𝑇𝑐𝑃𝐵_𝑐−14𝜂^𝑠𝐵^𝑇𝑐𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃𝑒_𝑐−12𝜂^𝑠𝐵^𝑇𝑐𝑃𝐵_𝑐𝑐^𝑇(𝑡)𝑊,(A.8)or𝑉₂̇≤12𝑒^𝑇𝑐𝑃𝐴^𝑐+𝐴^𝑇𝑐𝑃𝑒_𝑐+𝛾₁𝑒^𝑇𝑐𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃𝑒_𝑐++𝜂𝜂_𝑠𝑠−𝑦^𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢)+1𝑏(𝑧,𝑢)(𝑢−𝛼(𝑧,𝜉^𝑟))+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇𝑘₁𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊−𝜂_𝑠𝑐^𝑇(𝑡)𝑊12𝐵^𝑇𝑐+𝛾₂𝑃𝐵_𝑐𝜂^2𝑠+𝑘²4𝑒^𝑇𝑐𝐴^𝑇𝑐𝐴_𝑐𝑒_𝑐,(A.9)where 𝑘₁,𝑘₂, and 𝑘₃ are positive constants and𝛾₁=−1𝛾₂2+𝑘2¹+𝑘8³,=1𝑘₁+14𝑘₂𝐵^𝑇𝑐𝑃𝑃𝐵_𝑐+12𝐵^𝑇𝐶𝑃𝐵_𝑐+18𝑘₃𝐵^𝑇𝑐𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃𝐵_𝑐.(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 equation𝑃𝐴_𝑐+𝐴^𝑇𝑐𝑃+2𝛾₁𝑃𝐵_𝑐𝐵^𝑇𝑐𝑃+12𝑘²𝐴^𝑇𝑐𝐴_𝑐+𝑄=0(A.11)for some positive definite symmetric matrix 𝑄, then inequality (A.9) becomes𝑉₂̇≤−12𝑒^𝑇𝑐𝑄𝑒_𝑐+𝛾₂𝜂^2𝑠++𝜂1_𝑠𝑏(𝑧,𝑢)(𝑢−𝛼(𝑧,𝜉^𝑟))+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇𝑘₁𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊−𝜂_𝑠𝑐^𝑇(𝑡)𝑊12𝐵^𝑇𝑐+𝜂_𝑠𝑃𝐵_𝑐−𝑦^𝑟(𝜌)+𝑓(𝑧,𝜉,𝑢)−𝑓(𝑧,𝜉_𝑟,𝑢).(A.12)
Substituting (12), inequality (A.12) becomes 𝑉₂̇≤−12𝑒^𝑇𝑐𝑄𝑒_𝑐+𝛾₂𝜂^2𝑠+𝜂_𝑠×+1𝑏(𝑧,𝑢)𝑢−𝑊^𝑇𝑆(𝑧,𝜉_𝑟)−𝜇₁(𝑡)+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇𝑘₁𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊−𝜂_𝑠𝑐^𝑇(𝑡)𝑊12𝐵^𝑇𝑐−𝜂_𝑠𝑃𝐵_𝑐≤−1𝑦^𝑟(𝜌)+𝜂_𝑠𝐿₁‖‖𝜉‖‖+𝜂_𝑠𝑏(𝑧,𝑢)𝑊^𝑇𝑆(𝑧,𝜉_𝑟)2𝑒^𝑇𝑐𝑄𝑒_𝑐+𝛾₂𝜂^2𝑠+𝑘^𝜇2𝑏(𝑧,𝑢)²𝜂^2𝑠+12𝑘_𝜇𝜇₁(𝑥(𝑡))²+𝑘^𝑑2𝜂^2𝑠+12𝑘_𝑑𝑦^𝑟(𝜌)²+𝜂_𝑠+𝜂_𝑠𝐿₁‖‖𝜉‖‖+1𝑏(𝑧,𝑢)𝑢−𝑊^𝑇𝑆(𝑧,𝜉_𝑟)+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇𝑘₁𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊−𝜂_𝑠𝑐^𝑇(𝑡)𝑊12𝐵^𝑇𝑐+𝜂_𝑠𝑃𝐵_𝑐𝑏(𝑧,𝑢)𝑊^𝑇𝑆(𝑧,𝜉_𝑟),(A.13)where 𝑘_𝜇 and 𝑘_𝑑 are positive constants.
