A novel nonsingular integral terminal sliding mode control scheme in epilepsy treatment

Abstract

In this paper, a closed-loop brain stimulation control problem is investigated using the nonsingular integral terminal sliding mode (NITSM) control approach. First, the thalamocortical model of epilepsy seizure is given, which is composed of the cortical PY-IN subnetwork and the subcortical RE-TC subsystem. Then, a nonsingular integral terminal sliding mode surface is designed utilizing the derived output tracking error, and the stability of the sliding mode dynamics is proved by Lyapunov approach. Furthermore, a disturbance observer (DOB) based NITSM controller design approach is proposed for the established thalamocortical model, and the reachability of the closed-loop control system under the designed controller is analyzed using Lyapunov theory. Finally, simulation results are given to illustrate the effectiveness and superiority of the designed control scheme.

Keywords

Closed-loop brain stimulation epilepsy seizure nonsingular integral terminal sliding mode (NITSM)disturbance observer (DOB)

Introduction

Epilepsy is neurological disease that bothers the whole world very much (Jonathan et al., 2014; Shenoy and Carmena, 2014). About 50 million people worldwide have epilepsy and many patients suffer from the disease. Moreover, the disturbing is that many aspects of epilepsy patients’ lives are more or less affected by epilepsy, such as making friends, working, getting married and so on (Blumcke et al., 2017; Riasi et al., 2014). It cannot be denied that the successful development of effective epilepsy therapies is of extremely important significance and great benefit to epilepsy patients worldwide. At present, epilepsy patients all over the world still mainly use drugs to relieve the pain of seizures, and sometimes they are forced to undergo surgical treatment. However, many medical investigations show that epilepsy drugs are not effective in curing epilepsy and have a lot of adverse effects. For example, it is very easy to damage the function of body organs, especially the liver, if the patient takes too much epilepsy drugs. While the other normal treatment method, surgical treatment is not suitable for patients with general epilepsy either. It requires doctors with very high surgical skills since it is very dangerous and any small deviation or accident in surgical treatment process will cause permanent and irreversible damage to the human brain. Therefore, compared with the above two schemes, the brain stimulation therapy has distinct advantages in terms of treatment effect and safety (Kozak and Berenyi, 2017).

Many brain stimulation therapies are originally developed by adapting some control schemes used in clinical treatment. It has been proved that the deep brain stimulation (DBS) therapy has better stimulation effects on many neurological diseases, including epilepsy, primary dystonia, essential tremor, medication intractable Parkinson’s disease, and so on (Jonathan et al., 2014; Shenoy and Carmena, 2014). The DBS therapy can be interpreted as inserting electrodes into specific nucleus of the brain, then these electrodes release therapeutic electrical stimulation signals, mainly continuous high-frequency pulses, to inhibit abnormal brain nuclei discharge, thereby achieve a therapeutic effect. DBS can be divided into two types, and the open-loop brain stimulation is the most common followed by the closed-loop brain stimulation therapy. According to a large number of medical investigations, it is found that the open-loop brain stimulation technology in seizure abatement cannot stably and effectively suppress the pathological peak discharge, so a more efficient, stable and applicable closed-loop control strategy is needed (Johnson et al., 2016; Saillet et al., 2013). Shortening seizure time and decreasing seizure intensity are important signs of seizure relief, which can be achieved using the high-frequency pulse stimulation string released by brain stimulation device. However, the study on closed-loop control strategy for brain stimulation is still in its infancy and there is very few results (Kozak and Berenyi, 2017). Therefore, the task of this study is to design an efficient closed-loop brain stimulation control scheme for epilepsy seizure abatement (Ge et al., 2019), which can continuously adjust the stimulus signals online by the wired or wireless connection between the computer and the brain.

There is a cerebral cortex area with normal background activity when an epilepsy patient is healthy. The normal neural electric signals generated by healthy brain neural activity are collected by recording electrodes, and then a controller is developed so that the thalamocortical area with pathological characteristics can learn the behavior of normal neural electric signals, that is, the stimulation signals can make the patient return to normal from the diseased state. The smaller gap between pathological dynamics and normal dynamics, the better therapeutic effect for seizures abatement. Therefore, the reduction or even elimination of seizures can be transformed into tracking control problem of the thalamocortical system.

In recent years, the sliding mode control technique has been widely concerned and developed rapidly for its robustness against the uncertainties of parameters and the dynamics of disturbances on the specified sliding surface (Miao et al., 2019; Songa and Lewis, 2020; Zhang et al., 2016). For example, the robust closed-loop controller proposed in the Ge’s study is suitable for the neural system with nonlinearities, loss of model dynamics and disturbances, and has a relatively good control effect on seizures. However, the slow tracking speed is its obvious disadvantage in the closed-loop control scheme (Ge et al., 2019). The terminal sliding mode control strategy can be employed to raise tracking rate (Labbadi and Cherkaoui, 2019; Mobayen, 2018), while the integral sliding mode technique plays an important role in improving control accuracy (Zhang et al., 2016; Riani et al., 2018). Recently, observer-based sliding mode control technique has also obtained some remarkable results, and the observer is applied in approximating the unmeasured states, uncertainties, and disturbances of controlled systems (Rabiee et al., 2019; Tuan, 2019; Tuan, et al., 2021). Considering the unmodeled dynamics of the thalamocortical system, ubiquitous electromagnetic disturbances, and the oscillations of seizures, an applicable disturbance observer can be adopted to estimate the effect of lumped disturbances. Here the considered lumped disturbances contain the practical unmodeled dynamics as well as electromagnetic disturbances.

