Sliding-mode-based chaos suppression of coronary artery systems

Abstract

Since coronary artery disease is the leading global cause of mortality and morbidity, this paper investigates the chaos suppression of coronary artery systems. The motivation of the paper is to discuss and analyze coronary artery disease in the field of dynamics. Firstly, the mathematic model of coronary artery systems is formulated and the properties of this model are illustrated by bifurcation diagram, information entropy analysis, phase plane trajectory, and Poincaré section. With regard to the uncertainties of coronary artery systems, the disturbance observer technique is adopted. Meanwhile, the smooth second-order sliding mode controller is designed to suppress the chaos phenomenon. In light of the combination of the controller and observer, the stability of such a closed-loop system is proven in the sense of Lyapunov. Finally, some numerical simulations demonstrate the feasibility and validity of such design.

Introduction

Coronary artery systems are two blood vessels that branch from the aorta. They are close to the point of departure from the heart. They take charge of carrying oxygen-rich blood to the myocardium. Blockage of any branch of the coronary arteries will cause death of a portion of the heart tissue because it is deprived of oxygen-rich blood. Such blockage leads from coronary artery disease. ¹ It is reported that the coronary artery disease is the leading global cause of mortality and morbidity. ²

From the aspect of medical treatment, the coronary artery disease is characterized by an inadequate supply of oxygen-rich blood to the myocardium because of narrowing or blocking of a coronary artery by fatty plaques. If the oxygen depletion is extreme, the effect may be a myocardial infarction, that is, heart attack. If the deprivation is insufficient to cause the death of a section of heart muscle, the effect may be angina pectoris, that is, pain or discomfort in the chest. Both the effects can be fatal because they can lead to heart failure, ventricular fibrillation as far as sudden death. ³ Therefore, effective management and treatment of the coronary artery disease rely on risk assessment stratify. In order to assess the risk, mathematical modeling is powerful. ⁴

When heart failure or ventricular fibrillation takes place, it is characterized by an uncontrolled and uncoordinated contraction of the coronary artery systems. From the aspect of mathematical modeling, such contraction can be treated as the phenomenon of chaos. ⁵ With regard to the chaotic phenomena, such abnormal and fatal chaos can induce the coronary artery disease because chaotic systems are very sensitive to perturbations. Unfortunately, there are just many perturbations in the coronary artery systems, that is, body temperature, blood pressure, blood viscosity, and so forth. This fact indicates that the effective management and treatment can be mathematically considered as the chaos suppression problem of the coronary artery systems. ⁶

Since the coronary artery systems are inherently nonlinear, they are rather complicated. ⁷ Such characteristics challenge the chaos suppression very much. So far, many methods have been presented to suppress the chaos of coronary artery systems, including sliding mode or variable structure control methods, ⁸ time-delay based control design,^9,10 adaptive fuzzy control approach, ¹¹ to name but a few. The idea behind these methods is to suppress the fatal chaos so as to avoid serious health problems and illness development. ¹² For the purpose of effective treatment, the motion states in a vessel with pathological changes have to synchronize with the motion states of a normal vessel through the designed chaos-suppression methods. Therefore, two problems from the aspect of chaos suppression rise up for coronary artery systems. One is how to gain some insight into the chaotic behaviors. The other is how to develop the effective chaos-suppression methods.

As a powerful design tool, the methodology of sliding mode control (SMC) is advocated and investigated for the chaos-suppression problem of the coronary artery systems because the SMC methods exhibit better control performances than the conventional robust control methods.^13,14 Although the SMC methods suffer from the chattering drawback, many sliding mode-based methods have been reported for the chaos-suppression problem, that is, terminal sliding mode control, ¹⁵ higher order sliding mode adaptive control ¹⁶ and fractional-order sliding mode control. ¹⁷ There is a variety of SMC design methods, wherein one is called second-order SMC. Such second-order SMC method not only possesses the property of resisting uncertain disturbances, but also attenuates the adverse effects of chattering.^18–20

Concerning the coronary artery systems, the achievement of the chaos-suppression problem indicates that the symptoms of the coronary artery disease are disappeared or relieved, which will be beneficial for patients a lot. Further, doctors argue that the chattering should be as less as possible during the process of chaos suppression, which will inspire and encourage us to develop more effective SMC methods with high performance. Motivated by the merits of the second-order SMC method, this paper investigates a smooth second-order SMC method for the chaos-suppression problem of the coronary artery systems. Once the abnormal and fatal chaotic behaviors take place in the coronary artery systems, the smooth second-order SMC method is able to reject them despite uncertainties. Meanwhile, the kind of SMC method is able to avoid the chattering.