Consider the control structure shown in (15), we choose the following controller:𝑢=𝑊^𝑇𝑆(𝑧,𝜉_𝑟)+𝑒_𝑠−𝑘⁴−1𝑏₀𝛾₂−𝑘^𝜇2𝑏¹(𝑧)−𝑘^𝑑2𝑏1₀−𝑘^𝑤2𝑏¹(𝑧)²𝑏₀=𝑊_𝑇𝑆(𝑧,𝜉_𝑟)^𝑇𝑆(𝑧,𝜉_𝑟)𝑆(𝑧,𝜉_𝑟)+𝜂_𝑠−𝑘⁴−1𝑏₀𝛾₂−𝑘^𝜇2𝑏¹(𝑧)−𝑘^𝑑2𝑏1₀−𝑘^𝑤2𝑏¹(𝑧)²𝑏₀−𝑐_𝑇𝑆(𝑧,𝜉_𝑟)^𝑇𝑆(𝑧,𝜉_𝑟)(𝑡)𝑊𝑘_𝑡,(A.14)where 𝑘₄>0 is a constant, 𝑏₀ and 𝑏₁(𝑧) are the lower and upper bound in the inequality (13), and 𝑘_𝑡 is given by𝑘_𝑡=−𝑘₄−1𝑏₀𝛾₂−𝑘^𝜇2𝑏¹(𝑧)−𝑘^𝑑2𝑏1₀−𝑘^𝑤2𝑏¹(𝑧)²𝑏₀𝑆(𝑧,𝜉_𝑟)^𝑇𝑆(𝑧,𝜉_𝑟).(A.15)The above control action constitutes filtered tracking error and the approximated nonlinearity.
The weight 𝑊 satisfies ‖𝑊‖≤𝑤_𝑚, where the upper bound, 𝑤_𝑚, is guaranteed in the design of an adaptive law (16).
Substitution of the controller equation (A.14) in the inequality (A.13) gives𝑉₂̇≤−12𝑒^𝑇𝑐+𝜂_𝑠𝑄𝑒_𝑐−𝑘₄𝑏𝜂^2𝑠+𝜂_𝑠𝑏𝑐^𝑇(𝑡)𝑊𝑘_𝑡+𝑐̇^𝑇(𝑡)𝑊+𝑐^𝑇(𝑡)𝑊̇+1𝑏(𝑧,𝑢)𝑊^𝑇𝑆(𝑧,𝜉_𝑟)+𝜂_𝑠𝐿₁‖‖𝜉‖‖𝑘₁𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊−𝜂_𝑠𝑐^𝑇(𝑡)𝑊12𝐵^𝑇𝑐+1𝑃𝐵_𝑐2𝑘_𝜇𝜇₁(𝑥(𝑡))²+12𝑘_𝑑𝑦^𝑟(𝜌)²−𝑘^𝑤2𝑏¹(𝑧)²𝑏₀𝑏𝑆(𝑧,𝜉_𝑟)^𝑇𝑆(𝑧,𝜉_𝑟)𝜂^2𝑠,(A.16)because of the inequality (13).
The adaptive law (16) is chosen such that ‖𝑊‖≤𝑤_𝑚, and−𝑊^𝑇𝑐(𝑡)𝑒_𝑠+1𝛾_𝑤𝑊^𝑇𝑊̇≤0.(A.17)It takes the following form:𝑊̇=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝛾^𝑤𝑐(𝑡)𝑒_𝑠,if‖‖‖‖𝑊𝑊‖‖‖‖=<𝑤𝑤_𝑚𝑚,or𝑊,and_𝑇𝛾𝑐(𝑡)𝑒_𝑠≤0,_𝑤𝑐(𝑡)𝑒_𝑠−𝛾_𝑤𝑊𝑊^𝑇𝑐(𝑡)𝑒_𝑠‖‖𝑊‖‖²,if𝑊‖‖𝑊‖‖=𝑤_𝑚and_𝑇𝑐(𝑡)𝑒_𝑠>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), gives𝑉₂̇≤−12𝑒^𝑇𝑐𝑄𝑒_𝑐−𝑘₄𝑏𝜂^2𝑠+𝛾_𝑤𝑐^𝑇(𝑡)𝑐(𝑡)𝜂^2𝑠+𝑘^𝑤2(𝑏−𝑏¹)²𝜂^2𝑠𝑆^𝑇𝑆−𝑘^𝑤2𝑏𝑏²¹₀𝑏𝑆^𝑇𝑆𝜂^2𝑠−+(1𝑏₁−𝑏)𝑘_𝑡+𝛾_𝑤𝑐^𝑇(𝑡)𝑐(𝑡)𝑐^𝑇(𝑡)𝑊𝜂_𝑠+𝜂_𝑠𝐿₁‖‖𝜉‖‖2𝑘_𝜇𝜇₁(𝑡)²+12𝑘_𝑑𝑦^𝑟(𝜌)²+12𝑘_𝑤𝑊^𝑇𝑊+1𝑘₁𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊,(A.19)where 𝑏₁=𝑏₁(𝑧)𝑏=𝑏(𝑧,𝑢), and 𝑆=𝑆(𝑧,𝜉_𝑟) due to space limits.