Motivated by the above discussion, a closed-loop brain stimulation control scheme is developed for Taylor’s epilepsy thalamocortical model (Taylor et al., 2014) in this paper, which guarantees that thalamocortical signal of seizures quickly and accurately converges to the normal level. In light of the specific finite-time stability, a nonsingular integral terminal sliding mode controller on the basis of a disturbance observer is proposed in this study, and the satisfactory convergence speed and tracking precision of the thalamocortical system can be furthermore achieved. Compared with other recently proposed approaches and results, the main contributions of this control scheme are summarized as follows:

Based on epilepsy patients’ thalamocortical mechanism model, a novel closed-loop brain stimulation control scheme is proposed. Compared with most open-loop treatment strategy, our closed-loop control scheme can greatly improve the effectiveness of epileptic seizures abatement.

To eliminate the effect of lumped disturbances, a disturbance observer (DOB) is designed for the thalamocortical system. The lumped disturbances including the unmodeled dynamics and electromagnetic disturbances are estimated accurately so that the behavior of thalamocortical area with pathological characteristics tracks normal neural electric signals.

The DOB-based nonsingular integral terminal sliding mode controller is designed in this study, which guarantees that EEG signal returns to the normal level within a preset time. Moreover, the required convergence speed and tracking performance can be improved. Therefore, in terms of the timely and quick online response for seizures, the proposed control scheme is more suitable for clinical treatment of epilepsy than the other recently research results.

Noting that $| Λ |$ means the norm of $Λ$ , where $Λ$ is a matrix or a constant.

Problem formation

Neural mass models can well characterize the macroscopic dynamics of neuronal networks and provide a potential way to illustrate the mechanisms underlying absence seizures. In this paper, we use a macroscopic model of thalamocortical system consisting essentially of the synaptic circuitry of thalamus and cortex to investigate the effects of stimulation on SWDs. Figure 1 shows the connectivity scheme of the thalamocortical system, where the cortical subsystem includes an excitatory pyramidal neuronal population $(PY)$ and an inhibitory interneuronal population $(IN)$ , and the subcortical thalamus subsystem consists of the specific relay nucleus $(TC)$ and thalamic reticular nucleus $(RE)$ . The interactions between different populations are represented by connectivity parameters $c_{ee}, c_{ei}, c_{et}, c_{ie}, c_{te}, c_{tr}, c_{re}, c_{rt}$ , and $c_{rr}$ . All populations are interconnected in agreement with experimentally known connections.

Thee interactions within the thalamocortical system are described as the following differential equations (Taylor et al., 2014)

{\begin{matrix} \frac{dTC}{dt} = τ_{t} (h_{t} - TC + c_{te} f [PY] - c_{tr} s [RE]) \\ \frac{dRE}{dt} = τ_{r} (h_{r} - RE + c_{re} f [PY] + c_{rt} s [TC] - c_{rr} s [RE]) \\ \frac{dIN}{dt} = τ_{i} (h_{i} - IN + c_{ie} f [PY]) \\ \frac{dPY}{dt} = τ_{e} (h_{e} - PY + c_{ee} f [PY] - c_{ei} f [IN] + c_{et} f [TC]) \end{matrix}

(1)

where $PY$ , $IN$ , $TC$ , and $RE$ are the state variables, representing the fractional firing activity in each neuronal population. $h_{e, i, t, r}$ are input parameters and $τ_{e, i, t, r}$ are time scale constants mediated by different excitatory and inhibitory neuro-transmitters. $f [x] = 1 / (1 + ε^{- x})$ is the sigmoid transition function describing the cortical dynamics, and $s [x] = α x + β$ is the linear activation function describing the thalamic subsystem. $c_{ee}, c_{ei}, c_{et}, c_{ie}, c_{te}, c_{tr}, c_{re}, c_{rt}$ , and $c_{rr}$ are the connectivity strengths between different neuronal populations.

