Boundary Layer-Based Event-Triggered Sliding Mode Controller for Nonlinear Systems with Uncertainty

The conventional feedback control is commonly using sampling criterion that rely on time-periodic decision rules, which can sometimes be overly conservative. However, there are circumstances in which centralized sampling is impractical or even deleterious. When dealing with large-scale systems, this becomes especially important because of the lack of resources and transmission bandwidth for decentralized control or because of the lack of suitable processing capacity in rapid systems. A new method has arisen to deal with these problems by only sampling when it is absolutely required to do so. The stability and convergence of the system must be maintained even if the periodicity for computing the control law is relaxed.

As a result, research on mechanisms for sampling that do not rely on periodicity or time-triggering procedures is gaining momentum. Because of this, event-driven feedback and sampling methods have been developed in recent years.

The defining feature of these methods is that the decision to execute the control task is not made ad-hoc but is instead based on a specific condition of the system's state [1].

Event-triggered Control (ETC) was first introduced by Åström et al. [2] and received significant attention from researchers. The methodologies of ETC have since been applied to many continuous-time systems, as demonstrated in recent works for example, event triggered input is used to compensate actuators failure by Xing et al. [3], adaptive control that is based on event triggered sampling is used to control saturated nonlinear systems with time varying state constraints [4], and uncertain stochastic nonlinear systems with actuator failures is controlled successfully using event trigger fuzzy adaptive controller by Liu et al. [5]. The aforementioned studies mainly focus on the case from the controller to the actuator in continuous-time systems. The primary design strategy involves constructing an ideal continuous control input signal approach and then applying the corresponding triggering mechanism to the continuous controller to derive the ETC system. This approach can also be observed in self-triggered control for linear systems [6], event-triggered and self-triggered control over actuators and sensor networks [7], and self-triggered-full information H-infinity controller [8].

Utilizing limited network bandwidth sufficiently is a significant challenge in networked control systems, as widely acknowledged in the literature. To address this challenge, numerous communication protocols have been developed, such as moving-horizon estimation for linear dynamic networks using binary encoding schemes [9], master-slave synchronization of heterogeneous systems under scheduling communication [10]. Resilient control design based on a sampled-data model for a class of networked control systems under denial-of-service attacks is discussed by Zhang et al. [11]. A survey of the trends and techniques of networked control systems is given by Zhang et al. [12]. In addition, a popular protocol that is based on event-triggered strategy, is demonstrated in several works [13-17], in which the communication access is released only when the pre-designed event occurs through the event generator. Furthermore, robust energy efficiency control for wireless sensor networks can be achieved via unity sliding mode control is considered by Raafat et al. [18].

The Sliding Mode Control (SMC) approach has been widely studied due to its capability to stabilize dynamical systems subject to external disturbances and nonlinearities, receiving good attention in both theoretical research [19-25] and engineering applications [26-29]. However, none of the above-mentioned works have provided a clear study and analysis of using SMC for both system types, namely continuous and discrete, and very few of them consider a nonlinear process with its model complications and the uncertainty that occurs in its parameter values.

Recently, researchers have introduced the event-triggered mechanism into SMC, resulting in a new approach named event-triggered SMC (ET-SMC). In addition to the main features of ET technique in reducing the control effort, and communication overload, hence reducing energy consumption in the system. Combining event-triggered control with sliding mode control produces a novel strategy that can offer several benefits of sliding mode and solve various issues in nonlinear systems. The most important one is that it makes the whole system more resilient to uncertainties and disturbance and well-suiting nonlinear systems with complicated dynamics and unidentified environments. However, this added feature increases the system's complexity and requires careful consideration of several aspects in the design process. Such as affecting the reachability of the sliding surface and the stability of the sliding-mode dynamics, requiring a deeper understanding of modeling techniques to optimize their potential and conquer associated challenges [30]. Although continuous event-triggered SMC can stabilize closed-loop systems subject to external disturbances with less communication frequency, it can only ensure practical sliding mode but not achieve normal sliding mode [31, 32]. Various continuous event-triggered SMC techniques have been reported in the literature, such as static event-triggered SMC [30, 33], self-triggered SMC [34], periodic event-triggered SMC [35], dynamic event-triggered SMC [36], and invariant sets in SMC theory [37], depending on different event-triggering conditions.