Since (𝑏(𝑧,𝑢)/𝑏₁(𝑧)−1)²≤1, it follows that𝑉₂̇≤−12𝑒^𝑇𝑐𝑄𝑒_𝑐−𝑘₄𝑏𝜂^2𝑠+𝛾_𝑤𝑐^𝑇(𝑡)𝑐(𝑡)𝜂^2𝑠−(𝑏₁−𝑏)𝑘_𝑡+𝛾_𝑤𝑐^𝑇(𝑡)𝑐(𝑡)𝑐^𝑇(𝑡)𝑊𝜂_𝑠+𝜂_𝑠𝐿₁‖‖𝜉‖‖+12𝑘_𝜇𝜇₁(𝑡)²+12𝑘_𝑑𝑦^𝑟(𝜌)²+12𝑘_𝑤𝑊^𝑇𝑊+1𝑘₁𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊.(A.20)
Let 𝑉=𝑉₂+(1/2)𝑊^𝑇𝑊, and let 𝑘_𝑐=(𝑏₁−𝑏)𝑘_𝑡, we have𝑉̇≤−12𝑒^𝑇𝑐𝑄𝑒_𝑐−𝑘₄𝑏𝜂^2𝑠−𝛾_𝑤+𝑘^𝑤2(𝑘^𝑐+𝛾_𝑤𝑐^𝑇(𝑡)𝑐(𝑡)−𝛾_𝑤)²𝑐^𝑇(𝑡)𝑐(𝑡)𝜂^2𝑠+1𝑘₁−𝛾_𝑤𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊+12𝑘_𝑤+𝜂_𝑠𝑊^𝑇𝑊𝐿₁‖‖𝜉‖‖+12𝑘_𝜇𝜇₁(𝑥(𝑡))²+12𝑘_𝑑𝑦^𝑟(𝜌)².(A.21)
Let 𝑘₄=𝑘₄𝑏₀−(𝛾_𝑤+(𝑘_𝑤/2)(𝑘_𝑐+𝛾_𝑤𝑐^𝑇(𝑡)𝑐(𝑡)−𝛾_𝑤)²)𝑐^𝑇(𝑡)𝑐(𝑡)>0, inequality (A.21) becomes𝑉̇≤−12𝑒^𝑇𝑐𝑄𝑒_𝑐−𝑘₄𝜂^2𝑠+1𝑘₁+1−𝛾_𝑤𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊𝑘_𝑤𝑊^𝑇𝑊+𝜂_𝑠𝐿₁‖‖𝜉‖‖+12𝑘_𝜇𝜇₁(𝑥(𝑡))²+12𝑘_𝑑𝑦^𝑟(𝜌)².(A.22)
Noting that, by assumption, |𝜇_𝑙(𝑥(𝑡))|≤𝜇_𝑙,‖𝑦^𝑟(𝜌)‖≤𝜈₁, and using the fact that𝑊^𝑇𝑊=𝑊−𝑊^∗≤‖‖𝑊‖‖₂^𝑇𝑊−𝑊^∗+2|||𝑊^𝑇𝑊^∗|||+‖‖𝑊_∗‖‖²≤4𝑤^2𝑚,(A.23)we obtain𝑉̇≤−12𝑒^𝑇𝑐𝑄𝑒_𝑐−𝑘₄𝜂^2𝑠+𝜂_𝑠𝐿¹‖‖𝜉‖‖+12𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚+1𝑘₁−𝛾_𝑤𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊.(A.24)
Consider the following composite Lyapunov function for the closed-loop system (15), (16), (6), and (7):𝑉_𝑐=𝑉+𝛼𝑈,(A.25)where 𝛼 is a positive design parameter. The derivative of the Lyapunov function 𝑉_𝑐 is given by𝑉_𝑐̇=≤𝑉−̇+1𝛼𝑈̇2𝑒^𝑇𝑐𝑄𝑒_𝑐−𝑘₄𝜂^2𝑠+𝑘⁵2𝜂^2𝑠+12𝑘₅𝐿²¹‖‖𝜉‖‖²−𝑐₃𝛼‖‖𝜉‖‖²+Γ𝑘⁶2𝑒^𝑇𝑐𝑒_𝑐+Γ𝑘⁶2𝑒^2𝑠+12𝑘₆𝑐²⁴𝛼²‖‖𝜉‖‖²+12𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚+1𝑘₁−𝛾_𝑤𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊,(A.26)where Γ=(1+𝐵^𝑇𝑐𝑃𝑃𝐵_𝑐), since, by definition, the trajectory error vector is such that𝑒^𝑇𝑒=𝑒^𝑇𝑐𝑒_𝑐+𝑒^2𝑛≤(1+𝐵^𝑇𝑐𝑃𝑃𝐵_𝑐)𝑒^𝑇𝑐𝑒_𝑐+𝑒^2𝑠,(A.27)or𝑒^𝑇𝑒≤(1+𝐵^𝑇𝑐𝑃𝑃𝐵_𝑐)(𝑒^𝑇𝑐𝑒_𝑐+𝑒^2𝑠).(A.28)
