Improved Dynamic Event-Triggered Robust Control for Flexible Robotic Arm Systems with Semi-Markov Jump Process

In this paper, we investigate the problem of a dynamic event-triggered robust controller design for flexible robotic arm systems with continuous-time phase-type semi-Markov jump process. In particular, the change in moment of inertia is first considered in the flexible robotic arm system, which is necessary for ensuring the security and stability control of special robots employed under special circumstances, such as surgical robots and assisted-living robots which have strict lightweight requirements. To handle this problem, a semi-Markov chain is conducted to model this process. Furthermore, the dynamic event-triggered scheme is used to solve the problem of limited bandwidth in the network transmission environment, while considering the impact of DoS attacks. With regard to the challenging circumstances and negative elements previously mentioned, the adequate criteria for the existence of the resilient H∞ controller are obtained using the Lyapunov function approach, and the controller gains, Lyapunov parameters and event-triggered parameters are co-designed. Finally, the effectiveness of the designed controller is demonstrated via numerical simulation using the LMI toolbox in MATLAB.


Introduction
With the development of information technology, physical systems in practical applications are often composed of a large number of interacting elements, which are called complex systems, such as robotic arm control [1], power system [2], and multi-agent system [3,4]. As a special class of complex systems, Markov jump systems (MJSs) are widely studied by scholars for their superior modeling capacity to deal with unexpected changes, which may include environmental disruptions, random component mistakes, or even human influences during regular operation [5,6]. The authors in [7][8][9] reviewed recent advancements in modeling, stability analysis, filter design and sliding model control of MJSs. A new controller was proposed to ensure the stabilization of generalized MJSs in [10] to solve the random stabilization problem. Zhuang et al. [11] considered both the stability analysis and stabilization of generalized Markov jump time-delay systems, with the relevant state-feedback controller derived by means of LMI.
However, there exists a limitation in practical applications whereby the sojourn time of the Markov chain follows an exponential probability distribution, that is, results on MJSs are unfortunately conservative for the constant transition rates as a memoryless property as follows: (1) We consider the H ∞ performance of the flexible robotic arm with phasetype semi-Markov process; (2) The necessary conditions are found for robust controllers to exist by using the Lyapunov function method, furthermore, the controller and the dynamic event-triggering scheme are co-designed; (3) A simulation is conducted on the flexible robotic arm using the LMI toolbox in MATLAB, to estimate the usefulness of the designed controller gain with the comparison diagram of the time-trigger mechanism and the dynamic event-trigger mechanism.
The remainder of this paper is structured as follows. Section 2 provides a description of the system and a statement of the problem, along with some fundamental ideas and lemmas. Principal conclusions from a stability analysis and time-/event-triggered robust controller for flexible robotic arm systems with semi-Markov jump process are rendered in Section 3. In Section 4, a numerical simulation method is used to demonstrate the functional operation of the suggested controller design process.
Notation. There are various notations used throughout this paper. In the upper-right section of the matrix, −1 and T represent the inverse and transpose of matrix, respectively, and * in the symmetric matrix denotes the omitted term, and diag{·} means a block diagonal matrix. Notation 0 n×m represents the zero matrix of dimension n × m. Furthermore, the 2−norm of vectors in this paper are represented by || * || 2 , and R 0+ denotes the set of positive real numbers. Finally, E{·} represents the mathematical expectation. The remaining notations are less general in nature and are shown after their respective formula.