Clinically, many risk factors are imperceptible to patients but pivotal to the coronary artery systems, such as, the perturbation of blood pressure, the concentration of serum cholesterol, the change of ambient temperature, even the fluctuation of emotion. ²¹ These risk factors can hardly be measurable, predictive or known in advance. They can be treated as the disturbances of the coronary artery systems, which make the chaos suppression problem extremely tough. It is reported that the disturbance-observer technique is powerful to resist disturbances.^22,23 This paper fuses the technique with the smooth second-order SMC method for this chaos-suppression problem.

Medically, the developed second-order SMC method means a certain clinical pharmacy and medical device for therapeutic purposes. The proposed strategy can promote the research and development because therapies can be treated as a kind of control input to suppress the chaotic vasospasm. Considering a certain therapy, it can be treated as the control input of this chaotic system. Once the therapy is effective, the chaotic phenomenon has to achieve state synchronization of the blood vessels with pathological changes and the normal blood vessels. In this sense, the designed control can be utilized by a pharmacodynamics or a medical device, which is also the potential application of the presented control design.

This paper faces with the chaos-suppression problem of the coronary artery systems. The remainder of this paper is organized as follows. Modeling and chaotic behavior analysis of the coronary artery systems are addressed in Section 2. Section 3 designs the smooth second-order SMC method in the sense of Lyapunov. In Section 4, the control design is implemented via a numerical platform and some comparisons are illustrated to support the control design. Finally, Section 5 concludes this paper.

Modeling and analysis

Bio-mathematical model

Coronary arteries extend over the surface of heart and branch into smaller capillaries. They move in a complex pattern during each cardiac cycle. Characteristically, such arteries are categorized as muscular blood vessels. The lumped-parameter model that is applicable to the interpretation of a coronary artery has the form of

$$\begin{array}{c}{\stackrel{·}{x}}_1=-b{x}_1-c{x}_2\hfill \\ {\stackrel{·}{x}}_2=-\lambda (1+b)x_1-\lambda (1+c)x_2+\lambda x_1^3+E\cos \omega t\hfill \end{array}$$

(1)

Here $x_1$ means the change of inside radius in the vessel, $x_2$ indicates the pressure change in the vessel, $t$ is the time variable, $b$ , $c$ , and $\lambda$ denote the lumped parameters in this model, and $E\cos \omega t$ represents a periodical disturbance term.

Model-based analysis

No matter what coronary artery disease is, one of the clinic symptoms is vasospasm. Since the coronary artery spasm means an uncontrolled and uncoordinated contraction of the blood vessels that deliver blood to heart muscle, many serious consequences may be believed to be due to partial or complete occlusion of the blood vessels. (1) is just a window that gains some insight into such spasm from the aspect of dynamics.

In (1), define the initial condition at $t=0$ as $[x_1(0)x_2(0){]}^T=[0.20.2{]}^T$ , consider the parameters in the disturbance term as $E=0.3$ and $\omega =1$ , and set the time series as $0$ , $2\pi$ , $4\pi$ , $6\pi$ , $8\pi$ , $10\pi$ , and $12\pi$ where the period is $2\pi$ . Now, take the changes of the parameters $\lambda$ , $b$ , and $c$ into account. In order to visualize the effects of varying a certain parameter at a time and keeping others fixed, the bifurcation diagrams of the system (1) are employed.

Bifurcation diagram

The bifurcation diagrams of the coronary artery system are illustrated in Figure 1. In Figure 1(a), the diagram shows the changes of $x_1$ and $x_2$ with respect to the change of $\lambda$ . Here $\lambda$ is considered in the closed interval $[-1,0]$ and the other two parameters are fixed as $b=0.15$ and $c=-1.7$ . Similarly, the diagram in Figure 1(b) reveals the changes of $x_1$ and $x_2$ with respect to the change of $b$ . Here $b$ is considered in the closed interval $[0,1]$ and the other two parameters are fixed as $\lambda =-0.65$ and $c=-1.7$ . The diagram in Figure 1(c) displays the changes of $x_1$ and $x_2$ with respect to the change of $c$ . Here $c$ is considered in the closed interval $[-2,-1.5]$ and the other two parameters are fixed as $\lambda =-0.65$ and $b=0.15$ . The closed interval of each parameter in Figure 1 is equally divided into 4999 portions. Then, 5000 endpoints can be obtained in each closed interval. On each endpoint, $x_1$ and $x_2$ can be calculated according to the given time series. Finally, the curves can be plotted.