Define𝜆=min𝜆^min𝑄2𝜆_max𝑃−Γ𝑘2⁶,𝑘⁴−𝑘⁵𝑐₃2−Γ𝑘2⁶,−𝐿²¹2𝑘₅𝛼𝑐₂−𝑐²⁴𝛼2𝑘₆𝑐₂,(A.29)Γ𝑘₆+(1/𝑘₁)+𝜆<𝛾_𝑤, and the gains of the two interconnected system (6) and (7) are such that 𝑐₃−𝐿²¹/2𝑘₅𝛼𝑐₂−𝑐²⁴𝛼/2𝑘₆𝑐₂>0, we have𝑉_𝑐̇≤+−𝜆1𝑒_𝑐𝑃𝑒_𝑐−𝜆𝜂^2𝑠−𝜆𝛼𝑈−𝜆𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊2𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚≤−𝜆2𝑒^𝑐𝑃𝑒_𝑐−𝜆2𝑒^2𝑠−𝜆2𝛼𝑈+21𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚,(A.30)or𝑉_𝑐̇≤−𝜆𝑉_𝑐+12𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚.(A.31)
It follows that the error vector, 𝑒, variable 𝜂_𝑠, and the parameter estimation errors, 𝑊, are bounded. Since the tracking trajectory 𝑦_𝑟 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 ‖𝜂_𝑠‖ given by‖‖𝜂_𝑠‖‖≤𝛼_𝜂𝑒^{−𝜆𝜂(𝑡−𝑡0)}+1√𝑘_𝜂𝜇^2𝑙+𝜈²¹+𝑤^2𝑚,(A.32)where 𝛼_𝜂=√2𝑉_𝑐(0),𝜆_𝜂=𝜆/2, and 𝑘_𝜂=min[2𝑘_𝑑,2𝑘_𝜇,𝑘_𝑤/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 by𝑐^𝑇(𝑡)=𝑒^{−𝐾(𝑡)(𝑡−𝑡0)}𝑐^𝑇(𝑡₀)+^𝑡𝑡₀𝑒^{−𝐾(𝑡)(𝑡−𝜏)}𝐵(𝜏)𝑑𝜏.(A.33)
It is obvious that the differential equation𝑐̇^𝑇(𝑡)=−𝐾(𝑡)𝑐^𝑇(𝑡)(A.34)is globally exponentially stable.
The element 𝐵(𝑡) is a bounded function of time. Note that‖‖𝐵(𝑡)‖‖²=𝑏₁(𝑧)²𝑆^𝑇𝑆.(A.35)For the particular choice of basis functions proposed in this paper, we have ‖𝑆‖≤√𝑁, 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 𝐵_𝑀, that is,‖‖𝐵(𝑡)‖‖≤𝐵_𝑀.(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:‖‖𝑐^𝑇(𝑡)‖‖≤𝐶𝑒^{−𝜆𝑐(𝑡−𝑡0)}+𝐶𝐵^𝑀𝜆_𝑐,(A.37)where 𝐶=‖𝑐^𝑇(𝑡₀)‖>0 and 𝜆_𝑐>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., 𝜂_𝑠≡0) dynamics of the parameter estimation errors are exponentially stable.