Both the unmodeled dynamics and electromagnetic signals lead to extensive lumped disturbances in the brain stimulation control system, which has a huge impact on neural electric signals and network connections. Therefore, the interaction of the thalamic cortex system of the epilepsy patient can be mathematically modeled by describable differential equations as follows (Ge et al., 2019; Taylor et al., 2014)

{\begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + Bu (t) + g (x (t)) + η (x, t) \\ y (t) = Cx (t) \end{matrix}

(2)

where $x = [x_{1}, x_{2}, x_{3}, x_{4}]^{T} = [TC, RE, IN, PY]^{T}$ represents the system state variable and $x \in R^{4}$ , $u \in R^{r \times 1}, 0 < r < 4$ is the control input and represents external stimulus signal applied to the brain. $y \in R^{m}$ is the output and represents the discharge state of the thalamocortical system, the output matrix $C = [c_{1}, c_{2}, c_{3}, c_{4}] \in R^{m \times 4}$ is a matrix with row full rank. $B \in R^{4 \times r}$ is the control matrix, $η \in R^{4}$ represents the lumped disturbances comprising all uncertainties and electromagnetic disturbances. The uncertainties caused by the internal parameter variations of the thalamocortical system, include the cerebral blood flow, glucose metabolism, blood oxygen level dependence, oscillations of seizures, and so on. The electromagnetic disturbances caused by the external disturbances of the thalamocortical system are composed of the power line disturbance, heart beating, or muscle movement electromagnetic disturbance, such as blink and swallow. The known nonlinear term $g \in R^{4}$ is as follows

g (x) = [\begin{matrix} τ_{t} (h_{t} - c_{tr} β + c_{te} {(1 + ε^{- x_{4}})}^{- 1}) \\ τ_{r} (h_{r} + (c_{rt} - c_{rr}) β + c_{re} {(1 + ε^{- x_{4}})}^{- 1}) \\ τ_{i} (h_{i} + c_{ie} {(1 + ε^{- x_{4}})}^{- 1}) \\ τ_{e} (h_{e} + c_{et} {(1 + ε^{- x_{1}})}^{- 1} - c_{ei} {(1 + ε^{- x_{3}})}^{- 1} \\ + c_{ee} {(1 + ε^{- x_{4}})}^{- 1}) \end{matrix}],

A = [\begin{matrix} - τ_{t} & - τ_{t} c_{tr} α & 0 & 0 \\ τ_{r} c_{rt} α & - (τ_{r} + τ_{r} c_{rr} α) & 0 & 0 \\ 0 & 0 & - τ_{i} & 0 \\ 0 & 0 & 0 & - τ_{e} \end{matrix}] .

where $τ_{i}, τ_{t}, τ_{e}, τ_{r}$ represent the proportional constant mediated by different neurotransmitters. $c_{ie}, c_{te}, c_{tr}, c_{ee}, c_{ei}, c_{et}, c_{re}, c_{rt}$ , and $c_{rr}$ represent the connection strength between different parts of different neurons. The linear activation function of the thalamus subsystem is defined by the appropriate $α, β$ and the cortical dynamics is expressed by the appropriate constant $ε > 0$ . More detailed parameters explanation can be checked in Taylor et al. (2014). Here, we use a normal minicolumn to measure the desired output $y_{d} \in R^{m}$ .

The following Assumptions and Lemmas are introduced, which will be used for deriving our main results in the sequel.

Assumption 1. (Ge et al., 2019) The matrix pair $(A, B)$ is controllable and the generalized inverse matrix of $CB$ exists.

Assumption 2. The vector form of the system output is $y (t) = [y_{1} (t), \dots, y_{m} (t)]^{T}$ and the initial value $y_{i} (0) (i = 1, \dots, m)$ is a positive constant.

Lemma 1. (Zhang et al., 2014) Consider the system: $\overset{\cdot}{x} = f (x, t), f (0, t) = 0, x \in U \subset R^{n}$ , where $f (\cdot) : R^{n} \to R^{n}$ is continuous. Assume that there exists a Lyapunov function $V (x)$ in the range of the neighborhood $U \subset R^{n}$

\overset{\cdot}{V} (x) + λ V^{α} (x) \leq 0, \forall x \in U \ {0}

(3)

where $λ > 0, 0 < α < 1$ . Then, a finite time stable equilibrium point of the system is the origin. The time needed for the system to reach equilibrium $V (x) \equiv 0$ is defined as $Treach$ , then system (3) is finite time stable, and the reach time can be expressed as

T_{reach} \leq \frac{V^{(1 - α)} (x_{0})}{λ (1 - α)}

(4)

Lemma 2. (Shen et al., 2015) Consider the system: $\overset{\cdot}{x} = f (x, t), f (0, t) = 0, x \in U \subset R^{n}$ , where $f : U \times R^{+} \to R^{n}$ is continuous on an open neighborhood $U$ of the origin. If there is a Lyapunov function $V (x)$ with the characteristics of the value in the domain of the origin, and it satisfies that

\overset{\cdot}{V} (x) + λ_{1} V (x) + λ_{2} V^{α} (x) \leq 0

(5)

then the system $\overset{\cdot}{x} = f (x, t)$ is locally stable in a finite time, and the settling time is

T \leq \frac{1}{λ_{1} (1 - α)} \ln \frac{λ_{1} V^{(1 - α)} (x_{0}) + λ_{2}}{λ_{2}}

(6)

where $λ_{1} > 0, λ_{2} > 0$ , $α \in (0, 1)$ .

Lemma 3. (Zirkohi, 2019) Young’ inequality: assume that $a, b$ is non-negative real scalars, $p > 1$ , $\frac{1}{p} + \frac{1}{q} = 1$ , then

ab \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}

(7)

where the equal sign holds if and only if $a^{p} = b^{q}$ .