Figure 1.

Bifurcation diagrams: (a) obtained by the change of $\lambda$ from $-1$ to $0$ , (b) obtained by the change of $b$ from $0$ to $1$ , and (c) obtained by the change of $c$ from $-2$ to $-1$ .

Information entropy analysis

According to its concept, information entropy refers to the degree of chaos in a system. As an indicator, the more orderly the system is, the lower value its information entropy has. On the contrary, the more chaotic the system is, the higher value its information entropy has.

Here the data in Figure 1 are employed. For each parameter, 5000 numbers in its closed interval are available. In order to calculate the information entropy of the coronary artery system, each 10 consecutive numbers in the closed interval are divided into a sub-interval so that 500 sub-intervals are gotten. Apparently, the maximum and minimum can be found in the 5000 numbers.

Define another closed interval that is between the extreme values. Then, divide this interval into 10 equal sub-intervals. Concerning each 10 consecutive numbers, count the quantity that they fall into each sub-interval. Finally, the value of information entropy $H$ can be calculated by

$$H=-{\sum }_{i=1}^{10}{p}_i{\log }_2{p}_i$$

(2)

Here $p_i$ is the probability that the 10 consecutive numbers fall into the ith sub-interval, formulated by the quantity that the 10 consecutive numbers falls into the ith sub-interval over 10.

Figure 2 illustrates the diagrams of information entropy of the coronary artery system. It can be calculated that the maximum of information entropy is ${\log }_{2}10$ . When each number in the 10 consecutive numbers falls into one sub-interval constructed by the extreme values, the maximum of information entropy is obtained by $H_{\max }=-\sum _{i=1}^{10}\frac{1}{10}{\log }_2\frac{1}{10}$ . By comparison, both Figures 1 and 2 reveals that the coronary artery system in (1) tends to be chaotic as $\lambda \in [-0.7,-0.1]$ , $b\in [0,0.2]$ , and $c\in [-1.7,-1.65]\cup [-1.5,-1]$ . Without loss of generality, the parameters $\lambda =-0.65$ , $b=0.15$ , and $c=-1.7$ are picked up as representative ones.

Figure 2.

Diagrams of information entropy: (a) calculated by the change of $\lambda$ from $-1$ to $0$ , (b) calculated by the change of $b$ from $0$ to $1$ , and (c) calculated by the change of $c$ from $-2$ to $-1$ .

Dynamic behaviors

Consider the coronary artery system described by (1) with the selected representative parameters. Figure 3 illustrates the dynamic behaviors of the coronary artery system. Figure 3(a) in the phase plane demonstrates that the trajectory of this coronary artery system is very complex. It is well known that the technique of Poincaré section is a general method to explain complex dynamics because the technique can help to visualize the complexity. Here the Poincaré section analysis is carried out as insight into the complexity of the coronary artery system. Since the system (1) possesses two variables, an extended system should be defined in order to depict the system dynamics in 3 dimension. For this purpose, define $x_3=E\cos \omega t$ and substitute $x_3$ into (1). Then, let $x_3=0$ as the section. Finally, the Poincaré section in three dimension is displayed in Figure 3(b), where the lime-green plane is the defined section $x_3=0$ , the blue curve is the trajectory of the extended system in three dimension, and the red curve on the lime-green plane is just the Poincaré section. Figure 3(c) shows the Poincaré section in two dimension. Apparently, no matter what analysis tools are adopted, the dynamic behaviors of the coronary artery system revealed by such tools are very complex.

Figure 3.

Diagrams of dynamic behaviors: (a) phase plane trajectory, (b) Poincaré section in three dimension, and (c) Poincaré section.

In clinic, these dynamic behaviors referring to the vasospasm of the coronary artery system implicate a large number of coronary artery diseases. Unfortunately, such complexity challenges the chaos suppression problem of the coronary artery system very much.