Lemma 1 establishes that the origin of the differential equation𝑊̇=−𝛾_𝑤𝑐(𝑡)𝑐^𝑇(𝑡)𝑊(A.38)is an exponentially stable equilibrium. In fact, it follows from the proof of Lemma 1 that the Lyapunov function 𝑉_𝑤=(1/2𝛾_𝑤)𝑊^𝑇𝑊 is such that𝑉_𝑤̇=−𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊≤−𝑐_𝑤‖‖𝑊‖‖²(A.39)for 𝑐_𝑤>0 a positive constant.
It follows from the property of the projection algorithm that the rate of change of 𝑉_𝑤 along the trajectories of (A.18) is given by𝑉_𝑤̇≤−𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊−𝑊^𝑇𝑐(𝑡)𝜂_𝑠.(A.40)Completing the squares, we get𝑉_𝑤̇≤−12𝑊^𝑇𝑐(𝑡)𝑐^𝑇(𝑡)𝑊+12𝜂^2𝑠.(A.41)Substitution of (A.32) yields𝑉_𝑤̇≤−𝑐_𝑤𝛾_𝑤𝑉_𝑤+𝛼^2𝜂𝑒^{−2𝜆𝜂(𝑡−𝑡0)}+1𝑘_𝜂𝜇^2𝑙+𝜈²¹+𝑤^2𝑚.(A.42)By integrating, we get𝑉_𝑤≤max𝑉_𝑤(𝑡₀),|||𝛼^2𝜂𝑐_𝑤𝛾_𝑤−2𝜆_𝜂|||,1𝑐_𝑤𝛾_𝑤𝑘_𝜂×exp−min𝑐_𝑤𝜇^2𝑙+𝜈²¹+𝑤^2𝑚+1𝛾_𝑤,2𝜆_𝜂𝑡−𝑡⁰𝑐_𝑤𝛾_𝑤𝑘_𝜂𝜇^2𝑙+𝜈²¹+𝑤^2𝑚.(A.43)Consequently, the parameter estimation error is guaranteed to decay exponentially as‖‖𝑊‖‖≤𝛼_𝑤𝑒^−𝜆𝑤(𝑡−𝑡₀)+2𝑘_𝜂𝑐_𝑤𝜇^2𝑙+𝜈²¹+𝑤^2𝑚^1/2.(A.44)
Taking the limit as 𝑡→∞ confirms that the estimation error converges to a small adjustable neighborhood of the origin given bylim_𝑡→∞‖‖𝑊‖‖≤2𝑘_𝜂𝑐_𝑤𝜇^2𝑙+𝜈²¹+𝑤^2𝑚^1/2.(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, 𝜂_𝑠, 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 as𝑉_𝑐̇≤−12𝜆^min𝑃𝑒^𝑇𝑐𝑒_𝑐−𝜆2𝑒^2𝑠−𝜆2𝛼𝑈+21𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚,(A.46)where 𝜆_min{𝑄} is the smallest eigenvalue of 𝑄. Integration of inequality (A.46) gives𝑉_𝑐(𝑡)≤−𝑉−_𝑐(0_𝑡0)+𝑡21𝜆^min𝑃𝑒_𝑐(𝜎)^𝑇𝑒_𝑐(𝜎)+𝜆2𝑒^𝑠(𝜎)²+𝜆2𝛼𝑈(𝜎)²𝑑𝜎221𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚.(A.47)This result implies that the following inequality holds:1𝑡^𝑡0𝑒₁(𝜎)²𝑑𝜎≤2𝑉^𝑐(0)𝑡𝜆_min𝑃+𝑘1𝜇^2𝑙+𝜈²¹+𝑤^2𝑚,(A.48)where 𝑘=(1/𝜆_min{𝑄})min{𝑘_𝜇,𝑘_𝑑,𝑘_𝑤/8}. Hence, the boundedness of the mean square error stated in inequality (23) is achieved with 𝛼₀=2/𝜆_min{𝑃} as required.