Remark 1. The cortical macroscopic dynamics are described by the simulated electroencephalogram (EEG) signal $y (t)$ , which is calculated as the combination of an excitatory pyramidal neuronal population (PY) and an inhibitory interneuronal population (IN) in cortex, that is $y = c_{3} IN + c_{4} PY$ with the scaling factors $c_{4} \geq c_{3} > 0$ and $c_{3} + c_{4} = 1$ (Ge et al., 2019; Taylor et al., 2014). Although pyramidal cells have a greater open dipolar field influence than interneurons to the surface EEG (Markram et al., 2015), a general demonstration of the effects of $c_{3}$ and $c_{4}$ given in Figure 1 shows that, the shape of the simulated EEG $y (t)$ in different cases is almost consistent. For comparison with previous results (Ge et al., 2019), the parameters are chosen as $c_{1} = c_{2} = 0, c_{3} = c_{4} = 0.5$ in the following simulations.

Main results

The objective of this article is to design a DOB-based integral sliding mode control strategy such that all the tracking errors of system (2) can converge to zero in presence of unmodeled dynamics and electromagnetic disturbances. The closed-loop brain stimulation sliding control scheme is shown in Figure 2.

Design for sliding mode surface

In this part, a novel sliding manifold is presented for system (2), and the stability and convergence of its tracking control system are theoretically proved.

For an epilepsy patient, if the discharging activities of system (2) return to the normal level within a predefined time, the control objective of seizure abatement achieves. Meanwhile, the discharging activities can be described by the output of system (2) as well as the desired output $y_{d}$ represents the normal discharging activities. It is assumed that $y_{d}$ is differentiable and the tracking error is defined as $e (t) = y (t) - y_{d} (t)$ , then the differential equation of tracking error can be written as

\overset{\cdot}{e} = C (Ax + Bu + g + η) - {\overset{\cdot}{y}}_{d}

(8)

In order to achieve the goal that the dynamics of the thalamocortical cortex can track the desired normal activities well, namely $y \to y_{d}$ , a new sliding variable is introduced

s = e (t) + \int_{0}^{t} (μ_{1} si g^{1 + \frac{1}{γ}} (e) + μ_{2} si g^{1 - \frac{1}{γ}} (e)) dt

(9)

where $s \in R^{m}$ , $γ > 1$ is a designed constant, $μ_{1} = diag {μ_{11}, \dots, μ_{1 m}}$ , $μ_{2} = diag {μ_{21}, \dots, μ_{2 m}}$ , $μ_{ij} > 0 (i, j = 1, \dots, m)$ , $si g^{γ} (e) = [| e_{1} |^{γ} sign (e_{1}), \dots, | e_{m} |^{γ}$ $sign (e_{m})]^{T}$ are m-dimensional vectors. Note that $s = 0$ is referred to as a sliding mode surface (Shtessel et al., 2014).

Theorem 1. Consider the control system (8). On Assumptions 1 and 2, the sliding motion of system (8) corresponding to the sliding surface (9) is finite time stable. The time needed for the system (8) to reach equilibrium $T_{1}$ is given by

T_{1} \leq \frac{γ | e {(0) |}^{\frac{1}{γ}} 2^{\frac{1}{2 γ}}}{{\underline{μ}}_{2}}

(10)

where {\underline{μ}}_{1} = \min_{i = 1, \dots, m} {μ_{1 i}} > 0, {\underline{μ}}_{2} = \min_{i = 1, \dots, m} {μ_{2 i}} > 0 .

Proof. In the case of sliding motion $s = 0$ , then

e + \int_{0}^{t} (μ_{1} si g^{1 + \frac{1}{γ}} (e) + μ_{2} si g^{1 - \frac{1}{γ}} (e)) dt = 0

(11)

The derivative of both sides of equation (9) with respect to time is

\overset{\cdot}{e} = - μ_{1} si g^{1 + \frac{1}{γ}} (e) - μ_{2} si g^{1 - \frac{1}{γ}} (e)

(12)

Consider the Lyapunov function candidate

V_{e} = \frac{1}{2} \sum_{i = 1}^{m} e_{i}^{2}

(13)

Substituting (8) into the derivative of (13) with respect to time yields

\begin{matrix} {\overset{\cdot}{V}}_{e} = \sum_{i = 1}^{m} e_{i} {\overset{\cdot}{e}}_{i} = - \sum_{i = 1}^{m} e_{i} (μ_{1 i} si g^{1 + \frac{1}{γ}} (e_{i}) + μ_{2 i} si g^{1 - \frac{1}{γ}} (e_{i})) \\ = - \sum_{i = 1}^{m} (μ_{1 i} | e_{i} |^{2 + \frac{1}{γ}} + μ_{2 i} | e_{i} |^{2 - \frac{1}{γ}}) = - \sum_{i = 1}^{m} μ_{1 i} | e_{i} |^{2 + \frac{1}{γ}} \\ - \sum_{i = 1}^{m} μ_{2 i} | e_{i} |^{2 - \frac{1}{γ}} \\ \leq - {\underline{μ}}_{2} \sum_{i = 1}^{m} (| e_{i} |)^{2 - \frac{1}{γ}} . \end{matrix}