Control design

According to the bio-mathematical model (1), a coronary artery system with uncertainties can have the form of

$$\begin{array}{c}{\stackrel{·}{x}}_1=-b{x}_1-c{x}_2+d_1\hfill \\ {\stackrel{·}{x}}_2=-\lambda (1+b)x_1-\lambda (1+c)x_2+\lambda x_1^{3}n+E\cos \omega t+u(t)+d_2\hfill \end{array}$$

(3)

where $u(t)$ means the control input injected into the coronary artery system and it is generated by the designed controller. $d_1$ and $d_2$ indicate the uncertain factors that may affect the inside radius and pressure in the vessel, that is, body temperature, blood pressure, blood viscosity, and environmental temperature. In (3), define $\mathbf{x}=[x_1,x_2{]}^T$ as the state vector and $\mathbf{d}=[d_1,d_2{]}^T$ as the disturbance vector. With regard to $\mathbf{d}$ , the following two assumptions are considered.

• Assumption 1: $\mathbf{d}$ is bounded, depicted by $||\mathbf{d}|{|}_{\infty }\le d^{*}$ where $d^{*}$ is a positive constant.

• Assumption 2: $\mathbf{d}$ is slowly time varying, formulated by $\stackrel{·}{\mathbf{d}}\simeq 0_{2\times 1}$ where $0_{2\times 1}=[0,0{]}^T$ .

Both Assumption 1 and Assumption 2 originate from the biological properties of the coronary artery system because violent disturbances may cause some irreversible damages of the coronary artery system. From the perspective, both the assumptions are mild.

The nominal system (4) with no control input can be drawn in light of the uncertain model (3).

$$\begin{array}{c}{\stackrel{·}{\overline{x}}}_1=-b{\overline{x}}_1-c{\overline{x}}_2\hfill \\ {\stackrel{·}{\overline{x}}}_2=-\lambda (1+b){\overline{x}}_1-\lambda (1+c){\overline{x}}_2+\lambda {\overline{x}}_1^3+E\cos \omega t\hfill \end{array}$$

(4)

Here $\overline{\mathbf{x}}=[{\overline{x}}_1,{\overline{x}}_2{]}^T$ is the state vector of the nominal system. Apparently, (4) seems similar to (1) very much. In fact, they depict the coronary artery system from the two aspects. (1) formulates the bio-mathematical model and (4) is derived from the coronary artery system with uncertainties (3). Therefore, the different variables are adopted in (1) and (4).

According to the aforementioned model-based analysis about (1), not only the nominal system (4) exhibits the complex dynamic behaviors, but also it is sensitive to the system parameters. For the chaos suppression problem of the uncertain coronary artery system (3), the developed smooth second-order sliding mode controller carried out in (3) needs to possess the ability of forcing the error vector $\mathbf{e}$ in (5) to $0_{2\times 1}$ once the uncertain coronary artery system becomes disordered.

$$\underset{t\to \infty }{lim}\mathbf{e}=0_{2\times 1}$$

(5)

Here $\mathbf{e}=[e_1,e_2{]}^T$ , $e_1=x_1-{\overline{x}}_1$ and $e_2=x_2-{\overline{x}}_2$ .

Substituting (3) and (4) into the definition of $\mathbf{e}$ yields

$$\begin{array}{c}{\stackrel{·}{e}}_1=-b{e}_1-c{e}_2+d_1\hfill \\ {\stackrel{·}{e}}_2=-\lambda (1+b)e_1-\lambda (1+c)e_2+\lambda e_1^{3}n+3\lambda x_1{\overline{x}}_1{e}_1+u+d_2\hfill \end{array}$$

(6)

Let ${\tilde{x}}_1=e_1$ and ${\tilde{x}}_2={\stackrel{·}{e}}_1$ . Then, differentiate ${\tilde{x}}_1$ and ${\tilde{x}}_2$ with respect to $t$ . Substituting (6) into ${\stackrel{.}{\stackrel{~}{x}}}_2$ yields

$$\begin{array}{c}{\stackrel{·}{\stackrel{~}{x}}}_1={\tilde{x}}_2\hfill \\ {\stackrel{·}{\stackrel{~}{x}}}_2=\lambda (c-b){\tilde{x}}_1-(b+\lambda +c\lambda ){\tilde{x}}_2-\lambda c{\tilde{x}}_1^3-3c\lambda x_1{\overline{x}}_1{\tilde{x}}_1\hfill \\ -\mathrm{cu}-c{d}_2+{\stackrel{·}{d}}_1+\lambda (1+c)d_1\hfill \end{array}$$

(7)

Considering Assumption 2, there exists ${\stackrel{·}{d}}_1\simeq 0$ in (7). Then, (7) can be written by (8) in the form of state space.

