Suppression of Chaotic Behaviors in a Complex Biological System by Disturbance Observer-based Derivative-Integral Terminal Sliding Mode

I. Introduction

BIOLOGICAL systems are diverse and they are different from each other. This paper focuses on a kind of biological systems, called coronary artery systems. The coronary artery systems are very complex and they are directly related with heart health. Coronary artery disease has dominated mortality for most of the past century worldwide [1]. Its treatments, both pharmaceutical and surgical, have become leading sectors of the healthcare economy. In 2015, coronary artery disease affected 110 million people and resulted in 8.9 million deaths [2]. In order to explore new treatment options, both the medical and engineering communities try to provide some insight into coronary artery disease.

From the aspect of medicine, coronary arteries act as the blood vessels that supply oxygen-rich blood to the heart muscle. In case plaque builds up inside the coronary arteries, the buildup of plaque gradually obstruct the coronary arteries. Such obstruction may lead to vascular spasm. The vasospasms constitute the basic cause of a variety of coronary artery disease. Some drugs have been developed via the mechanism [3]–[7].

In the field of engineering, a coronary artery system consists of the bio-mathematical model of blood vessels, where the vasospasms are the chaotic states of the blood vessels [8]. Since chaotic systems are sensitive to perturbations, the chaotic phenomena may result in fatal chaos in the coronary arteries [9]. The chaotic phenomena must be suppressed to avoid serious health problems and illness development. In order that treatments can be effective, the chaos suppression has to achieve state synchronization of the blood vessels with pathological changes and the normal blood vessels.

The problem of state synchronization of a chaotic system was proposed by Pecora and Carroll [10]. Since then, the solution of this problem has become emerging because of the potential applications. Many synchronization approaches have been developed for various systems. The approaches can be roughly cataloged into two classes, that is, linear or partial linearisation methods [11], [12] and nonlinear approaches [13], [14]. Especially, linear or partial linearisation methods are just appropriate for stable linear systems that are characterised by a few distinct peaks in their power spectrum.

Concerning the coronary artery system, its complex bio-system is inherently nonlinear and calls for nonlinear methods to achieve its state synchronization. The research topic has been paid more and more attention and some reports can be seen. Some synchronization methods have been reported. Gong et al. [15] developed a backstepping-based controller to synchronize the spastic vessel with a normal vessel. Li [9] designed a nonlinear tracking controller and the controller could robustly and adaptively drive a chaotic coronary artery system into the normal orbit. Wu et al. [16] considered the drug absorption time and presented the time-delay state feedback control synthesis.

As a nonlinear design tool, the methodology of sliding mode control (SMC) is advocated for its invariance, meaning that a SMC system on the sliding-mode stage is completely insensitive to parametric uncertainties and external disturbances under certain matching conditions [17]. The characteristic makes the SMC exhibit better performances than the ordinary conventional robust control methods. In practice, Lin et al. [18] have investigated that the SMC-based method can be carried out for the chaos suppression problem of coronary artery systems. Zhao et al. also have applied the higher-order sliding mode control [19] and the terminal sliding mode control [20] for the same chaos suppression problem.

One of the SMC design methods is entitled “terminal sliding mode control (T-SMC)”. Such a design method is characterized by its terminal sliding surface. Usually, the design method has faster convergence speed and higher steady-state accuracy than other SMC design methods [21]. As an extension, the integral T-SMC method introduces sign and fractional integral terminal sliding mode concepts and can track relative-degree-one systems with uncertainty and disturbance [22]. The integral T-SMC design method allows a system to start on the integral terminal sliding surface. Further, the derivative-integral T-SMC design method, a progress of the extension, is derived to track high-order systems because it can guarantee the exact estimation of finite error convergence time [23].

Motivated by the merits of the progress, this paper investigates the derivative-integral T-SMC design method for the chaos suppression problem in a coronary artery system. The purpose of the paper is to reduce the occurrence of coronary artery disease by suppressing the abnormal chaotic behaviors of the coronary artery system. Even when they have different initial conditions, the abnormal chaotic behaviors can still synchronize with the normally unstable periodic orbit. For this purpose, the derivative-integral T-SMC design method is adopted to achieve the behavior synchronization.