(14)

The above inequality can be transformed into

\begin{matrix} {\overset{\cdot}{V}}_{e} \leq - {\underline{μ}}_{2} \sum_{i = 1}^{m} (| e_{i} |)^{2 - \frac{1}{γ}} \leq - {\underline{μ}}_{2} {(\sum_{i = 1}^{m} (| e_{i} |)^{2})}^{1 - \frac{1}{2 γ}} \\ = - {\underline{μ}}_{2} (2 V_{e})^{1 - \frac{1}{2 γ}} = - 2^{1 - \frac{1}{2 γ}} {\underline{μ}}_{2} V_{e}^{1 - \frac{1}{2 γ}} \end{matrix}

(15)

Thus, ${\overset{\cdot}{V}}_{e} < 0$ for $t \in (0, \infty)$ . It is shown that the sliding motion of system (8) corresponding to the sliding surface (9) is finite time stable. From Lemma 1, the reach time is $T_{1} (T_{1} \leq (γ | e (0) |^{(1 / γ)} 2^{(1 / 2 γ)}) / {\underline{μ}}_{2})$ . Therefore, after the system reaches the designed sliding surface (9), it can be guaranteed that the tracking error converges to zero within $T_{1}$ , namely the output of system (2) can accurately track the desired signal $y_{d}$ when $t \geq T_{1}$ . Proof is completed.

Necessary conditions to ensure this sliding motion which is attained and maintained will further be developed.

Reachability analysis

Fast and precise tracking of the desired signal plays a decisive role in the treatment of epilepsy. However, the general sliding mode control approach is not ideal in terms of tracking accuracy and speed. In order to satisfy the above requirements, a finite time nonsingular integral terminal sliding mode (FNITSM) controller is proposed for the treatment of seizure abatement, as illustrated in Figure 2.

It is worth noting that the lumped disturbances and their upper bounds are unknown, which are effected by cerebral blood circulation, glucose conversion, BOLD, complex electromagnetic signals and so on. The lumped disturbances need to be identified by using various identification algorithms for the compensation control design. In this paper, the disturbance observer (DOB) technique, which has been proposed in Chen (2004) and Chen et al. (2000, 2016), is introduced to identify the lumped disturbances and has the following form

{\begin{matrix} \overset{\cdot}{l} = - h (x) [l + W (x) + Ax + Bu + g] \\ \hat{η} = l + W (x) \end{matrix}

(16)

where $\hat{η} \in R^{n}$ denotes the estimation of the lumped disturbance, $l \in R^{n}$ denotes the internal state of the disturbance observer, and $W (x) \in R^{n}$ is a vector-valued function to be designed. The observer gain $h (x) \in R^{n \times n}$ is determined by

h (x) = \frac{\partial W (x)}{\partial (x)}

(17)

It has been shown in Chen (2004) that the DOB asymptotically estimates the disturbance if the observer gain $h (x)$ is chosen such that

\overset{\cdot}{\tilde{η}} = - h (x) \tilde{η}

is asymptotically stable regardless of $x$ , where $\tilde{η} = η - \hat{η}$ is the disturbance estimation error.

Remark 2. There are many ways to choose the observer gain $h (x)$ and the vector-valued function $W (x)$ such that the above equation is stable and (17) is satisfied. A systematic approach was suggested by Chen (2004) and Chen et al. (2000). The convergence and the performance of the proposed DOB have been established for the unknown disturbance in Chen et al. (2016). It allows the wide application of the DOB, particularly for estimating the influence of lumped disturbances as long as the observer error dynamics are faster than the closed-loop dynamics. In addition, the internal state of the disturbance observer $l$ is introduced according to the detailed design steps of the DOB developed in Chen (2004) and Chen et al. (2000, 2016).

According to equation (9), one can get the differential equation as follows

\begin{matrix} \overset{\cdot}{s} = \overset{\cdot}{e} (t) + μ_{1} si g^{1 + \frac{1}{γ}} (e) + μ_{2} si g^{1 - \frac{1}{γ}} (e) \\ = C (Ax + Bu + g + η) - {\overset{\cdot}{y}}_{d} + μ_{1} si g^{1 + \frac{1}{γ}} (e) + μ_{2} si g^{1 - \frac{1}{γ}} (e) \end{matrix}

(18)

The corresponding integral terminal sliding mode control law can be designed as

u = (CB)^{- 1} (u_{eq} + u_{es})

(19)

where the equivalent control law can be

u_{eq} = {\overset{\cdot}{y}}_{d} - C (Ax + g + \hat{η}) - μ_{1} si g^{1 + \frac{1}{γ}} (e) - μ_{2} si g^{1 - \frac{1}{γ}} (e)

(20)

and the approaching control law is

u_{es} = - λ_{1} s - λ_{2} | s |^{p} sign (s)

(21)

where $(CB)^{- 1}$ represents the generalized invertible matrix of $CB$ . $λ_{1}$ , $λ_{2}$ , and $p$ are designed positive constants, $0 < p < 1$ .