$$\begin{array}{c}\stackrel{·}{\stackrel{~}{\mathbf{x}}}=F(\tilde{\mathbf{x}},d)+\mathrm{Bu}\hfill \\ y=H\left(\tilde{\mathbf{x}}\right)\hfill \end{array}$$

(8)

Here $\tilde{\mathbf{x}}=[{\tilde{x}}_1,{\tilde{x}}_2{]}^T$ , $d=-c{d}_2+\lambda (1+c)d_1$ , $H\left(\tilde{\mathbf{x}}\right)={\tilde{x}}_1$ , $B=[0,-c{]}^T$ , $F(\tilde{\mathbf{x}},d)=[{\tilde{x}}_2,f(\tilde{\mathbf{x}})+d{]}^T$ , and $f(\tilde{\mathbf{x}})=\lambda (c-b){\tilde{x}}_1-(b+\lambda +c\lambda ){\tilde{x}}_2-\lambda c{\tilde{x}}_1^3-3c\lambda x_1{\overline{x}}_1{\tilde{x}}_1$ .

Take Assumption 1 into consideration. In (8), there exists $d=-c{d}_2+\lambda (1+c)d_1\le |c{d}_2|+|\lambda (1+c)d_1|\le |c|·||\mathbf{d}|{|}_{\infty }+|\lambda (1+c)|·||\mathbf{d}|{|}_{\infty }=d_0^{*}$ . This indicates

$$d\le d_0^{*}$$

(9)

Design of disturbance observer

(9) indicates that $d$ is also bounded concerning Assumption 1. Without loss of generality, the boundary of $d$ is unknown such that $d_0^{*}>0$ is unknown. Unfortunately, the value of $d_0^{*}$ is affiliated with the system stability and the control performance very much. In order to solve this issue concerning the unknown $d_0^{*}$ , the disturbance-observer technique is adopted and a disturbance observer is designed to estimate $d$ . Such a disturbance observer can be described by

$$\begin{array}{c}\stackrel{·}{z}=-\mathbb{L}{\mathbb{B}}_{2}z-\mathbb{L}({\mathbb{B}}_2\mathbb{L}\tilde{\mathbf{x}}+\mathbb{A}\tilde{\mathbf{x}}+{\mathbb{B}}_{1}u+\mathbb{M})\hfill \\ \hat{d}=z+\mathbb{L}\tilde{\mathbf{x}}\hfill \end{array}$$

(10)

Here $z\in {\Re }^{1\times 1}$ is the internal state variable of this observer, $\hat{d}\in {\Re }^{1\times 1}$ is the the estimate of $d$ , $\mathbb{L}\in {\Re }^{1\times 2}$ is the observer gain vector and $\mathbb{L}$ is pre-defined by designers, the matrix $\mathbb{A}$ is written by $\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{\lambda (c-b)}& -(b+\lambda +c\lambda )\hfill \end{array}\right]$ , the vectors ${\mathbb{B}}_1$ , ${\mathbb{B}}_2$ , and $\mathbb{M}$ are determined by $\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-c}\end{array}\right]$ , $\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$ , and $\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-\lambda c{\tilde{x}}_1^3-3c{x}_1{\overline{x}}_{11}{\tilde{x}}_{11}}\end{array}\right]$ .

Define the estimate error of this disturbance observer $e_d$ as

$$e_d=d-\hat{d}$$

(11)

• Assumption 3: $e_d$ is bounded, that is, $|e_d|\le e_d^{*}$ , where $e_d^{*}>0$ is unknown.

In (11), the derivative of the estimate error can be written by

$${\stackrel{·}{e}}_d=\stackrel{·}{d}-\stackrel{·}{\hat{d}}$$

(12)

Compared with the dynamic characteristics of the observer (10), Assumption 2 indicates that the uncertainties in (2) are slowly time varying such that $\stackrel{·}{d}\simeq 0$ in (12). Finally, (13) can be obtained by substituting (10) into (12).

$$\begin{array}{c}{\stackrel{·}{e}}_d\simeq \mathbb{L}{\mathbb{B}}_2\mathbb{P}+\mathbb{L}({\mathbb{B}}_2\mathbb{L}\tilde{\mathbf{x}}+\mathbb{A}\tilde{\mathbf{x}}+{\mathbb{B}}_{1}u+\mathbb{M})\hfill \\ -\mathbb{L}(\mathbb{A}\tilde{x}+{\mathbb{B}}_{1}u+{\mathbb{B}}_{2}d+\mathbb{M})\hfill \\ =\mathbb{L}{\mathbb{B}}_2(\hat{d}-\mathbb{L}\tilde{\mathbf{x}})+{\mathbb{L}}^2{\mathbb{B}}_2\tilde{\mathbf{x}}-\mathbb{L}{\mathbb{B}}_{2}d\hfill \\ =\mathbb{L}{\mathbb{B}}_2(\hat{d}-d)\hfill \\ =-\mathbb{L}{\mathbb{B}}_2{e}_d\hfill \end{array}$$

(13)

The solution of (13) can be calculated by

$$e_d=\exp (-\mathbb{L}{\mathbb{B}}_2)e_d(0)$$

(14)

In (14), the vector $\mathbb{L}$ can be picked up by designers such that the eigenvalue of $\mathbb{L}{\mathbb{B}}_2$ is a positive constant. Such design means that the estimate error $e_d$ is able to exponentially converge as $t\to \infty$ if $e_d(0)$ is not infinite.

Apparently, the condition that $e_d(0)$ is finite is not strict, that is, Assumption 3 is quit mild. Consequently, the output of this disturbance observer $\hat{d}$ can be treated as the estimate of $d$ and $\hat{d}$ will be adopted to replace the unknown $d$ in (8) in the following controller design. Inevitably, such replacement may affect the stability of the coronary artery system, which should be discussed in theory.

Controller design

In order to synchronize (8) by the smooth second-order SMC method, a sliding surface needs to be defined firstly.

$$s={\tilde{x}}_1+\beta {\tilde{x}}_2$$

(15)

Here the predefined constant $\beta$ is positive. Accordingly, the derivative of $s$ has the form of

$$\stackrel{·}{s}={\stackrel{·}{\stackrel{~}{x}}}_1+\beta {\stackrel{·}{\stackrel{~}{x}}}_2$$

(16)

Substituting (8) into (16) yields

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}={\tilde{x}}_2+\beta \{\lambda (c-b){\tilde{x}}_1-(b+\lambda +c\lambda ){\tilde{x}}_2}\\ -\lambda c{\tilde{x}}_1^3-3c\lambda x_1{\overline{x}}_1{\tilde{x}}_1-\mathrm{cu}\}+\beta d\hfill \end{array}$$

(17)

In (17), the disturbance term $d$ is unknown such that the output of the disturbance observer in (10) is employed to replace the term. By such replacement, (17) can be re-written by

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}={\tilde{x}}_2+\beta \{\lambda (c-b){\tilde{x}}_1-(b+\lambda +c\lambda ){\tilde{x}}_2}\\ -\lambda c{\tilde{x}}_1^3-3c\lambda x_1{\overline{x}}_1{\tilde{x}}_1-\mathrm{cu}\}+\beta \hat{d}\hfill \end{array}$$

(18)

Since the coronary artery system (8) is biological, not only may the dramatic change badly damage the coronary artery system, but also such change may cause the coronary artery diseases. This indicates that the control input $u$ in (8) should be as smooth as possible. For the purpose, the smooth second-order SMC law is defined as

$$\begin{array}{c}u=\frac{1}{c\beta }\{{\kappa }_1{s}^{\frac{2}{3}}\mathrm{sgn}(s)-v+{\tilde{x}}_2+\beta \lambda (c-b){\tilde{x}}_1\hfill \\ -\beta (b+\lambda +c\lambda ){\tilde{x}}_2-c\lambda \beta {{\tilde{x}}_1}^3-3c\beta x_1{\overline{x}}_1{\tilde{x}}_1+\hat{d}\}\hfill \\ \stackrel{·}{v}=-{\kappa }_2{s}^{\frac{1}{3}}\hfill \end{array}$$

(19)

Here ${\kappa }_1$ and ${\kappa }_2$ are positive and they are predefined, $\mathrm{sgn}(·)$ is the sign function.

Stability analysis

In (19), the output of the disturbance observer is involved in the smooth second-order sliding mode controller to achieve the chaos suppression of this coronary artery system. Due to the combination of the controller and the observer, the stability of this chaos suppression problem should be taken into consideration.

Replacing $u$ in (8) by (19) gives

$$\begin{array}{c}\stackrel{·}{s}=-{\kappa }_1{s}^{\frac{2}{3}}\mathrm{sgn}(s)+v+e_d\hfill \\ \stackrel{·}{v}=-{\kappa }_2{s}^{\frac{1}{3}}\hfill \end{array}$$

(20)

Then, the stability of the coronary artery system (8) by the smooth second-order sliding mode controller (19) with the disturbance observer (10) is equivalent to the stability of (20).