Physical Model of Flexible Robotic Arm
In this paper, a single link manipulator driven by DC motor with flexible joint is used as the controlled object (as shown in Figure 1a). The dynamic characteristics of a flexible joint can be approximated by a linear torsion spring with an elastic coefficient k. Inspired by [27], the structure is shown in Figure 1b. The ideal dynamic equations of the flexible robotic arm systems are given by where θ(t) is the rotation angle, J is the equivalent moment, µ 1 (t) represents external disturbance of motor; q(t), I, µ 2 (t) represent the same parameters of the connecting rod. k represents the equivalent spring elastic coefficient; u 0 (t) is the input torque of the motor; M is the weight of the connecting rod; H is the center of mass and the distance between the axis of rotation, the acceleration of gravity is represented by g. Let us define the following state variables: where x 1 (t) denotes the rotation angle of the DC motor, x 2 (t) is the angular velocity of the motor, x 3 (t) is the rotation angle of connecting rod, and x 4 (t) denotes the angular velocity of the connecting rod.
Assumption 1. The nonlinear term sin q(t) is presented in the dynamic equation mentioned above.
We make the assumption that sin q(t) = q(t), q(t) ∈ (− π 2 , π 2 ) to remove the nonlinear term's influence. In other words, the connecting rod's accessible range is partially constrained.
It is well known that the moment of inertia depends on: (a) the shape of the object, (b) the position of the axis of rotation, and (c) the distribution of mass. When grasping objects of different masses, where (a,b) remain unchanged and (c) changes with the endeffector grasping different objects, the moment of inertia of the connecting rod has different modes in Figure 2. A flexible joint manipulator system is a complex system with continuous time and discrete state. Take into consideration the subsequent continuous-time stochastic systems over a probability space ( , F , P r ) with Semi-Markov properties, add the dynamical equation as shown below.
where T , ω(t) means the external disturbance and ω T (t)ω(t) ≤ I, u(t) represents the input signal of the closed-loop system, and and C = [1 0 0 0].r(t) represents a finite state semi-Markov jump process, which accepts discrete values inside the specified finite set {1, 2, · · · , m + 1}, the state m + 1 is absorbing and other states are transient. Abbreviated symbols,Â i forÂ(t, r(t)),B 1i forB 1 (t, r(t)). The infinitesimal generator is where matrix T m×m = (T ij ) satisfies T ij < 0, T ij ≥ 0, i = j, the non-negative column vector T 0 meets Te + T 0 = 0. While (a, m + 1) is represented as the initial distribution vector, where a = (a 1 , a 2 , . . . , a m ), ae + a m+1 = 1, and e stands for an m-dimensional column vector made up only of 1. Before moving on, let us review the following fundamental claims and definitions: Proposition 1. The rate at whichr(t) is absorbed inr(t) is distributed as Definition 1. The phase of the distribution F(·) at time t is the state thatr(t) reaches at that instant. Its representation of order m is (a, e), and the distribution F(·) described in version (5) on [0, ∞) is known as a phase-type (PH) distribution.
If G is a finite set, then, Markov chain is the denumerable phase semi-Markov process, where F i (t)(i ∈ G) has a negative exponential distribution. However, the characteristic of Markov chain that the sojourn time obey negative exponential distribution can be overcome by a denumerable phase semi-Markov process. The main issue is whether or not the denumerable phase semi-Markov process can be changed into a Markov chain. Moreover, a finite Markov chain can be created from a finite phase semi-Markov process. The remainder of this paragraph will provide evidence supporting the validity of the aforementioned statement.

Definition 2.
The pair of the semi-Markov processr(t) is referred to as {P, (a, T)}. Define χ(t) = the phase of Fr (t) (·) at time t − ι n for every n(n = 0, 1, · · · ), ι n ≤ t < ι n+1 . For any i ∈ G, we define We can readily obtain the following conclusions from the aforementioned analysis.

Lemma 1.
Markov chain Z(t) = (r(t), χ(t)) has state space U . The pair ofr(t) given by P, (a, T) determines the infinitesimal generator of Z(t) exclusively, as shown in the following Suppose that U has s = ∑ i∈G m i elements, which results in s elements in the state space of Z(t). The s elements are numbered using the following procedure: the number of Adding the letter ψ to this transformation as well, and one has Furthermore, we define The infinitesimal generator Ξ = (υ im , 1 ≤ i, m ≤ s) with a state space S = {1, 2, · · · , s} make r(t) be a Markov chain. Above all, the associated Markov chain ofr(t) is the name given by the Markov chain r(t). The latest transition probability matrix satisfies that where π ij ≥ 0 (i = j) and ∑ s j=1,i =j π ij = −π ii (i ∈ S) and o(δ)/δ → 0 for δ > 0. When δ → 0, r 0 ∈ S is the initial condition for the continuous state.

Network Control Based on Dynamic Event-Triggered Scheme
Benefit from the rapid development and wide application of network communication technology, the combination of control system and network transmission has attracting more and more attentions of scholars [28,29]. The traditional time-triggered/periodictriggered control scheme has the shortcoming of wasting network resources (CPU for computing) which information are less important, the static event-triggered control in Figure 3 is introduced to solve this problem, which greatly alleviates the pressure of network bandwidth and reduces the waste of communication resources while implementing effective control. In this article, a dynamic event-triggered scheme in Figure 4 is proposed to further save the network bandwidth.  An intelligent sensor is composed of a sensor, sampler, event generator and memory. The sensor is responsible for collecting the sensitive information of the controlled object, and the sampler collects the continuous signals transmitted by the sensor at a fixed period. The event generator screens the received sampled data according to the preset event trigger conditions, and the sampled signals that meet the trigger conditions will be sent to the controller, otherwise they will not be sent. Sampling sequence H 1 = {0, h, 2h, · · · , kh}, k ∈ {0, 1, 2, 3, · · · }, the time series of successful signal release is H 2 = {t 1 , t 2 , t 3 , · · · , t k }, k ∈ {0, 1, 2, 3, · · · }. Clearly H 2 ∈ H 1 , the real control signal u(t) for the controller is provided by For t ∈ t k + τ t k , t k+1 + τ t k+1 , motivated by [30], we can easily prove that there must be a positive integer d M (≥ 1) such that According to the above equation: Furthermore, let us define an error vector as following: Then, dynamic event-triggered conditions will be written as follows.
Furthermore, the internal variable κ(t) that fulfills the following differential equation is where κ 0 ≥ 0, ς is a function of class K ∞ with Lipschitz continuous [24]. Compared with static event triggering scheme, dynamic event triggering scheme is less conservative that do not have to satisfy σx T (t)Wx(t) − e T (t)We(t) ≥ 0.