On the other hand, the coronary artery system suffers from external disturbances such as the perturbations of blood pressure and body temperature in reality. The external disturbances challenge the chaos suppression problem very much because they are neither known nor predictive. As proven, the disturbance-observer technique can effectively deal with disturbances with guaranteed robustness [24], [25]. Here the technique is employed in order to attack the challenging issue.

In the paper, both the derivative-integral T-SMC design method and the disturbance-observer technique are integrated to suppress the abnormal chaotic behaviors in the coronary artery system with disturbances. In the sense of Lyapunov, the stability of the synchronization control system is analyzed. Compared with some published results, the feasibility and effectiveness of the integration solution is illustrated via some numerical results.

The remainder of this paper is organized as follows. Section II introduces the bio-mathematical model of the coronary artery system. Section III formulates the control problem. The synchronization control via the derivative-integral T-SMC-based controller and the disturbance observer is derived in Section IV for the coronary artery system. To show the synchronization performance, numerical simulations are carried out in Sections V. Finally, some conclusions are drawn in Section VI.

II. Bio-mathematical Model

Inherently, coronary arteries are a kind of muscular blood vessels. Concerning a coronary artery system, the lumped-parameter model that describes its dynamics [18] has the form of

where $x_1$ is the inradius change of the vessel, $x_2$ means the pressure change in the vessel, $t$ indicates the time variable, $b$, $c$ and $\lambda$ denote lumped parameters of the coronary artery system and $E \cos\omega t$ represents a periodical disturbance term.

As one factor that results in myocardial infarction, coronary artery spasm is caused by partial or complete occlusion of coronary arteries [1]. Equation (1) reveals the medical phenomenon from the perspective of mathematics. In (1), the initial condition $[x_1(0)\ x_2(0)]^T = [0.2\ 0.2]^T$ is taken into consideration, where the parameters $E$ and $\omega$ are determined by 0.3 and 1, respectively.

Fig. 1 reveals the the bifurcation diagram of the coronary artery system (1) with respect to the change of $\lambda$, where another parameters are set by $b = 0.15$, $c = −1.7$, $E = 0.3$ and $\omega = 1$. Fig. 2 reveals the the bifurcation diagram of the coronary artery system (1) with respect to the change of $b$, where another parameters are set by $\lambda = -0.65$, $c = −1.7$, $E = 0.3$ and $\omega = 1$. Fig. 3 reveals the the bifurcation diagram of the coronary artery system (1) with respect to the change of $c$, where another parameters are set by $\lambda = -0.65$, $b = 0.15$, $E = 0.3$ and $\omega = 1$. All the bifurcation diagrams share the same initial condition $[x_1(0), x_2(0)]^T = [0.2,\ 0.2]^T$. The time series are set by 0, 2π, 4π, 6π, 10π and 12π, where the system period is 2π.

Figure  1.  Bifurcation diagrams with respect to the change of $\lambda$ from –1 to 0.

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Figure  2.  Bifurcation diagrams with respect to the change of $b$ from 0 to 1.

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Figure  3.  Bifurcation diagrams with respect to the change of c from –2 to –1.5.

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The bifurcation diagrams in Figs. 1–3 illustrates that the coronary artery system under some parameters is disordered and will lead to chaos. As far as the coronary artery system is concerned, the following lumped parameters are considered, that is, $\lambda = -0.65$, $b = 0.15$, $c = -1.7$, $E = 0.3$ and $\omega = 1$. With the group of lumped parameters, Fig. 4 illustrates the coronary artery system has very complex dynamics in the phase plane.

Figure  4.  Phase plane of the coronary artery system.

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From Figs. 1–4, the coronary artery system is apparently disordered with respect to the changes of these lumped parameters. The coronary artery system may descend into chaos under some initial conditions with respect to the change of the lumped parameters. The chaotic behaviors in clinic medicine indicate that the vascular spasm of the coronary artery system may exhibit a series of coronary artery disease, including but not limited to angina, myocardial infarction, and sudden cardiac death. The chaos extremely hazard health and it must be suppressed immediately. Since the chaos suppression problem of such a coronary artery system is challenging and interesting, this paper explores the problem from the aspect of control design via derivative-integral terminal sliding mode.

III. Control Problem Formulation

From the bio-mathematical model (1), the coronary artery system with uncertainties (2) can be drawn

Here ${ x} = [x_1,\ x_2]^T$ is defined as the state vector and $u(t)$ generated by the designed controller is the control input. The vector ${ d} = [d_1,\ d_2]$ describes the uncertainties, meaning unmodelled dynamics, external disturbances and structural variations.

Assumption 1: ${ d}$ is bounded, that is, $||{ d}||_\infty\leq d^*$, where $d^*$ is positive but unknown.

Assumption 2: ${ d}$ is slowly time varying, that is, ${ {\dot{d}}}\simeq {{\textit{0}}}_{2\times1}$, where ${{\textit{0}}}_{2\times1} = [0,\ 0]^T$.

From (2), the nominal system of the coronary artery system with no control input can be written by

Concerning the bio-mathematical model (1), (3) can display different dynamics, highly sensitive to its parameters. This fact means that even small differences of the parameters yield widely diverging outcomes of the chaotic behaviors. Consequently, the purpose of chaos suppression in the coronary artery system is to synchronize the uncertain coronary artery system (2) with the nominal system (3) via derivative-integral terminal sliding mode, that is

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

Considering (2), (3) and (4), the error system can be described by

Further, define ${\tilde x_1} = {e_1}$ and ${\tilde x_2} = {\dot e_1}$ and differentiate ${\tilde x_1}$ and ${\tilde x_2}$ with respect to the time variable $t$. Then, substituting (5) into $\dot{{\tilde x}}_2$ has the form of

According to Assumption 2, ${{\dot d}_1}\approx 0$ in (6). Then, (6) can be written by (7) in the form of state space.

Here $\tilde {{ x}} = [\tilde{x}_1,\ \tilde{x}_2]^T$, $d = - c{d_2} + \lambda \left( {1 + c} \right){d_1}$, $H\left( {\tilde {{ x}}} \right) = {\tilde x}_1$, B = $[0,\ -c]^T$, $F\left( {\tilde {{ x}},d} \right) = [{\tilde x}_2,\ f(\tilde {{ x}}) + d]^T$ and $f(\tilde {{ x}}) = \lambda \left( {c - b} \right){{\tilde x}_1} -$ $\left( {b + \lambda + c\lambda } \right){{\tilde x}_2} - \lambda c\tilde x_1^3 - 3c{x_1}{{\bar x}_1}{{\tilde x}_1}$.

Assumption 3: The functions $F(\tilde{{ x}}, d)$ and $H(\tilde{{ x}})$ are sufficiently smooth so that all Lie derivative calculations are well-defined.

Proven by Chiu [23], the derivative-integral T-SMC design can be available for high relative-degree systems that have higher relative degree than 1. According to Assumption 3, the relative degree of (7) can be calculated as (8), (9) and (10) in order to develop the derivative-integral T-SMC design for (7).

From (8), (9) and (10), the relative degree of (7) can be determined as 2 so that it is possible to develop a derivative-integral terminal sliding mode controller for the chaos suppression [23]. Furthermore, the input-output dynamics can be expressed as

Here $L_F^2H\left( {\tilde {{ x}},d} \right) = \dfrac{{\partial \left( {{L_F}H\left( {\tilde {{ x}},d} \right)} \right)}}{{\partial \tilde {{ x}}}}F = f\left(\tilde {{ x}}\right) + d$. Then, (11) can be written by

IV. Control Design

A Derivative-Integral T-SMC Design

This paper investigates the derivative-integral T-SMC method for the chaos suppression problem. The so-called derivative-integral terminal sliding mode means that the derivative and integral terms exist in a sliding surface. Motivated by this purpose, the following sliding surface $s$ is taken into consideration.

Here $\alpha>0$ is a design parameter, ${e_{D1}} = {\dot e}_{D0}^{{p_{11}}/{q_{11}}} + {\beta}{e_{D0}}$, ${e_{D0}} = {e_1}$, ${e_{I1}} =\displaystyle\int_0^te_{D1}^{{q_{21}}/{p_{21}}}(\tau)d\tau$ and ${e_{I1}}(0) = - \dfrac{{e_{D1}}(0)}{{\alpha}}$, where $\beta>0$ is pre-defined and $p_{11} > q_{11}>0$ chosen by designers are odd integers, as well as $p_{21} > q_{21}>0$.

Theorem 1: If the sliding surface is defined as (13), then $[\tilde{x}_1,\ \tilde{x}_2]^T$ in (6) will reach ${{\textit{0}}}_{2\times1}$ in the finite convergence time $\tau_{DI}$, where $\tau_{DI}$ is formulated by

Here $t_{11}$ is the reaching time of the sliding mode $e_{D1} = 0$.

Proof: From (13), the sliding mode starts at $t = 0$. From then on, ${e_{D1}} = - {\alpha}{e_{I1}}$ can always hold true via control design. Subsequently, substituting ${e_{D1}} = - {\alpha}{e_{I1}}$ into ${\dot e_{I1}} = e_{D1}^{{q_{21}}/{p_{21}}}(t)$ yields

where ${e_{I1}}(0) = - {e_{D1}}(0)/{\alpha}$.

Solving (15), the convergent time of $e_{I1}$ can be gotten as

On the sliding surface $s = 0$, ${e_{D1}} = - {\alpha}{e_{I1}}$ indicates that the sliding mode of $e_{D1}$ takes place at the same time $t_{11}$ to zero. When $e_{D1} = 0$, $e_{D0}$ will successively converge to zero. At $t = t_{11}$, $e_{D0}$ can be formulated by

From (17), the time spent from $e_{D0}(t_{11})$ to $e_{D0} = 0$ can be calculated as

According to (16) and (18), the time spent from $s(0) = 0$ to $e_1 = 0$ is their summation because the integral and derivative terms in (13) are independent. Consequently, the finite convergence time can have the form of (14), that is, $\tau_{DI} = t_{11}+t_{01}$. ■

In order to obtain the derivative-integral T-SMC law, a Lyapunov candidate is selected as $V_0 = \dfrac{1}{2}s^2$. Differentiate $V_0$ with respect to the time variable $t$. $\dot{V}_0 = s\dot{s}$ can be gotten. Further, the derivative of $s$ can be formulated by

Substituting (12) into (19) yields

where $\psi = {\beta}{\dot e_{D0}} + {\alpha}{\left( {\dot e_{D0}^{\textstyle \frac{{p_{11}}}{{q_{11}}}} + {\beta}{e_{D0}}}\right)^{\textstyle \frac{{{q_{21}}}}{{{p_{21}}}}}}$.

In the sense of Lyapunov, a derivative-integral T-SMC law can be deduced from $\dot{V}_0<0$, determined by

where $A = \left\{ {\begin{aligned} & {\dot e_{_{D01}}^{\left( {\frac{{p_{11}}}{{q_{11}}} - 1} \right)}, \quad \left| {\dot e_{_{D01}}^{\left( {\frac{p_{11}}{q_{11}} - 1} \right)}} \right| \ge\varepsilon } \\ & \varepsilon , \quad\quad\quad\;\; {\rm{otherwise}} \end{aligned}} \right.$, $\kappa>0$ and $\eta>0$ are chosen by designers, $d_0^* = [|c|+|\lambda (1+c)|]d^*$ and $k_e>0$ is also a design parameter, determined later.

Concerning the expression of $A$, a parameter $\phi>0$ exists such that

In order to have $\dot{V}_0<0$, ${k_e} > \dfrac{1}{1 - \phi}$ is picked up.

Replace $u$ by (21) in (20) and consider the condition (22). $\dot{V}_0<0$ can be guaranteed if $\kappa>d_0^*$ holds true. Unfortunately, $d^*$ in Assumption 1 is unknown so that $d_0^*$ is also unknown, meaning that designers can not guarantee $\dot{V}_0<0$ by selecting an appropriate value of $\kappa$.

From another conservative perspective, the value of $\kappa$ can be assigned as large as possible in order to have the guaranteed stability. However, such a solution is not prosperous because a large value of $\kappa$ may induce the chattering phenomenon and hazard the control performance.

B Disturbance Observer-based Derivative-Integral T-SMC Design

To deal with issue raised by the unknown $d^*_0$, a disturbance observer is adopted in order that it can estimate $d^*_0$. Meanwhile, the control system can also possess the guaranteed stability as well as the good control performance. The disturbance observer is formulated by

Here $z \in {\mathbb R} ^{1\times1}$ is the internal state variable of the observer, $\hat d \in {\mathbb R} ^{1\times1}$ is the the estimate of $d$, ${\mathbb L}\in {\mathbb R} ^{1\times2}$ pre-defined by designers is the observer gain vector, the matrix ${\mathbb A}$ is written by $\left[ {\begin{array}{*{20}{c}} 0&1 \\ {\lambda \left( {c - b} \right)}&{ - \left( {b + \lambda + c\lambda } \right)} \end{array}} \right]$, the vectors ${\mathbb B}_1$, ${\mathbb B}_2$ and ${\mathbb M}$ are determined by $\left[ {\begin{array}{*{20}{c}} 0 \\ { - c} \end{array}} \right]$, $\left[ {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right]$ and $\left[ {\begin{array}{*{20}{c}} 0 \\ { - \lambda c\tilde {{x}}_1^3 - 3c{x_1}{{\bar{ {x}}}_1}{{\tilde {{x}}}_1}} \end{array}} \right]$.

Define an estimate error as

Assumption 4: $e_d$ is bounded, that is, ${\left| {{e_d}} \right|} \leq e_d^*$, where $e_d^*>0$ is unknown.

The derivative of the estimate error can be written by

From Assumption 2, the assumption indicates that the uncertainties in (2) are slowly time varying compared with the dynamic characteristics of the observer (23). Further, it is concluded that $\dot d \approx 0$ from Assumption 2. Finally, (26) can be obtained by substituting (23) into (25).

The solution of (26) can be calculated by

Select a vector ${\mathbb L}$ such that the eigenvalue of ${\mathbb L}{{\mathbb B}_2}$ is a positive constant. This selection means that the estimate error $e_d$ is able to exponentially converge as $t \to \infty$. Since $e_d$ is exponentially convergent from (27), Assumption 4 is mild. Consequently, the unknown $d_0^*$ in (21) can be replaced by the disturbance observer output $\hat d$, that is, the estimate of $d$.

Theorem 2: Consider the bio-mathematical model of the coronary artery system (2); take Assumptions 1–4 into account; define the error system (5) and have the input-output dynamics (12); formulate the derivative-integral terminal sliding surface (13); design the disturbance observer (23). If the derivative-integral T-SMC law is formulated by (28), the closed-loop chaos suppression system is then asymptotically stable in the presence of uncertainties.

where $\kappa> e_d^*$.

Proof: The aforementioned Lyapunov candidate function $V_0 = \dfrac{1}{2}s^2$ are taken into consideration again. Differentiate $V_0$ with respect to the time variable $t$ and replace $\dot s$ by (20). Then, we can obtain

Substituting (28) into (29) yields

Owing to ${p_{11}} > 0$, ${q_{11}} > 0$ and ${p_{11}}>{q_{11}}$, $\dfrac{{{p_{11}}}}{{{q_{11}}}} > 1$ exists such that $\dot e_{D01}^{\left( {\frac{p_{11}}{q_{11}} - 1} \right)}>0$ holds true in (30) for all $\dot e_{D01}^{\left( {\frac{p_{11}}{q_{11}} - 1} \right)} \ne 0$. Consequently, the first and second terms in (30) can have the form of

In (31), ${k_e}>(1-\phi)$ can be picked up in advance such that $s\psi - s \dot e_{D0}^{\left( {\frac{{p_{11}}}{{q_{11}}} - 1} \right)} {k_e}\left\| \psi \right\|{A^{ - 1}}\frac{s}{\left\| s \right\|}<0$.

Further, the third term in (30) can be written by

In (32), $\kappa > e_d^*$ can be selected by designers such that $s \frac{{{p_{11}}}}{{{q_{11}}}}\dot e_{D0}^{\left( {\frac{{p_{11}}}{{q_{11}}} - 1} \right)}\left[ d-\hat d - \left(\kappa\operatorname{sgn} (s) + \eta s \right) \right]<0$.