The majority of cited works exhibit a flaw in their methodology, which can be attributed to the researchers' reliance on linear systems in their applications. In addition, most researchers utilize continuous design as opposed to discrete systems, which are considered fundamental for event-triggered control. The main aims of this article can be summarized in the following points:

1. Incorporate sliding-mode control (SMC) with event-based technique to produce event-based siding mode control (EB-SMC). a powerful methodology that can provides an efficient solution to finding the suitable control action in dealing with complicated nonlinear and uncertain systems with high robustness control properties. The technique shows high resistance to external disturbances and unpredictable parameter variations. Also, it reduces the CPU's computational processes, thereby conserving energy.
2. To highlight the above mentioned features a continuous-time SMC (CT-SMC) and discrete-time SMC (DT-SMC) is developed for comparison purposes.
3. The proposed design also addresses two inevitable system problems: The chattering problem that comes with sliding mode technique, which is violated using the sat function, and Zeno phenomenon, which is avoided by limiting the time between consecutive triggering events.

The remainder of this paper is as follows: Section 2 explains the fundamentals of (ET-SMC) and triggering mechanism design, while Section 3 describes the mathematical modeling of the pendulum system. In Section 4, the design SMC and closed-loop stability analysis are presented. In Section 5, the results and analysis are presented, using four examples to evaluate the validity of the proposed methodology. Comparison between the two proposed controller is given in Section 6. The paper concludes with a summary of the main conclusions and suggestions for future research.

Considering the pendulum system illustrated in Figure 1 and the model is given in Eq. (3) below:

$\ddot{\theta}=-\operatorname{asin}(\theta)-b \dot{\theta}+c \dot{T}+c d(t)$       (3)

By considering the torque $(\mathrm{T})$ as the control input $(\mathrm{u})$. Suppose that the input ($u$) is applied at the pinned end of the pendulum and that is required to stabilize the pendulum at an angle $\theta=\theta_f$ by choosing the state variables as:

$x_1=\theta-\theta_f, \quad x_2=\dot{\theta}$       (4)

Knowing that $\theta_f$ is constant, and the control variable $(u=T)$. Then Eq. (3) can be rewritten as:

$\begin{gathered}\dot{x}_1=x_2=\dot{\theta} \\ \dot{x}_2=-\operatorname{asin}\left(x_1+\theta_f\right)-b x_2+c u+c d(t)\end{gathered}$      (5)

The controller design aims to translate the pendulum to the angle position $\theta_f$, and maintain it at this angle in the presence of the disturbance $(d(t))$.

3.1 Sliding mode controller design

By forcing the system's state trajectory to a predetermined sliding surface where the dynamics become more predictable and insensitive to perturbs and uncertainties, SMC can manage uncertain and nonlinear systems reliably [38]. In sliding mode control, the control law consists of two important phases: (i) reaching phase, and (ii) sliding mode phase. When the system state is driven from any initial state to reach the switching manifold in finite time, then such phase is called reaching phase, and when the system is induced into the sliding motion on the switching manifolds, then such phase is defined as sliding mode phase. The two phases of sliding mode control can be observed in Figure 3, where s represents the continuous-time sliding variable:

$s=s(x)$     (6)

The sliding manifold can be defined by:

$\Omega_s=s=\{x \mid s(x)=0\}$       (7)

The following characteristics should exist to induce the sliding mode. Two conditions must be satisfied for sliding mode control: (i) the stability of the system should be constrained to the sliding surface, and (ii) the sliding motion must begin within a finite time. A sufficient condition for the system to slide along the specified surface is expressed as follows:

$s \dot{s}<0$        (8)

where, $\dot{s}$ the rate of change of the sliding variable $(s)$. Let us consider the continuous-time pendulum system is given by:

$\dot{x}_1=x_2, \dot{x}_2=-a \sin \left(x_1+\theta_f\right)-b x_2+c u+c d(t)$

The first step in SMC design is the selection of a sliding variable:

$s(x)=\lambda x_1+x_2, \lambda>0$         (9)

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Figure 3. The sliding mode control phases

The controller $u$ designed according to the following design considerations:

The pendulum system parameters are certain, and a perturbation is acting on the pendulum. When sliding variable (s) equals zero ($s=0)$, the origin is an asymptotically stable point. In the present work, the system parameters are taken as follows: $a_0=\left(1 \_10\right), b_o=\left(0.5 \_1\right), c_o=\left(5 \_10\right)$ and let $d(t)=1$, by differentiating the function $s$ with respect to time $t$, this yields:

$\dot{s}=-a_o \sin \left(x_1+\theta_f\right)-b_o x_2+c_o u+\delta(t, x, u)+\lambda x_2$            (10)

where, $\delta$ is the perturbation.