Theorem 2. Consider the control system (8) and disturbance observer (16). On the Assumptions 1 and 2, the designed control law (19)–(21) can drive system (8) to the composite sliding mode surface (9) in finite time and maintain a sliding motion on it thereafter, if the observer parameter $| h (x) | > (1 / 2)$ is satisfied.

Proof. Choose the following Lyapunov function

V = \frac{1}{2} s^{T} s + \frac{1}{2} {\tilde{η}}^{T} \tilde{η}

(22)

Taking the time derivative of the above formula to yield

\begin{matrix} \overset{\cdot}{V} = s^{T} \overset{\cdot}{s} - {\tilde{η}}^{T} [\overset{\cdot}{l} + \overset{\cdot}{W} (x)] \\ = s^{T} [C (Ax + Bu + g + η) - {\overset{\cdot}{y}}_{d} + μ_{1} si g^{1 + \frac{1}{γ}} (e) + μ_{2} si g^{1 - \frac{1}{γ}} (e)] \\ - {\tilde{η}}^{T} [h (x) (\overset{\cdot}{x} - l - W (x) - Ax - Bu - g)] \end{matrix}

(23)

Substituting equations (19)–(21) into equation (23) and using equation (2), one can obtain

\begin{matrix} \overset{\cdot}{V} = s^{T} [η - \hat{η} + u_{es}] - \tilde{η} h (x) [η - l - W (x)] \\ = s^{T} [\tilde{η} - λ_{1} s - λ_{2} | s |^{p} sign (s)] - {\tilde{η}}^{T} h (x) \tilde{η} \\ = s^{T} \tilde{η} - s^{T} [λ_{1} s + λ_{2} | s |^{p} sign (s)] - {\tilde{η}}^{T} h (x) \tilde{η} \end{matrix}

(24)

According to Lemma 3, the above inequality can be transformed into

\begin{matrix} \overset{\cdot}{V} \leq (\frac{| s |^{2}}{2} + \frac{| \tilde{η} |^{2}}{2}) - λ_{1} | s |^{2} - λ_{2} | s |^{p + 1} - | h (x) | | \tilde{η} |^{2} \\ = - (λ_{1} - \frac{1}{2}) | s |^{2} - (| h (x) | - \frac{1}{2}) | \tilde{η} |^{2} - λ_{2} | s |^{p + 1} \end{matrix}

(25)

When $| h (x) | > 1 / 2$ and the designed parameter $λ_{1} > 1 / 2$ , one can obtain $\overset{\cdot}{V} < 0$ , namely both $s$ and the disturbance observer error $\tilde{η}$ converge to zero asymptotically.

Let $ρ = (1 / 2 V) | \tilde{η} |^{2}, k_{1} = λ_{1} - (1 / 2), k_{2} = | h (x) | - (1 / 2)$ . Then $0 < ρ < 1$ , and equation (25) can be rewritten by

\begin{matrix} \overset{\cdot}{V} \leq - 2 k_{1} (1 - ρ) V - 2 k_{2} ρ V - λ_{2} [2 (1 - ρ)]^{\frac{p + 1}{2}} V^{\frac{p + 1}{2}} \\ = - 2 (k_{1} (1 - ρ) + k_{2} ρ) V - λ_{2} [2 (1 - ρ)]^{\frac{p + 1}{2}} V^{\frac{p + 1}{2}} \end{matrix}

(26)

Let $φ_{1} = 2 (k_{1} (1 - ρ) + k_{2} ρ), φ_{2} = λ_{2} [2 (1 - ρ)]^{(p + 1) / 2}, ξ = (p + 1) / 2$ , and equation (26) can be rewritten by

\overset{\cdot}{V} + φ_{1} V + φ_{2} V^{ξ} \leq 0

(27)

where $φ_{1} > 0$ , $φ_{2} > 0$ , and $0 < ξ < 1$ .

According to Lemma 2, the tracking error system (8) can reach the designed sliding surface (9) within $T_{2}$ and keep sliding motion on it. In addition, the reaching time satisfies

T_{2} \leq \frac{1}{φ_{1} (1 - ξ)} \ln \frac{φ_{1} V^{(1 - ξ)} (0) + φ_{2}}{φ_{2}}

(28)

Taking into account classic sliding mode control theory, Theorem 2 together with Theorem 1 shows that the tracking error $e$ of the closed-loop system (2) converges to zero in finite time, namely, $y$ accurately tracks the desired signal $y_{d}$ when $t \geq T_{1} + T_{2}$ . The proof is completed.

Remark 3. The normal sliding mode (NSM) controller can solve the tracking control problem of system (8), but the convergence speed of the tracking error cannot satisfy online adjusting requirement (Ge et al., 2019). In this study, a nonsingular integral terminal sliding mode manifold is proposed, which guarantees $y$ can accurately track the desired signal $y_{d}$ within a preset time. Thus, a finite time sliding mode control scheme in this paper can obtain better effect of epilepsy seizures abatement than (Ge et al., 2019).