From (31) and (32), $\dot V_0 <0$ holds true by the derivative-integral T-SMC design, that is, the control design can asymptotically stabilize the uncertain coronary artery system by the chaos suppression. ■

From Assumption 4, $e_d^*$ is unknown as well, indicating that $\kappa$ in (28) has to be assigned in advance. From the perspective of the system stability, the value of $\kappa$ should be large enough. In this sense, (28) makes no difference over (21) at a first glance. However, $e_d$ is able to exponentially converge via the disturbance observer (23). It is concluded that its maximum exists at $t = 0$, i.e., $e_d(0)$. Correspondingly, $d$ is just bounded according to Assumption 1. It is completely unknown when, where and how many its maximum is. Consequently, $\kappa$ in (21), associated with $d_0^*$, should be more conservative in order to guarantee the stability, while $\kappa$ in (28), concerned to $e_d^*$, can contribute to the alleviation of the chattering phenomenon.

V. Simulation Results

This section presents some numerical simulation results that illustrate the feasibility and validity of the proposed strategy in comparison with a benchmark. Concerning the uncertain system (2) and the nominal system (3), the lumped parameters are fixed to $\lambda = −0.65$, $b = 0.15$, $c = −1.7$, $E = 0.3$ and $\omega = 1$, which are kept unchanged from the model presented by Zhao et al. [19], [20]. The initial states are assigned to $[\bar{x}_1(0),\ \bar{x}_2(0)]^T = [0.2,\ 0.2]^T$, which are also the same as the initial states presented in [19], [20]. Considering the derivative-integral T-SMC law, the controller parameters are determined by $\alpha = 2$, $\beta = 2$, $p_{11} = p_{21} = 11$, $q_{11} = q_{21} = 13$, $k_e = 2$, $\eta = 0.1$ and $\kappa = 2$. As far as the disturbance observer is concerned, its sole parameter ${\mathbb L}$, the gain vector, is given by $[0,\ 5]^T$.

A Matched Uncertainties

The uncertain system (2) contains two disturbance sources, that is, $d_1$ and $d_2$. From the viewpoint of sliding mode control, the two disturbances have different characteristics that $d_2$ is matched and $d_1$ is mismatched. Further, $d_1$ and $d_2$ are merged into $d$ in (6). From the expression of $d$, both $u$ and $d_2$ share the same coefficient $-c$. This fact indicates that $d_2$ is matched. Meanwhile, $d_1$ can hardly be matched because its coefficient $\lambda (1+ c)$ is almost impossible to be $-c$.

Pointed out by Utkin [17], the matched uncertainties enter the coronary artery system through the control channel $u$, so that the control system is insensitive to the uncertainties when the sliding mode is reached. Fig. 5 displays the proposed strategy can effectively suppress the chaotic phenomena under the effects of matched uncertainties, where $d_2$ is set by $2\sin(t)$ and $d_1$ is assigned to $0$.

Figure  5.  Error states and control input in (7) in comparison with the benchmark under the effects of matched uncertainties.

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The sliding surface variable is shown in Fig. 6. From Fig. 6, the sliding mode of the proposed strategy can be reached at the limited time as proven in Theorem 1. Thereafter, the coronary artery system can suppress the uncertainties in light of the invariance of SMC [17]. Consequently, the chaos suppression by the proposed strategy in Fig. 5 has the best performance, compared with the benchmark and the derivative-integral T-SMC without observer.

Figure  6.  Sliding surface variable in comparison with the benchmark under the effects of matched uncertainties.

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Fig. 7 depicts the phase plane trajectories of the error system (7), the nominal system (3) and the uncertain system (2). As proven in Theorem 2, the uncertain coronary artery system with matched uncertainties can synchronize the nominal coronary artery system by the proposed strategy in Fig. 7, while the tracking errors can be convergent to zero in the limited time given in (14).

Figure  7.  Phase plane trajectories of the error system (7), the nominal system (3) and the uncertain system (2) under the effects of matched uncertainties.