Let,

$v=-a_o \sin \left(x_1+\theta_f\right)-b_o x_2+c_o u+\lambda x_2$        (11)

$s=\dot{s} s[v+\delta]$       (12)

Let

$v=-k \operatorname{sign}(s), k>0$,        (13)

Then we have,

$s \dot{s}=-\mathrm{k}|\mathrm{s}|+\delta \mathrm{s} \leq-|\mathrm{k}||\mathrm{s}|+|\mathrm{s}||\delta|=-|\mathrm{s}|[\mathrm{k}-|\delta|]$          (14)

Finally, the control action can be obtained by taking $(k>$ $\delta)$ and then:

$u=\frac{1}{c_0}\left[-a_o \sin \left(x_1+\theta_f\right)+\left(b_o-\lambda\right) x_2-k \operatorname{sign}(s)+d(t)\right]$        (15)

The main issue with continuous-time SMC is that a high frequency switching, discontinuous control term is commenced once the closed-loop system states reach to the sliding surface this results in chattering phenomena. This phenomenon happened due to many reasons where switching time delay, controller computation delay, dynamics of plant elements such as actuator and sensor are among the main causes. Chatter does not have a desirable effect on plant performance as it can cause heat loss and wear on moving parts of machines.

3.2 Stability analysis

Since traditional approaches, may not be relevant to nonlinear and uncertain systems, Lyapunov functions are sometimes used instead. Lyapunov stability is indicated by the existence of a positive-definite function whose time derivative along the system's trajectory is negative-definite [37]. Before diving further into the stability analysis, it is convenient to introduce the following definitions:

Definition 1 (attractiveness)

The set $\left(\Omega_{\varepsilon}\right)$ which defined as $\{x:|s|<\varepsilon\}$ is said to be attractive if the sliding variable $s$ satisfies the $\eta$-reaching condition of Eq. (8)

$s \dot{s} \leq-\eta|\mathrm{s}|$

where, $\eta$ is a positive design scalar. The expression above is called the $\eta$-reachability condition. The reachability requirement is a crucial criterion in the sliding mode literature since it ensures the existence of the sliding mode [38].

Moreover the $\Omega_{\varepsilon}$ is a positively invariant set [37]. Theoretically, any effective controller will attempt to move the state back to the origin, or at least to a positive invariant set that does include the origin. In addition, the controller will designate a certain area, the "area of attraction," that centers around the origin. The Definition 1 can be proved by using the Lyapunov function as follows:

Let $V=\frac{1}{2} s^2$ be the candidate Lyapunov function, also let $t_{r \varepsilon}$ is the reaching time to the $\varepsilon$ boundary layer then:

$\dot{V}=s \dot{s}<\eta|s|<0 \Rightarrow V(t)<V\left(t_{r \varepsilon}\right)$      (16)

or, $\frac{1}{2} s^2<\frac{1}{2} \varepsilon^2$, which implies that $|s|<\varepsilon, \forall t>t_{r \varepsilon}$ and that means that $\Omega_{\varepsilon}$ is a positively invariant set.

4.png

Figure 4. Behavior of the pendulum system

Considering Figure 4, we can explain the concept of stability for the model (pendulum system) through which this strategy was implemented. The Event triggered scheme is applied when the state is inside $\Omega_{\varepsilon}$ via the following rule:

$u_{i+1}=u_i$ if $e_{i+1} \in \Omega_{\varepsilon}$.

Else $u_{i+1}$ is updated according to the derived control law in Eq. (15). Moreover, due to the discretization, the actual positively invariant set is given by $\hat{\Omega}_{\varepsilon}=\{x:|| s \mid<\delta\}$, where, $\delta>\varepsilon$. Eventually, the steady-state error is ultimately bounded by $|\mathrm{e}|<(\delta / \lambda)$ [37].

Before diving into the design of the various controllers proposed in this paper, it is important to note that sign function produce a discontinuity in the control input. This can result in the undesirable chattering phenomenon, for which the saturation function sat() is one possible remedy. A saturation function that approximates the sign() term in a boundary layer of the sliding surface replaces the discontinues control law. This solution is investigated in greater detail in the subsequent sections on designed controllers [29].