Integration of inequality (A.31) gives𝑉_𝑐(𝑡)≤𝑉_𝑐(0)𝑒^{−𝜆(𝑡−𝑡0)}+12𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤×_𝑡𝑡04𝑤^2𝑚≤𝑉_𝑐𝑒^{−𝜆(𝑡−𝜏)}𝑑𝜏(0)𝑒^{−𝜆(𝑡−𝑡0)}+12𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚.(A.49)
Given (A.7), we have12𝑝_𝑐𝑒^2𝑠≤𝜂^2𝑠+‖‖𝑊‖‖²,(A.50)where 𝑝_𝑐=sup_𝑡>0‖𝑐(𝑡)‖². With the positive constant𝑝_𝑚=min𝜆_min𝑃,12𝑝_𝑐,(A.51)inequality (A.49) simplifies to12𝑒^𝑇𝑐𝑒_𝑐+12𝑒^2𝑠≤𝑉^𝑐(0)𝑝_𝑚𝑒^{−𝜆(𝑡−𝑡0)}+1𝑝_𝑚21𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚.(A.52)
It follows from inequality (A.52) that𝑒^𝑇𝑒≤Γ𝑉^𝑐(0)𝑝_𝑚𝑒^{−𝜆(𝑡−𝑡0)}+Γ1𝑝_𝑚21𝑘_𝑑𝜈²¹+12𝑘_𝜇𝜇^2𝑙+1𝑘_𝑤4𝑤^2𝑚,(A.53)where Γ=2(1+𝐵^𝑇𝑐𝑃𝑃𝐵_𝑐). As a result, the adaptive learning tracking control guarantees that the tracking error, 𝑒, fulfills the following inequality:‖‖𝑒‖‖≤𝛼₁𝑒^{−𝛽1(𝑡−𝑡0)}+𝛼₂1√𝑘𝜇_𝑙+𝜈₁+𝑤_𝑚,(A.54)where 𝛼₁=Γ𝑉(0)𝑝_𝑚^1/2𝛽₁,𝛼=𝜆,₂=Γ𝑝_𝑚^1/2𝑘=min2𝑘_𝑑,,2𝑘_𝜇,𝑘^𝑤4.(A.55)This completes the proof.

Proof of Theorem 2. We consider the Lyapunov function 𝑉₂=(1/2)𝑒^𝑇𝑐𝑃𝑒_𝑐+(1/2)𝑒^2𝑠+(1/2𝛾_𝑤)𝑊^𝑇𝑊. We first note that 𝑒̂=𝑒−𝐷(𝜖)𝜂, where 𝐷(𝜖)=diag[𝜖^𝑛−1,…,1] and write the closed-loop system as follows:𝑊𝑒̇̇==𝐴⎧⎪⎪⎪⎨⎪⎪⎪⎩−⁰𝑒𝛾+Υ_𝑠𝐵₀𝑓(𝑥,Ψ^𝑠(𝑒−𝐷(𝜖)𝜂,𝑊,𝑌_𝑑)−𝑦^𝑑(𝑛),,if‖‖‖‖𝑊𝑊‖‖‖‖=<𝑤𝑤_𝑚𝑚,or−𝛾Υ_𝑠,𝑊^𝑇Υ_𝑠≤0,+𝛾𝑊𝑊^𝑇Υ_𝑠‖‖𝑊‖‖²𝜖𝜂̇=𝐴₀,if‖‖𝑊‖‖=𝑤_𝑚,𝑊^𝑇Υ_𝑠>0,𝜂+𝜖𝐵₀𝑓𝑥,Ψ^𝑠𝑒−𝐷(𝜖)𝜂,𝑊,𝑌^𝑑−𝑦^𝑑(𝑛).(A.56)By Lipschitz continuity of the projection algorithm [28], it follows that for a sufficiently small 𝜖 the inequality‖‖ProjΥ_𝑠(𝑒−𝐷(𝜖)𝜂,𝑌_𝑑),𝑊−ProjΥ_𝑠(𝑒,𝑌_𝑑),𝑊‖‖≤𝛾𝐿₁‖‖𝜂‖‖(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 following‖‖Ψ_𝑠𝑒−𝐷(𝜖)𝜂,𝑌^𝑑,𝑊−Ψ_𝑠𝑒,𝑌^𝑑,𝑊‖‖≤𝐿₂‖‖𝜂‖‖(A.58)holds on Ω for some positive nonzero constant 𝐿₂ and all 0<𝜖≤𝜖₂.