Remark 4. It is worth emphasizing that the robust optimal control (OPC) in Songa and Lewis (2020) can also estimate the lumped disturbances based on the disturbance observer, but the controller designed in this paper combines the finite time integral terminal sliding mode (FNITSM) with disturbance observer technique, which can achieve better control performance than optimal control (OPC) in terms of convergence speed.

Remark 5. Compared with the fractional order sliding mode(FFOSM) control approach in Qian et al. (2020), the finite time nonsingular integral terminal sliding mode(FNITSM) control approach in this study improves the convergence speed and tracking performance. Therefore, in terms of the timely and quick online response for epilepsy seizures, this control scheme is more suitable for clinical treatment of epilepsy than Qian et al. (2020).

Simulation

In the section, to verify the effectiveness of the DOB-FNITSM control scheme proposed in this paper, the control performance comparison between DOB-FNITSM scheme and RBFNN-NSM scheme in Ge et al. (2019), DOB-OPC scheme in Songa and Lewis, (2020) will be implemented. Here the thalamocortical system (2) including uncertainties and disturbances is considered as the actual controlled system.

The initial value of $(TC, RE, IN, PY)^{T} = (0.05, 0.05, 0.05, 0.05)^{T}$ , matrix $C = [0, 0, 0.5, 0.5]$ , coefficients $h_{e} = - 0.35$ , $h_{i} = - 3.4$ , $h_{t} = - 2$ , $h_{r} = - 5$ , $τ_{e} = 26$ , $τ_{i} = 32.5$ , $τ_{t} = 2.6$ , $τ_{r} = 2.6$ , $c_{ee} = 1.8$ , $c_{ei} = 1.5$ , $c_{et} = 1$ , $c_{ie} = 4$ , $c_{te} = 3$ , $c_{tr} = 0.6$ , $c_{re} = 3$ , $c_{rt} = 10.5$ , $c_{rr} = 0.2$ , $ε = 2 \cdot 10^{5}$ , $α = 2.8$ , $β = 0.5$ . Let $W (x) = [1.2 x]$ . In this experiment, the subpopulations on PY and IN are used as control variables because the relevant data of PY and IN represent external stimulation in the cortex, that is $B = [1, 0, 1, 1]^{T}$ . Since dominant frequency of the required normal signal is 12 Hz, therefore reference trajectory is set as $y_{d} = 0.3 + 0.05 \sin (75.4 t)$ . Other more detailed parameters of the thalamocortical model can be found in Ge et al. (2019).

In the view of practical clinical application, it is assumed that there exist the parameter uncertainties caused by blood flow change of cerebral cortex in the state matrix $A$ with respect to their nominal values $A_{0}$ . In this paper, the parameter uncertainty is selected as $Δ A = 1 / 6 A_{0}$ and then $A = A_{0} \pm Δ A$ . Simultaneously, suppose that the unknown disturbance $d (t)$ consists of two parts, namely randomly distributed pulses $dp (t)$ and Gaussian noise $dn (t)$ with a standard deviation of 1.2 and a mean value of 0. The total simulation time is 10 seconds, and the simulation step is 0.01 seconds. In addition, it is assumed that the abnormal discharge of the brain nucleus occurs at the beginning of the simulation and the controllers begin to work at different times.

In Case 1, controllers work from start to end of the simulation

dp (t) = {\begin{matrix} 20, & t = 3, 7 (s) \\ 0, & others \end{matrix}

(29)

where $dp (t)$ is single-pulse injected, which can approximately simulate larger electromyographic (EMG) disturbance signals of blinking.

In Case 2, Controllers work from 4 seconds to the end of the simulation

dp (t) = {\begin{matrix} 1.2, & t = 0.7 σ (s), σ = 1, 2, \dots, 14 \\ 0, & others \end{matrix}

(30)

where $dp (t)$ is randomly distributed pulses injected, which can approximately simulate a series of smaller electrocardiographic (ECG) disturbance signals. Considering that the heart rate of seizure patients is significantly faster, the value is considered as 85 beats per minute.

Note that the amplitude of the pulse here is to make the simulation effect more obvious, while the actual clinical pulse in EEG observed is from half to two times magnitude of the fluctuation of the ideal value $y_{d} (t)$ .

Table 1 lists the parameters used in the DOB-FNITSM controller:

Table 1.

Design of control parameters based on different control schemes.

Controller	$λ_{1}$	$λ_{2}$	$p$	$γ$	$μ_{1}$	$μ_{2}$
DOB-FFOSM	3	1	0.5	2	0.6	0.8

To compare our scheme with RBFNN-NSM of Ge et al. (2019), the sliding mode surface of RBFNN-NSM can be chosen as $s = e$ . Meanwhile, for comparison, the disturbance observer in Songa and Lewis (2020) can be selected as the same as the one in this study. Notice that all the corresponding simulation parameters of three different approaches are the same.

The control objective is that the actual output $y$ can accurately track the required output $y_{d}$ under the satisfactory tracking accuracy and speed. In the subsequent simulation, the different controllers in the same cases will be used to control the same thalamocortical system (2) and be discussed.