Low switching frequencies are typically necessary in discrete-time sliding mode control since there is limited sampling frequency, making it more practical for implementation. Also, in DT-SMC, the control signal is calculated at each sampling period and remains constant during that whole interval. Consequently, the system state trajectory cannot move along the sliding surface, but instead follows a zigzag motion about the sliding surface, known as quasi-sliding mode motion.

The procedure for designing DT-SMC involves the following steps: (i) designing the sliding surface, (ii) developing a reaching law, and (iii) creating a control law that guides the system states to slide along a predetermined sliding surface over a finite interval of time. Finally, after implementing the designed DT-SMC procedure, the system state-space in Eq. (17) can be rewritten as follows:

$\begin{gathered}u_{k \tau}=\frac{1}{c_o}\left[a_o \sin \left(x_1+\theta_f\right)+\left(b_o-\lambda\right) x_2-k \operatorname{sign}(s)\right. +d(\tau)]\end{gathered}$   (18）

Four cases are considered to study and analyze the DT-SMC developed in this section. Case 1 and Case 2 consider the design analysis of DT-SMC using the sign function with and without applying the event condition, while Case 3 and Case 4 consider the design of DT-SMC using the Sat function with and without applying the event condition. Further investigation of the results is given in Tables 3 and 4. All the simulation experiments are conducted with plant sampling time (dt) equal to 0.0001 and controller sampling time (d) equal to 0.01.

Case 1: Design of DT-SMC using sign function

In this case, control action obtained by applying Eq. (18) is considered. The system error behavior represented by the states $\left(x_1, x_2\right)$ is shown in Figure 15. It is obvious here how the error states approach the origin. Figure 16 shows the switching function behavior. The reaching time (time needed to reach the switching manifold) is clearly less than 0.8 sec. The remaining time, the system will move in a zigzag pattern until it reaches the origin. To elucidate this point, a magnified portion of Figure 16 is presented, illustrating how the sliding mode controller varies above and below the switching manifold. This is called the chattering behavior and will be eliminated in Case 3 by replacing the sign function in the control law by sat function. In Figure 17 the discontinuous part of the control action (u) will start functioning when the system states reach the switching manifold (about (0.9 sec). In Figure 18, it can be seen that the controller can derive the angle $(\theta)$ to track the reference angle (θ_f=45) in about (5.7 sec). Moreover, the number of updates for control action is found to be (3000) which is less than its value when CT SMC is used. Figure 19 depicts the phase plane of a closed-loop system; it is evident that the system begins at its initial value, reaches the switching manifold, and then slides to the origin (sliding mode).

15.png

Figure 15. Error states (x₁, x₂) for a SMC design Case 1

16.png

Figure 16. Switching function (s)

17.png

Figure 17. Time history for control action (u) Case 1

18.png

Figure 18. Time history of pendulum angle (θ) Case 1

19.png

Figure 19. Phase plane with SMC design for Case 1

Table 3. Comparison of Case 1 and Case 2

Case 2: Design of DT-SMC using sign function with applying event condition

The algorithm in Case 1 is repeated in this case but with the application of the event condition given in Eq. (1). The threshold value δ is chosen to be greater than 0.01. The results obtained in in this case are shown to be similar to Case 1 except for the number of updates occurred in the control action, which decreased significantly from 3000 to 90 due to the use of the event condition. It should be noted here, that the chattering phenomenon appeared in the control action behavior of Cases 1 and 2 will be processed in Cases 3 and 4. A comparison between the main outcomes of Case 1 and Case 2 is given in Table 3.

Case 3: Design of DT-SMC using sat function

In this case, sat function is used to replace the sign function in the control law given in Eq. (18) in order to eliminate chattering phenomena. Hence, Eq. (18) will be rewritten as follows:

$\begin{gathered}u_{k \tau}=\frac{1}{c_o}\left[a_o \sin \left(x_1+\theta_f\right)+\left(b_o-\lambda\right) x_2-k \operatorname{sat}(s)\right.+d(\tau)]\end{gathered}$      (19)

The previous algorithm is repeated here by using the control law given in Eq. (19) above with selecting best value of ($\emptyset$) which is chosen to be equal to 0.02 to reduce the steady-state error. Recall that $\emptyset$ is a parameter related to sat function.

It can be noticed that the controller succeeded in achieving its objective in driving the pendulum angle to its desired value as demonstrated in Figure 20-23. The chattering free response of the switching function s is depicted in Figure 21 this is due to the use of sat  function and as shown in Figure 21 it reaches the switching manifold in less than (0.8 sec). The control action is shown in Figure 22, where no zigzag behavior is shown due to the continuous part sat instead of the sign function. Furthermore, the number of updates for control action is found to be 3000. Finally, the phase plot of the system states is depicted in Figure 24.