Remark 1.
Notably, several factors that need to be carefully chosen for the communication strategy may be found in the event-triggered mechanism (16). Firstly, a bigger σ may accept a larger measured error e(t), which results in fewer data packets being triggered for the controller update. σ describes how tight the triggering process is. The internal dynamic variable κ(t) participation is then described by the parameter ρ.

Remark 2.
The internal variable κ(t) under the starting κ 0 ≥ 0 meets the following criterion for the dynamic event-triggered scheme (16) For the rest of this paper, we will consider the flexible manipulator subject to the semi-Markov jump process in the network environment, through the design of the corresponding event-trigger controller to make the system state asymptotically stable and meet the H ∞ performance of parameter γ. The event-trigger controller is selected as follows: Remark 3. Compared with the traditional static ETM, there are two advantages on the proposed DETM. First, the DETM can effectively avoid the transmission of redundant data and reduce the number of triggering compared with static ETM. Second, the event-triggered parameter ς(κ(t)) can be dynamically selected adaptively, which provides greater potential for optimization of the event-trigger mechanism. In this paper, the parameter ς(κ(t)) is assumed to be a direct proportional function which belongs to a function of class K ∞ with Lipschitz continuous. Furthermore, more options for the event-trigger parameter can be seen in existing results [31,32].
In the development of network technology, some hackers seek personal gains by launching network attacks, which represents both an immoral and illegal act. In this article, we consider a common form of network attack: DoS attacks. The Bernoulli process is used to describe the phenomena, the following is suggested for the affected controller: where the Bernoulli distributed variables h(t) display the frequency of DoS assaults and E{h(t) = h 1 }. Introduce into switch matrix L to describe DoS attacks: In summary, the manipulator's closed-loop control system is stated as where ϑ(t) is the initial function of x(t). Some necessary lemmas are presented to complete the following theorem.
Problem Statement: For a given semi-Markov flexible robotic arm system suffering from DoS attacks in (5), the dynamic event-triggered controller is designed in (19), so that the resulting system in (20) is asymptotically stable with a H ∞ performance. [34] For a given scalar 0 ≤ α ≤ 1,the matrix W 1 , W 2 ∈ R n × m, the positive matrix R ∈ R n , if there is a matrix N ∈ R n , satisfies S N * S > 0, then the following inequality is true

Stability Analysis
In this section, a sufficient condition of asymptotically stable is presented for the proposed flexible robotic arm system with an event-triggered controller in (20). Theorem 1. For given scalars τ 1 , τ 2 , σ ∈ (0, 1), the closed-loop system in (20) is asymptotically stable with a H ∞ performance γ if a matrix N can be found, in addition to positive definite symmetric matrices P i , Q 1 , Q 2 , R j (j = 1, 2, 3), W i (i = 1, 2, 3, 4) satisfying the following inequalities where , Proof. Let us build a Lyapunov-Krasovskii Functional (LKF) as follows: where Taking the derivative of t ∈ t k + τ t k , t k+1 + τ t k+1 along the system track, we get whereP i = ∑ s j=1 π ij P j , anḋ Using Lemmas 2 and 3, we can obtain the following inequalities with l = 1, 2: Combining (25)- (29) with the event-triggered condition in (16), we have where whereR and A i are defined in (23).

Remark 4.
In this section, the delay-dependent condition based on the LKF is construct. This is our first attempt to apply the stochastic theory and LMI technology directly to the control of flexible robotic arm systems in network communication. The LKF method has a more mature theoretical system and has been proved to be feasible. Furthermore, this method also can be easily extended to dissipative analysis, exponential stability and so on.

Co-Design of the Event-Triggered and Controller Parameters
In this section, the state controller of the proposed flexible robotic arm system using dynamic event-triggered scheme is presented in (20).
According to Theorem 1, let is assume that . Then, multiply both sides of this inequality by {X i , X i , X i , X i , X i , I, I, I, I, I, I} There exists −X iR −1 j X i (j = 1, 2, 3) nonlinear term in the equation, which is linearized by inequation −X iR −1 j X i < ϕ 2 jR j − 2ϕ j X i (j = 1, 2, 3. i ∈ S). In order to implement in the LMI toolbox of MATLAB, according to Schur complement lemmaΩ i < 0 is further restructured to (31).