Theorem 1: Take Assumptions 1, 2, and 3 into account; consider the bio-mathematical model of the coronary artery system (8); design the disturbance observer (10); formulate the sliding surface (15); the smooth second-order sliding mode controller has the form of (19). If ${\kappa }_1>0$ is as large as possible and ${\kappa }_2>0$ in (19), then the closed-loop system is asymptotically stable in the presence of uncertainties.

Proof: Select the following Lyapunov candidate $V$ as

$$V=\frac{1}{2}{v}^2+{\kappa }_2{\int }_0^s{\tau }^{\frac{1}{3}}\mathrm{d}\tau =\frac{1}{2}{v}^2+\frac{3}{4}{\kappa }_2{s}^{\frac{4}{3}}$$

(21)

The derivative of $V$ with respect to time $t$ in (21) has the form of

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{d}V}{\mathrm{d}t}=\frac{\partial V}{\partial v}\frac{\mathrm{d}v}{\mathrm{d}t}+\frac{\partial V}{\partial s}\frac{\mathrm{d}s}{\mathrm{d}t}}\\ =v\stackrel{·}{v}+{\kappa }_2{s}^{\frac{1}{3}}\stackrel{·}{s}\hfill \end{array}$$

(22)

Substituting (20) into (22) yields

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{d}V}{\mathrm{d}t}=-{\kappa }_1{\kappa }_2|s|+{\kappa }_2{e}_d{s}^{\frac{1}{3}}}\\ \multicolumn{1}{c}{\le -{\kappa }_1{\kappa }_2|s|+{\kappa }_2{e}_d^{*}{s}^{\frac{1}{3}}}\end{array}$$

(23)

Now, the inequation ${\kappa }_1|s|\ge e_d^{*}\left(s^{\frac{1}{3}}\right)$ will be discussed in the following five cases.

• Case 1: For $s<0$ , ${\kappa }_1|s|\ge e_d^{*}\left(s^{\frac{1}{3}}\right)$ is apparently held true such that $\frac{\mathrm{d}V}{\mathrm{d}t}<0$ .

• Case 2: For $s=0$ , ${\kappa }_1|s|=e_d^{*}\left(s^{\frac{1}{3}}\right)$ is also apparently held true such that $\frac{\mathrm{d}V}{\mathrm{d}t}=0$ .

• Case 3: For $0<s<1$ , there exists $|s|<s^{\frac{1}{3}}$ . In order to have $\frac{\mathrm{d}V}{\mathrm{d}t}\le 0$ , not only the value of ${\kappa }_1$ is larger than $e_d^{*}$ , but also ${\kappa }_1$ has to be as large as possible to assure ${\kappa }_1|s|\ge e_d^{*}\left(s^{\frac{1}{3}}\right)$ .

• Case 4: For $s=1$ , there exists $|s|=s^{\frac{1}{3}}$ . Similarly, the value of ${\kappa }_1$ needs to be larger than $e_d^{*}$ to assure ${\kappa }_1>e_d^{*}$ in order to have $\frac{\mathrm{d}V}{\mathrm{d}t}<0$ .

• Case 5: For $s>1$ , there exists $|s|>s^{\frac{1}{3}}$ . In order to have $\frac{\mathrm{d}V}{\mathrm{d}t}<0$ , the value of ${\kappa }_1$ should be picked up to guarantee ${\kappa }_1>e_d^{*}$ such that ${\kappa }_1|s|>e_d^{*}\left(s^{\frac{1}{3}}\right)$ .

In (23), $\frac{\mathrm{d}V}{\mathrm{d}t}\le 0$ will be held true if ${\kappa }_1>0$ , ${\kappa }_2>0$ , and ${\kappa }_1|s|\ge e_d^{*}\left(s^{\frac{1}{3}}\right)$ , that is, the closed-loop control system is of asymptotical stability in the sense of Lyapunov. ■

Remark: From the aforementioned discussions, for any $s$ , $\frac{\mathrm{d}V}{\mathrm{d}t}\le 0$ can be held true if ${\kappa }_1$ is large enough and ${\kappa }_1>e_d^{*}$ . However, $e_d^{*}>0$ is unknown according to Assumption 3, which means that the value of ${\kappa }_1$ does not have any reference at all. Fortunately, (14) indicates that $e_d$ is exponentially convergent to zero as $t\to \infty$ and that the maximum of $e_d$ in one dynamic process is $e_d(0)$ , which means that $e_d^{*}$ must be larger than or equal to $e_d(0)$ . Namely, the value of ${\kappa }_1$ must not be smaller than $e_d(0)$ from the aspect of the system stability.