In all the simulation figures below, the red line represents randomly distributed pulses $dp (t)$ , the blue solid line represents the actual output $y (t)$ , input $u (t)$ and tracking error $e (t)$ of the system, and the dark dashed line represents the desired trajectory $y_{d} (t)$ .

Comparisons between RBFNN-NSM andDOB-FNITSM

In Ge et al. (2019), a normal sliding mode controller with radial basis function neural networks (RBFNN-NSM) is used for seizure abatement, while this paper uses a DOB-FNITSM controller. Figures 3 –10 shows the comparisons between the control effect of two controllers. Under the case 1, Figures 3 and 4 show the output and input dynamics of RBFNN-NSM and DOB-FNITSM, respectively. In order to illustrate the tracking performance more clearly, the tracking error curves are exhibited in Figures 5 and 6, which obviously reveals that the tracking error of RBFNN-NSM and DOB-FNITSM converges to the small neighborhood of the origin within 1.5 and 0.2 seconds, respectively. Moreover, it can be observed from Figures 5 and 6 that the error dynamics of RBFNN-NSM is seriously effected by the single pulse, while the error dynamics of DOB-FNITSM is almost not effected. Under case 2, Figures 7 and 8 exhibit the output and input dynamics of two schemes. The closed-loop system under both the two controllers has the satisfactory tracking performance. However, the tracking errors of RBFNN-NSM and DOB-FNITSM shown in Figures 9 and 10 converge to the small neighborhood of the origin within 4.4 and 4.01 seconds, respectively, and error value of DOB-FNITSM is much smaller than RBFNN-NSM. Both two cases fully show that DOB-FNITSM controller is better than RBFNN-NSM controller in terms of convergence speed. All comparisons above reveal the better tracking precision and convergence speed of the finite time integral terms included in DOB-FNITSM controller, which is not available in normal sliding mode. The finite time nonsingular integral term in the sliding manifold can improve the dynamics of thalamocortical system. Moreover, the disturbance observer can identify the lumped effect of unknown uncertainties and disturbances. The proposed DOB-FNITSM fully illustrates the superiority of combination of the finite time integral terminal sliding mode technique and the disturbance observer.

Comparisons between DOB-OPC and DOB-FNITSM

Based on the research results in Songa and Lewis (2020), we compare DOB-OPC designed in Songa and Lewis (2020) with DOB-FNITSM control scheme in this study. Figures 11 –18 show the comparison results between two different controllers. It should be noted that two controllers can finally achieve effective tracking control, but the control performance is obviously different. Under the case 1, Figures 11 and 12 exhibit the output and input dynamics of DOB-OPC and DOB-FNITSM respectively. In order to show the tracking performance more clearly, the tracking errors are displayed in Figures 13 and 14, respectively. It is obvious from the Figures 13 and 14 that the tracking errors of DOB-OPC and DOB-FNITSM converge to a smaller area of origin within 1 and 0.2 seconds, respectively, but the tracking error of DOB-OPC is several times larger than that of DOB-FNITSM. Moreover, it can be observed that the tracking performance of DOB-OPC is severely effected by the single pulse at 3 and 7 seconds, but the tracking error of DOB-FNITSM is almost not effected by the single pulse. In case 2, Figures 15 and 16 display the output and input curves, which show that the system stability can be guaranteed by both two different controllers. Figures 17 and 18 exhibit that the tracking errors of DOB-OPC and DOB-FNITSM converge to a small neighborhood of origin within 4.1 and 4.01 seconds, respectively, and the tracking error of DOB-FNITSM is less than DOB-OPC. Based on the same disturbance observer, the difference between DOB-OPC and DOB-FNITSM is the difference between robust optimal controller and finite time integral sliding mode controller. In summary, the DOB-FNITSM can accelerate convergence process and improve tracking precision by adopting the nonsingular integral terminal sliding mode technique in the controller.

Conclusion

In this paper, a new integral sliding mode control scheme is proposed for the reduction or elimination of seizures, and the effective control of seizures is converted into a tracking control problem after reasonable analysis. First, the mathematical model of epilepsy seizures is transformed into a regular nonlinear control system, which considers the lumped disturbances comprised all uncertainties and electromagnetic disturbances. Second, the disturbance observer technique is used to estimate the lumped disturbances in the thalamocortical system such as glucose metabolism, electromagnetic disturbances, parameter uncertainties, and so on. Then, a novel finite time nonsingular integral terminal sliding mode controller based on the estimated information is developed, which is applied in adjusting the control signal of brain stimulation. Moreover, the theoretical proof of stability and reachability of the closed-loop control system are given rigorously by Lyapunov theory. Finally, the effectiveness and superiority of the proposed scheme are verified through sufficient simulation comparisons.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Post Doctoral Research Foundation of Jiangsu Province (1701140B), National Key Research and Development Project (2019YFB1705803), Natural Science Foundation of China (21727818), Postgraduate Research & Innovation Program of Jiangsu Province (KYCX21_1136), the GF Research, and Development Program of the Nanjing Tech University (201709).

ORCID iD

Guanghua Zhong

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