Using (A.31) and (A.42), the derivative of the Lyapunov function 𝑉₂ along the trajectories of the closed-loop system is such that𝑉₂̇≤×−𝑘Ψ₅𝑉₂+𝑐₂+𝑒_𝑠𝑏(𝑥,𝑢)_𝑠+1𝑒−𝐷(𝜖)𝜂,𝑌^𝑑,𝑊−Ψ_𝑠𝑒,𝑌^𝑑,𝑊𝛾𝑊^𝑇−PPrroojjΥΥ^𝑠_𝑠𝑒−𝐷(𝜖)𝜂,𝑌^𝑑,𝑊(𝑒,𝑌_𝑑),𝑊,(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 obtain𝑉₂̇=≤−−𝑘𝑘₅₅𝑉₂+𝑐₂+𝐿₂𝐿₃𝑏‖‖𝜂‖‖+2𝑤_𝑚𝐿₁‖‖𝜂‖‖𝑉₂+𝑐₂+𝑘₇‖‖𝜂‖‖,(A.60)where 𝑏=max_{𝑒∈Ω,𝑌𝑑∈𝑌}𝑏₁(𝑒,𝑌_𝑑), and 𝐿₃ is the maximum of 𝑒_𝑠 on Ω. Similarly, the derivative of 𝑊(𝜂) along the trajectories of the closed-loop system is given by𝜖𝑊̇≤−𝜂^𝑇𝜂+2𝜖𝜂^𝑇𝑃₀𝐵₀×𝑏(𝑥,𝑢)−Ψ_𝑠𝑊𝑒_∗−𝐷(𝜖)𝜂,𝑌^𝑑,𝑊𝑆𝑒+𝑌_𝑑−𝜇^𝑙𝑥(𝑡)−𝑦^𝑑(𝑛).(A.61)Since |Ψ_𝑠|≤𝑆_𝑢, ‖𝑊^∗‖≤𝑤_𝑚, ‖𝑆(𝑒+𝑌_𝑑)‖≤𝑆, and 𝜇_𝑙(𝑥(𝑡))≤𝜇_𝑙, we obtain𝜖𝑊̇=≤−−𝜂𝜂^𝑇_𝑇𝜂+2𝜖‖‖𝜂‖‖𝜆_max(𝑃₀)𝑏𝑆_𝑢+2𝑤_𝑚𝑆+𝜇_𝑙+𝜈₁𝜂+2𝜖‖‖𝜂‖‖𝜆_max(𝑃₀)𝑘₆.(A.62)Then, it follows that 𝑉₂̇≤0 on {𝑉₂=𝑐₁}×{𝑊(𝜂)≤𝜌𝜖²} as long as 𝜖≤(𝑘₅𝑐₁−𝑐₂)/𝑘₇√𝜌/𝜆_min(𝑃₀)=𝜖₃<min{𝜖₁,𝜖₂}. Similarly, 𝑊̇≤0 on {𝑉₂≤𝑐₁}×{𝑊(𝜂)=𝜌𝜖²} if 𝜌≤4𝜆_max(𝑃₀)³𝑘²⁶. As a result, we show that the set {𝑉₂≤𝑐₁}×{𝑊(𝜂)≤𝜌𝜖²} is positively invariant for all 0<𝜖≤𝜖₃.
Consider initial conditions {𝑒(0),𝑊(0)}∈𝑆, and {𝑒(̂0)}∈𝑂 where 𝑆 is a compact subset Ω_𝑏⊂Ω and 𝑂 is a compact subset of ℝ^𝑛. It follows that 𝜂(0)≤𝑘₄/𝜖^(𝑛−1), where 𝑘₄ is a positive constant which is related to the size of the compact sets 𝑆 and 𝑂.
Note that 𝑓(𝑒,𝑌_𝑑,𝑊,𝜇)=[𝑒₂,𝑒₃,…,𝑏(𝑥,𝑢)(Ψ_𝑠(⋅)−𝑊^∗𝑇𝑆(𝑒+𝑌_𝑑)−𝜇_𝑙(𝑥(𝑡))]^𝑇 is bounded on Ω. That is, ‖𝑓(𝑥,Ψ_𝑠(⋅))‖≤𝑘₅. By construction, the adaptive learning rate is such that ‖Proj(Υ_𝑠(⋅),𝑊)‖≤𝑘₆. As a result, the closed-loop trajectories of the system starting in 𝑆 are such that‖‖𝑊‖‖𝑒((𝑡𝑡))−−𝑒𝑊(0()0‖‖)‖‖≤≤𝑘₅𝑘𝑡₆,𝑡.(A.63)Therefore, there is a time 𝑇_𝑒 at which the closed-loop trajectory (𝑒(𝑡),𝑊(𝑡)) escapes the set Ω.