Simulation results

For the purpose of verifying the feasibility, the designed control method in this section will be carried out by a benchmark. Take the bio-mathematical model of a coronary artery system with uncertainties into consideration, where the system parameters in (3) are determined by $\lambda =-0.65$ , $b=0.15$ , $c=-1.7$ , $E=0.3$ , and $\omega =1$ . These parameters are kept unchanged from the model presented by Zhao et al.^10,11,16 The initial states of the system in (3) are depicted by $[\begin{array}{cc}{x}_1(0)\hfill & x_2(0)\hfill \end{array}{]}^T=[\begin{array}{cc}0.2\hfill & 0.2\hfill \end{array}{]}^T$ . Concerning the disturbance observer, the observer gain vector is assigned to $\mathbb{L}=[\begin{array}{cc}5\hfill & 5\hfill \end{array}{]}^T$ . With regard to the smooth second-order sliding mode controller, its parameter in the sliding surface (15) is picked up as $\beta =70$ and the parameters in the control law (19) are given by ${\kappa }_1=150$ and ${\kappa }_2=100$ .

In (3), both the equations contain uncertainties. But their properties of the uncertainties are inherently different. Apparently, the term $d_2$ enters the coronary artery system via the control channel such that it belongs to the matched uncertainties. Pointed out by Utkin, ¹⁴ the invariance means a sliding mode control system is insensitive to such matched uncertainties when the sliding mode is reached. On the other hand, the mismatched term $d_1$ is completely in the coronary artery system with no direct effects by the control channel, which is just indirectly effected by the control channel according to the dynamics (3).

In order to illustrate the validity of the designed control method, the mismatched term is set to $d_1=2\sin t$ due to its challenges of. Accordingly, the matched term is set to $d_2=0$ due to the invariance. Figure 4 demonstrates the phase plane trajectories of the uncertain system (3) under the effects of mismatched term, the error system (7) and the nominal system (4). From Figure 4, the uncertain coronary artery system (3) in Figure 4(a) can synchronize with the nominal system (4) in Figure 4(c) by the designed control method as proven in Theorem 1. Meanwhile, the tracking errors of the control system in Figure 4(b) can be asymptotically convergent to zero. Figure 5 displays that the designed control method can suppress the chaotic phenomena and overcome the mismatched uncertainties in the coronary artery system. The error of the disturbance observer in (11) is illustrated in Figure 6. The sliding surface in (15) and the control input in (19) are shown in Figure 7.

Figure 4.

Phase plane trajectories of the uncertain system (3) under the effects of mismatched uncertainties, the error system (7) and the nominal system (4). (a) Uncertain system, (b) error system, and (c) nominal system.

Figure 5.

Error states in (7) by the designed control method.

Figure 6.

Error of the disturbance observer in (11).

Figure 7.

Sliding surface and control input by the designed control method: (a) sliding surface and (b) control input.

The results in Figures 4 to 7 indicate that the designed control method possesses the ability of dealing with the adverse effects of the uncertainties in the coronary artery system. The method is able to synchronize the uncertain coronary artery system with the nominal one. In practice, therapy can be considered as a control input for any disease. Such method is potential to develop a machine or medicine for therapeutic purposes in order to synchronize chaotic blood vessels with a nominal one.

Conclusions

Abnormal and fatal chaos of coronary artery systems is able to induce the coronary artery disease. This paper has addressed the chaos-suppression problem by the sliding-mode-based design. The mathematic model of the coronary artery systems is described by dynamics. Some insight into the model has been illustrated by bifurcation diagram, information entropy analysis, phase plane trajectory, and Poincaré section. Due to the uncertainties the coronary artery systems tolerate, the disturbance-observer technique is employed to estimate the uncertainties. The smooth second-order sliding mode controller is also explored to achieve the chaos-suppression problem with the assistance of this observer. The stability of the closed-loop control system with the controller and the observer has been proven via the direct Lyapunov method. Finally, some numerical results have been displayed to verify the feasibility and validity of the presented control design. Concerning the therapeutic purposes, such design is potential and instructive to develop a clinical pharmacy or to invent a medical device. Patients are also able to benefit from such design a lot to avoid the risks of coronary artery disease.