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Further, the estimate error of the disturbance observer is given in Fig. 8. In Fig. 8, the estimate error is very large at the outset. But it can be exponentially convergent till the observer output can track the disturbances. The exponential convergence of $e_d$ makes the control performance of the proposed strategy superior to that of the sole derivative-integral T-SMC without observer.

Figure  8.  Estimate error of the disturbance observer under the effects of matched uncertainties.

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B Mismatched Uncertainties

Since the mismatched uncertainties can not be suppressed by the invariance of SMC, it is indispensable to illustrate their effects on the control performance. Here the controller parameters are kept unchanged, as well as the observer gain and the initial conditions. The sole change is that $d_1$ is set by $2\sin(t)$ and $d_2$ is assigned to $0$.

Fig. 9 shows that the proposed strategy can also suppress the chaotic phenomena even if the mismatched uncertainties exist in the coronary artery system. Although the invariance of SMC makes no difference to the mismatched uncertainties, the proposed strategy can deal with them via the integration of the derivative-integral terminal sliding mode controller and the disturbance observer.

Figure  9.  Error states and control input in (7) with mismatched uncertainties under the effects of mismatched uncertainties.

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The sliding surface variable is demonstrated in Fig. 10. Compared with Fig. 6 and Fig. 10, the sliding mode of the proposed strategy can be reached at the shortest time and keep on sliding on the surface even if the mismatched uncertainties have adverse effects.

Figure  10.  Sliding surface variable under the effects of mismatched uncertainties.

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Fig. 11 shows the phase plane trajectories of the error system (7), the nominal system (3) and the uncertain system (2) under the effects of mismatched uncertainties. Shown in Fig. 11, the coronary artery system with mismatched uncertainties can achieve the synchronization with the nominal coronary artery system as proven in Theorem 2.

Figure  11.  Phase plane trajectories of the error system (7), the nominal system (3) and the uncertain system (2) under the effects of mismatched uncertainties.

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The estimate error of the disturbance observer is given in Fig. 12. Similar to the curve in Fig. 8, the estimate error is very large at the outset but it exponentially converge as $t\to \infty$.

Figure  12.  Estimate error of the disturbance observer under the effects of mismatched uncertainties.

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No matter what kinds of the uncertainties are, the proposed strategy in Figs 5–12 can solve the adverse effects of the uncertainties and synchronize the uncertain coronary artery system with the nominal one. In clinical medicine, the solution can contribute to the reduction and elimination of chaotic vasospasms, where the mechanism is to synchronize any chaotic blood vessels with a nominal one. The proposed strategy can also benefit the research and development of clinical pharmacy for therapeutic purposes because medications can be treated as a kind of control input to suppress the chaotic vasospasms.

In the field of medicine, coronary artery disease in the early stage is often treated with pharmacies, meaning that a suitable pharmacodynamics has to be drawn for patients. Concerning the victims of advanced coronary artery disease, they have to suffer a lot from invasive procedures such as stent and bypass. These invasive procedures may result in extra pains and risks for patients.

The proposed strategy can benefit the research and development of clinical pharmacy and medical device for therapeutic purposes because therapies can be treated as a kind of control input to suppress the chaotic vasospasms. 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 the potential application of the presented control design.

VI. Conclusions

Coronary artery disease can be induced by blood vessel spasms. Since the vasospasms can be considered as chaos in the aspect of engineering, this paper has explored the chaos suppression problem of coronary artery systems. In order to have some insight into the bio-mathematical model of coronary artery systems, their nonlinear behaviors are presented by a series of bifurcation diagrams with respect to the changes of lumped parameters. The degree of complexity is described by the phase plane trajectory of coronary artery systems. The purpose is to synchronize any chaotic coronary artery with a nominal one. Motivated by the purpose, the method of derivative-integral terminal sliding mode control is taken into consideration. Since coronary artery systems suffer from uncertainties, the uncertainties trouble the control design. To attack this issue, the technique of disturbance observer is employed. The proposed strategy integrates the derivative-integral terminal sliding mode controller and the disturbance observer to achieve the chaos suppression. In the sense of Lyapunov, the stability of such an integrated control system has been proven. The simulation results have demonstrated the effectiveness and feasibility of the proposed strategy via some comparisons. The proposed strategy is beneficial to the research and development of clinical pharmacy.