20.png

Figure 20. Error states (x₁, x₂) for a SMC design Case 3

21.png

Figure 21. Switching function (s) Case 3

22.png

Figure 22. Time history for control action (u) Case 3

23.png

Figure 23. Time history of pendulum angle (θ) Case 3

24.png

Figure 24. Phase plane of system states with Case 3

Case 4: Design of DT-SMC using sat function with event condition

In this case, the same event condition given in Eq. (1) used before is applied here with the same threshold value (δ>0.01). The results obtained in this Case is almost the same as Case 3 except for the number of updates occurred at the control action, which decreased significantly from being 3000 to 90 due to the use of event based control approach. A brief comparison between Case 3 and Case 4 results are given in Table 4.

Table 4. Comparison of Case 3 and Case 4

5.1 DT-SMC results discussion and analysis

Simulation experiments were run for four different scenarios to examine the impact of incorporating an event trigger condition into discrete SMC and the impact of incorporating a sat function into the design to minimize chattering.

It should be noted that the implementation of the eventtriggered technique in Case 2 has reduced the values of the system states $x_1, x_2$ thereby reducing the system's error and resulting in improved performance. In terms of numbers, the state $x_1$ is decreased from $x_1=0.0089$ to $x_1=0.0014$, and the state $x_2$ decreased from $x_2=0.0149$ to $x_2=4.9949 \times$ $10^{-4}$ while $u=0.51$ remains nearly the same in Case 2 compared to Case 1. As shown in Table 3, The inclusion of an event-triggered condition in the design of a discrete control strategy reduces the number of control action updates from 3000 in Case 1 to 90 in Case 2, which is the primary objective of this study.

Comparing Case 3 and Case 4, which consider the design of DT-SMC using the sat function with and without the event condition, yields the same outcomes as shown in Table 4, while the number of updates were 3000 in Case 3 and 90 in Case 4, thereby reducing the number of updates significantly, which is main target of this work.

Considering Cases 2 and 4 highlights the effect of adding an event-triggered technique to DT-SMC, in which the number of control action updates is reduced by nearly 97%. Moreover, comparing Case 1 and Case 2 with Case 3 and Case 4 highlights the effect of using sat function instead of sign. Figure 16 depicts the behavior of the switching function. Clearly, the reaching time required to reach the switching manifold is less than 0.8 seconds. The system will continue to move in a zigzag pattern until it reaches the origin this is known as chattering behavior, and it will be eliminated in Case 3 and Case 4 by substituting sat function for sign function in the control law. Figure 21 displays the chattering-free response of the switching function s. This is due to the use of the sat function, and as shown in Figure 21 it reaches the switching manifold in less than (0.8 sec). The control action is shown in Figure 22, where there is no zigzag behavior because the sign function has been replaced by a continuous part sat. Tables 1 and 2 provide additional numerical information.

In this paper, the robustness and superiority of event-based control techniques are investigated and analyzed through the development of an event-triggered sliding mode controller. This approach is shown to be effective by applying it to a nonlinear model of a pendulum system and by considering both continuous and discrete time systems using CT-SMC and DT-SMC, respectively. A comparison of DT-SMC and CSMC with and without the application of the event-based technique to demonstrate the impact of this technique on both types of systems. The obtained results demonstrate that reduced control actions are required when event-based techniques are utilized without compromising control performance. This reduces the number of arithmetic operations conducted by the central processing unit and, consequently, the amount of energy used. Also, the proposed design considers two unavoidable system's problems and suggests effective solutions for minimizing their effect: the chattering problem that accompanies sliding mode technique, which is circumvented using the sat function in the design, and the well-known Zeno behavior, which is avoided by implementing minimum time intervals between consecutive triggering events.

Following is a list of relevant topics for future research:

1. Event based DT-SMC considered in this paper can serve as excellent solution for data congestion in network control systems due to its ability to deal with increased systems resources, which is major problem in such systems.
2. The mechanism considered in this paper to generate events (send of delta SoD mechanism) is the most popular mechanism. However, further improvement can be obtained by considering the development of this mechanism which is target for our next coming research.
3. To further minimize chattering without compromising control precision, smoother functions or approaches like boundary layer methods can be implemented.