From (A.62), it follows that𝑊̇≤−12𝜖𝜂^𝑇𝜂,(A.64)if 𝑊≥𝜌𝜖². Therefore, we have that𝑊̇≤−12𝜖𝜆_max(𝑃₀)𝑊,(A.65)or𝑊(𝑡)=𝑒⁻^{1/2𝜖𝜆max}^(𝑃0)𝑡𝑊(0),(A.66)for 𝑊≥𝜌𝜖². As a result, we can find an 𝜖<𝜖₃ and a 𝑇(𝜖)>0 such that 𝑊(𝑡)≤𝜌𝜖²,∀𝑡≥𝑇(𝜖). Moreover, it is always possible to pick 𝜖₃ small enough such that 𝑇(𝜖₃)≤𝑇₀. As a result, picking 𝜖^∗1=min{𝜖₁,𝜖₂,𝜖₃} ensures that the trajectories of the closed-loop process starting in 𝑆×𝑂 enter the compact set Λ={𝑉₂≤𝑐₁}×{𝑊≤𝜌𝜖²} in finite time 𝑇(𝜖)∀0<𝜖≤𝜖^∗1.
On the set Λ, it is shown that𝑉₂̇≤−𝑘₁𝑉₂+𝑐₂+𝑘₃𝜌𝜆_min𝜖.(A.67)Hence, we have𝑉₂̇≤−𝑘¹2𝑉²,(A.68)when {𝑉₂≥2𝑐₂+2𝑘₃√(𝜌/𝜆_min)𝜖} and, therefore, the trajectories of the system must be such that lim_𝑡→∞𝑉₂(𝑡)≤2𝑐₂+2𝑘₃𝜖. Define 𝜒=[𝑒,𝑊], if we pick 𝜖 such that𝑉²≤2𝑐₂+𝑘₃𝜖⊂‖‖𝜒‖‖≤𝜇2,(A.69)then there is a finite time 𝑇(𝜇) such that for all 0<𝜖≤𝜖₄‖‖𝜒‖‖≤𝜇2,∀𝑡≥𝑇(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 ‖𝜂‖≤𝜇/2∀𝑡≥𝑇(𝜇). Therefore, taking 𝜖^∗2=min{𝜖₄,𝜖₅} ensures that there is a finite time 𝑇₂=max{𝑇,𝑇} such that ‖𝜒‖≤𝜇/2 and ‖𝑒‖̂≤𝜇/2∀𝑡≥𝑇₂.
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)Its derivative is such that𝑉̇≤−𝑘₁𝑉₂+𝑐₂+𝑘₃‖‖𝜂‖‖−1𝜖‖‖𝜂‖‖²+2𝜆_max(𝑃₀)𝑘₂‖‖𝜂‖‖.(A.72)By completing the squares, we get𝑉̇≤−𝑘₁𝑉₂−1𝜖‖‖𝜂‖‖²+𝑘⁴2+𝑘2⁵‖‖𝜂‖‖²+𝑐₂+𝑘²³2𝑘₄+2(𝜆^max(𝑃₀)𝑘₂)²𝑘₅.(A.73)
Therefore, there exists 0<𝜖≤𝜖^∗3≤𝜖^∗2 such that−1𝜖+𝑘2⁴+𝑘2⁵≤−𝑘⁶<0.(A.74)As a result, we obtain𝑉̇≤−𝑘𝑉₂−𝑘₆‖‖𝜂‖‖²+𝑐₂,(A.75)or𝑉̇≤−𝑘_𝑚𝑉+𝑐₂,(A.76)where 𝑘_𝑚=min{𝑘₁,𝑘₆/𝜆_max(𝑃₀)}. Using the results above, we guarantee that every trajectory of the system starting in 𝑆×𝑂 enters a small neighborhood of the origin in finite time and converges exponentially fast to a small adjustable neighborhood of the origin. This completes the proof.