Remark 5.
Under the condition of ς(κ(t)) = κ(t), ∈ R 0+ , the inter-event intervals always have a positive lower bound, which rules out Zeno behavior. Furthermore, because the internal dynamic variable κ(t) is not negative, the dynamic event-triggered technique utilized might result in less data transfer, which helps conserve resources.
The implementation procedure of the dynamic event-triggered robust controller design for flexible robotic arm systems with continuous-time phase-type semi-Markov jump process is illustrated in Algorithm 1.

Simulation Results
The flexible robotic arm in (20) shown grasping objects of different weights is very common in production practice. In this section, a phase-type semi-Markov chain is used to simulate the grabbing of four different objects as shown in Figure 5.
The sojourn time in the first three states is a random negative exponential distribution variable of parameter λ j (j = 1, 2, 3). The sojourn time in the last state is split into two parts, namely two random variables with parameters λ 4 and λ 4 of negative exponential distribution. For example, if the processr(t) enters a functioning condition, it must first spend some time in the first part, then stay in the second part, and then enter state 1 at the end. Without loss of generality, we assume that Before presenting the result of the numerical simulation, the setup and software of the computer that we employ are given in Table 1: Similar to [17], some appropriate physical system parameters of (5) and parameters related to the dynamic event-trigger condition in (16) are set in the following Table 2. , and select 30 s as the total simulation time, and assume that the external disturbance is w(t) = exp(−t) sin(t). The dy-namic event-triggered weighting matrix W i and the corresponding controller K i (i = 1,2,3,4) are solved by the LMI program and given as follows:  Figure 6 displays the state-response diagram of the system under semi-Markov process without controller and shows that the open loop of the flexible robotic arm in (20) is unstable.
The state trajectory of the static event-triggered mechanism is shown in Figure 7. Furthermore, DoS attacks that obey the Bernoulli distribution of parameter property h 1 = 0.95 operate on the input signal u t . Contrapositive with the time-triggered mode, the eventtriggered mode achieves fewer times of sending at the expense of a part of the calming time, a total of 193 events are triggered. Compared with the time period sampling 3000 times, the data transmission rate of the event trigger is 6.43%, which greatly saves network resources. Figure 8 depicts the interval between the sampling time when the control system successfully triggered under the event triggering mechanism and the last two successful triggering moments. As can be seen from the figure, the trigger intervals under the event triggering mechanism are not equal, and all trigger intervals are greater than or equal to the sampling cycle. Note Event triggering can reduce the transmission frequency of system data and save limited resources.
The system state controlled by the dynamic event triggering mechanism is shown in Figure 9, and the matching triggering interval is shown in Figure 10. It has fewer trigger times (170, 5.67%) and exhibits a good calming effect for a limited time period. This demonstrates the effectiveness of the designed dynamic event-triggering controller in saving network resources.
In general, the proposed dynamic ETM can maintain the security and stability control of the flexible robotic arm systems while occupying far less network resources than the timetriggered mechanism and the static ETM. The minimum time-interval between two adjacent trigger instants is not less than the sampling period, to prevent the Zeno phenomenon. Furthermore, the threshold is dynamic, and thus the trigger condition can adaptively modify the transmission of the signal to fend off the negative effects of DoS attacks.

Conclusions
We herein present a type of physical model of flexible robot arm based on a semi-Markov jump process. For the corresponding linear system, the network control method is adopted, and the stabilization effect under the time-triggered mechanism and the dynamic event-triggered mechanism is considered, respectively. In order to obtain a full-state feedback controller satisfying the attenuation index γ of disturbance suppression, a series of matrix inequalities are obtained by constructing LKF and using Jensen inequality, reciprocal convex lemma and Schur complement lemma. The appropriate controller K i and weighted matrix W i are solved using the LMI toolkit in MATLAB. Finally, an effect diagram of the numerical simulation is given.
In this paper, for the first time, we take stochastic theory and LMI technologies into consideration in the control method of flexible robotic arm systems, and directly solve the controller parameters using LKF. However, this method still has some limitations. The angle q(θ) of the robot arm is assumed to be (− π 2 , π 2 ), which is somewhat conservative. To extend the rotation of the arm to (0, 2π), a neural-network-based approach will be considered in the future work.

Data Availability Statement:
The source codes and datasets used to support the findings of this study are available from the authors upon request via email: huiyanzhang@ctbu.edu.cn.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: