Time-Delay Luenberger Observer Design for Sliding Mode Control of Nonlinear Markovian Jump Systems via Event-Triggered Mechanism

: This paper is focused on the stabilization of Takagi–Sugeno fuzzy model-based Markovian jump systems with the aid of a delayed state observer. Due to network-induced constraints in the communication channel, a delay partition method combined with an event-triggered mechanism is proposed to design the observer. Then, a novel integral sliding surface is designed, based on which sliding mode dynamics is obtained. Further, according to stochastic stability theory, feasible conditions are provided to ensure the sliding mode dynamics and the error dynamics have an H ∞ attenuate level γ . The challenge is to deal with the issue that transition rates may be totally unknown. Moreover, an observer-based sliding mode controller is constructed to ensure the ﬁnite-time reachability of the predeﬁned sliding surface. Finally, a numerical example based on a robotic manipulator is given to verify the effectiveness of the proposed method.


Introduction
With the strong demands for modeling a physical system with high accuracy, for instance, the physical systems may have structural mutation due to changes of power, the shift of parameters and external disturbances, etc., which provoke the need to seek for more suitable mathematical models. Therefore, Markovian jump systems (MJSs) have attracted numerous attention because of its ability to model such kind of physical model with multimodal characteristics or intelligent control system with multi-controller switching [1]. So far, the researches on MJSs have been reported by many in both applications and theoretical aspects, such as the Markov jump models are widely applied in nuclear power systems, manufactory and network communication [2][3][4][5], etc. Theoretically, it also witnessed the issue of stochastic stability analysis, stabilization, filter design and so on. For example, in [6], the robust stability and control of uncertain discrete-time linear systems with Markovian jumping parameters was dealt with; in [7], the problems of stochastic stabilization and H ∞ control for 2-D MJSs were proposed; and systematic theory on stochastic differential equations with Markovian switching was presented in [8]; the H ∞ filtering problem for MJSs was studied in [9,10]; for more details, we may refer to [11][12][13][14][15] and some of their references. In another aspect, due to the existence of nonlinearity, the Takagi-Sugeno (T-S) fuzzy modeling approach has become one of the most popular and effective ways to handle the synthesis of complex nonlinear systems [16], and the investigation on T-S fuzzy model-based MJSs is also rich. In addition, the stabilization of nonlinear singular MJS finite-time reachability of sliding surface and keeps sliding motion of each sub-models in the presence of uncertain transition information and nonlinearities.
Notions: In this paper, the concept X > 0 (X ≥ 0) denotes X is a positive definite (positive semidefinite) matrix. I and 0 represent an identity matrix and zero matrix, respectively.E(·) represents the expectation operator about probability measures. · represents the Euclidean vector norm. * denotes Symmetric elements in a symmetric matrix. sym{P} expresses P T + P.

Model Establishing and Problem Statement
Let us consider the model of single-link robot arm model proposed in [43], where the equation of dynamic is given by ..
the meanings of parameters are defined in Table 1.  ϑ(t). According to the method proposed in [44] that under certain angle position, the nonlinearity sin(x 1 (t)) can be replaced by sin(x 1 (t)) = h 1 (x 1 (t))x 1 (t) + βh 2 (x 1 (t))x 1 (t) in which β = 0.1/π is a known parameter and the membership functions satisfy h 1 (x 1 (t)) + h 2 (x 1 (t)) = 1 with h 1 (x 1 (t)), h 2 (x 1 (t)) ∈ [0, 1]. By solving the above equation, the membership functions h 1 (x 1 (t)) and h 2 (x 1 (t)) are obtained as h 1 (x 1 (t)) = sin(x 1 (t))−βx 1 (t) x 1 (t) (1−β) , x 1 (t) = 0 1, Based on the membership function mentioned above, it is easy to see that if x 1 (t) is about 0 rad, then h 1 (x 1 (t)) = 1, h 2 (x 1 (t)) = 0, if x 1 (t) is about −π or π rad, then h 1 (x 1 (t)) = 0 and h 2 (x 1 (t)) = 1. Hence, the fuzzy model-based state-space description of the robotic system can be rewritten as: Particularly, as discussed in [43], the moment of inertia J may change due to the abrupt change of actual operating, for instance, the switching of the parameters follows the Markovian jumping rules. Therefore, based on the above model, let us consider the following general model: where x(t) ∈ R n denotes the state vector; ϑ 1 (t), · · · , ϑ p (t) are premise variables; F ij (i = 1, 2, · · · , r; j = 1, 2, · · · p) are fuzzy sets, u(t) ∈ R l is the control input; y(t) ∈ R q is the controlled output. A i (r t ) and C(r t ) are the system matrices with appropriate dimensions; B(r t ) is the input matrix with full column rank. The process of stochastic jumping {r t , t ≥ 0} is a continuous-time homogenous Markovian jumping process. The process of generator with values in a finite set S = {1, 2, · · · , s} given by Through fuzzy standard blending, the model can be easily derived: .
in which ϑ(t) = ϑ 1 (t) ϑ 2 (t) · · · ϑ p (t) T ,h i (ϑ(t)) is the membership function that the formula of which is where µ ij ϑ j (t) is the rank of membership of ϑ j (t) in µ ij . In addition, for all t > 0, it is satisfied that h i (ϑ(t)) ≥ 0 and ∑ r i=1 h i (ϑ(t)) = 1. For simplicity, r(t) m and h i (ϑ) is short for h i (ϑ(t)) in the following.
The following Definitions and Lemma are introduced.

Definition 1. ([45])
Given the Lyapunov functional candidate V(x(t), r t , t ≥ 0) with twice differentiable on x(t), then, LV(x(t), r t ) which is infinitesimal operator is given by Definition 2. ( [46]) The unforced stochastic system (2) (i.e., u(t) = 0 ) is said to be stochastically stable for any initial condition x(t 0 ) and t t 0 , as long as it satisfies 47]) Given any real number ε and any square matrix R, for any matrix F > 0, the matrix inequality ε R + R T ≤ ε 2 F + RF −1 R T holds.
Here, in this paper, an appropriate observer-based fuzzy SMC strategy for the model (2) on the communication networks with network-induced constraints has been designed to obtain good stochastic stability property with an H ∞ performance in sliding mode dynamics and error dynamics.

Main Results
Given that the state component may not be accessible in practice, a state observer needs to be designed first. Through digital communication medium, the signal is transmitted in the output channel of observer that may suffer from network-induced delays, where the output measurement y(t) sampled periodically with sampling period T. Here, y(iT) (i = 0, 1, 2, · · · , ∞) is the current measurement and y(i k T) (i k ∈ N, k = 0, 1, 2, · · · , ∞, i 0 = 0, k is triggering time) is the one latest transmitted, respectively. The judgement condition whether retrieving the transmission is determined by the following conditions: where Ω m is a positive definite weighting matrix decided by the error tolerance ρ ∈ [0, 1). The transmission scheme in the next transmission instant i i+1 T is determined by Remark 1. From the above triggered mechanism, only meeting the triggering condition that data packets sampled will be sent on the network; therefore, the network efficiency is improved. In particular, if ρ = 0, it is seen that the data transmission mechanism is simplified from event triggering to periodic triggering.

Network and ZOH
Due to network transmission speed, network transmission protocol and the load connected in the network, the network-induced delays are considered first, the released transmission delay in kth is marked as d k , that is, d k = t k − i k T, where t k is the instance that the measurement y(i k T) arrived ZOH. Let d m be the maximum transmission delay, i.e., d m = max{d k |k = 0, 1, 2, · · · , ∞}. Therefore, the interval of time [t k , t k + 1) can be divided into subintervals as follows: where j k is determined by j k = min{j|t k + (j − 1)T ≥ t k+1 , j = 1, 2, · · · }. Then which combines with (9) leads to Given the triggering situation (7), y((i k + j)T) satisfies that In addition, as shown in Figure 1 that the input y(t) in state observer satisfies that is the received signal equals to the signal released at instants of event-triggering t 0 < t 1 < t 2 < · · · < t k < · · · . The output signal from ZOH is piecewise constant but continuous from the right. Thanks to the characteristics of ZOH causal reconstruction, it is convenient to carry out the design of observer and the analysis of sliding motion.   Figure 3 plots the state response of observer system under control; the controller input is given in Figure 4.

Luenberger State Observer Design
The premise variable is also unavailable which is considered in the paper. Therefore, according to the measurement y(t) for t ∈ [t k , t k+1 ) (k = 0, 1, 2, · · · , ∞), it is easy to see as follows.
Observer Rule i: wherex(t) is the estimation of the state x(t). L i,m is the observer gain having appropriate dimensions to be determined. In addition, the dynamics of fuzzy observer (14) after fuzzy blending is inferred as It can be seen from (11) and (13) that y(t) = e k (t) + y(t − τ(t)). Moreover, the formula of the fuzzy observer is given as follows: Define e(t) = x(t) −x(t) as estimated error. Combining (3) with (16), it is easy to obtain: where (16) and (17), it is easy to obtain that the addition of time-delay term makes the effect of sliding mode control better and can better suppress the error.

Sliding Surface Design
On the basis of (16), we propose a fuzzy integral sliding surface function: In view of the systems (16) and (18), as shown as follows .
After arriving the sliding surface s(t) = 0, i.e., . s(t) = 0, the equivalent control variable can be obtained as follows: Then, by substituting (20) into (16), it is easy to obtain: where

Remark 3.
By selecting appropriate λ m such that I m is invertible. The benefit is that we can obtain the observer gain matrices by solving the optimal problem of the following conditions rather than by given in the design process.
Here, designing a sliding mode controller based on observer for the system (2) is the purpose of this paper and the controller can meet two conditions as follows:
The measurement of H ∞ performance with the condition of zero-initial will be satisfied as follows: where γ is a positive constant.

Remark 4.
By selecting this integral sliding mode functional, the linear LMI is obtained for the obtained sliding mode stochastic stability analysis, so that the gain matrix of the observer can be obtained instead of artificial design, which reduces the conservatism.

Stochastic Stability and H ∞ Performance Analysis
Remark 5. Given positive scalars d m , γ, a 1m , a 2m and ρ ∈ [0, 1), the error dynamics (17) and the sliding mode dynamics (21) are stochastically stable with an H ∞ attenuation level γ, if matrices and Y im with appropriate dimensions exist, the following condition is satisfied for each m ∈ S: The observer gain matrices are computed by Proof. First, the overall closed-loop system is stochastically stable with w(t) = 0 will be proved. Therefore, choose the following stochastic Lyapunov-Krasovskii functional:

e(s)dsdθ
By employing Definition 1, it obtains It is obvious from the Leibniz-Newton formula that where S 1m , S 2m are free weighting matrices. Moreover, Similarly, On the other hand, it holds that where α 1m and α 2m are chosen parameters. In summary, we have and Φ i,m = Φ i,m + {∑ s n=1 π mn P n , 0, ∑ s n=1 π mn P n , 0}, where Letting P m L i,m = Y i,m and Φ i,m < 0 can be known from (22) by Schur complement. So, if a scaler µ λ min −Φ i,m > 0 is denoted, we will know Hence, after using Dynkin's formula, some conclusions can be drawn, that is, for t > 0 Then, according to Definition 1, it is stochastically stable when w(t) = 0 for the sliding mode dynamics (21), and it also can be proved for error system (17) in the same way.
Next, the H ∞ performance of overall closed-loop system will be considered.

EV(t) = E
+∞ 0 LV(s)ds ≥ 0 in the condition of zero-initial. Therefore, By utilizing Schur's complement and the inequality (24), obviously,Φ i,m < 0 means J < 0. Therefore, it is stochastically stable for the sliding mode dynamics (21) with error dynamics (17) with an H ∞ disturbance attenuation level γ. Remark 6. Due to various environmental constraints, the TRs information is often not obtained as expected in practice. Therefore, TRS may encounter three situations, where π mn is completely known, partially known and completely unknown.
where ∆π mn ∈ [−δ mn , δ mn ] with δ mn ≥ 0. According to the above transformation, the following sets are defined: I m,k = I m,kn ∪ I m,ukn , where I m,kn {n :π mn is known for n ∈ S} I m,ukn {n :π mn is not known for n ∈ S} Based on the above sets, the following situations are considered: 1. m ∈ I m,kn andπ mn for ∀n ∈ I m,kn are known, that is I m,kn are known, that is I m,kn = S; 2. m ∈ I m,kn andπ mn for ∀n ∈ I m,kn are partially known, that is I m,kn = S while I m,kn is also not empty; 3. m ∈ I m,ukn andπ mn for ∀n ∈ I m,kn are partially known, that is I m,kn = S while I m,kn is also not empty; 4. m ∈ I m,ukn andπ mn for ∀n ∈ I m,kn are all unknown, that is I m,kn = φ.
It is known that the above cases 1-3 have been investigated in other works, while the cased 4 was neglected, where is the main difficulty lies. Therefore, in this paper, the following method brought from [48] is introduced.
  It is seen from the above matrix that I 3,kn is empty, while the unknown TR π 33 can be estimated by π 11 or π 22 . Based on TRs matrix mentioned above, the theorem will be obtained as follows.

Theorem 2.
Given positive scalars d m , γ, a 1m , a 2m and ρ ∈ [0, 1). The sliding mode dynamics (21) with the error dynamics (17) is stochastically stable with an H ∞ attenuation level γ, if matrices P m > 0, Q 1 > 0, Q 2 > 0, R 1 > 0, R 2 > 0, U mm > 0, W mm > 0, Ω m > 0, free weighting matrices S km (k = 1, 2, 3) and Y im with appropriate dimensions exist, the following conditions are satisfied for each m ∈ S If m ∈ I m,kn and I m,kn = S, then If m ∈ I m,ukn , there exists n = m such that n ∈ I n,kn , for ∀l ∈ I m,ukn , where Γ k1 i,m is defined as: and, Θ k1 i,m is defined as: The observer gain matrices are computed by L i,m = P −1 m Y i,m .
Proof. (Case I): i ∈ I i,kn and I i,kn = S.
According to the partition of TRs, letting ∑ s n=1,n =m π mn = −π mm . Therefore, ∑ s n=1 π mn (h)P j also is By using Schur's complement theory, it seems known that (26) supports Theorem 2 holds in this case.
(Case II): Since m ∈ I m,uk , and I m,kn = ∅, while there exists a n = m such that n ∈ I n,kn . Here, π mm is estimated by a m π nn . Denoting λ m,kn π mm .Therefore ∑ s n=1 π mn P n can be written as s ∑ n=1 π mn P n = π mm P m + ∑ n∈I m,ukn π mn P n = π mm P m − λ m,k ∑ n∈I m,ukn π mn −λ m,k P n (29) Noting that ∑ n∈I m,ukn π mn = −π mm = −λ m,k > 0. For ∀l ∈ I m,ukn , it satisfies that In the above formula (30), it has a m π nn (P m − P l ) = a mπnn (P m − P l ) + a m ∆π nn (P m − P l ) (31) Then, with the help of Lemma 1 and for any W mm > 0, it can be obtained that In combination with (28)-(32), by using Schur complement theory, Theorem 2 holds from (27) in this case.

Remark 7.
When analyzing the conditions in Theorem 2, this is an important issue that how to determine a i . Therefore, proposing the maximum optimization problem to settle this problem, that is max. ∑ a i , subject to P i , . . . ,Ω m in (26), (27).
Therefore, the H ∞ performance is ensured with unknown TRs.

Reachability of Sliding Surface
In the section, to guarantee the accessibility of the sliding surface s(t) = 0, it will be confirmed that the control scheme proposed will make the estimated state to the predesigned sliding surface in the limited range of time.

Theorem 3.
Assuming that the conditions in Theorem 2 are solvable and (18) is proposed. By the fuzzy SMC law synthesized as follows, the state trajectories of (16) will be driven onto the sliding surface s(t) = 0 in the limited range of time: in which δ is a small positive tuning scalar, and Proof. Choose Lyapunov function as follows: Then By substituting (33) into (35), we can obtain: Noting that s(t) ≤ |s(t)|, then Now, consider the equation as follows: where V −1/2 (t)dV(t) = − √ 2γdt from 0 to t * , we integrate both sides it will yield from this, it is known that a constant t * = 2V(0)/γ exists such that EV 1/2 (t * ) = 0 (for all t ≥ t * , that is Es(t) = 0). In the case where dV(t) < − √ 2γV 1/2 (t), because of monotonicity, t * is much smaller. As a result, the reachability is almost guaranteed in the limited range of time. The theorem is completely proved.

Remark 8.
Compared with the traditional Markov jump system with completely known transfer rate, this study gives a stochastic stability criterion with completely unknown transfer rate of a certain mode, which extends the theoretical depth of this kind of system. Remark 9. Theorem 3 not only illustrates the finite time reachability of sliding mode, but also proves the upper bound of the arrival time.

Numerical Example
Considering the dynamic equation of the single-link robot arm model as mentioned before ..
In detail, g = 9.81 and L = 0.5, the time invariant D(t) = D 0 = 2. M and J have three different modes as shown in Table 2. Following the fuzzy approach in Part II, the state-space can be described as follows: Plant Rule 2: IF x 1 (t) is "about π rad or −π rad", THEN . where

and
First, let us check the SMC theory with fully known TR information, and the related TR matrix of the three operation modes is given by: In this way, we can check the effectiveness of the proposed results based on Theorem 1. Suppose T = 0.1s and d m = 0.2. In addition, letting the gain matrices K 1m = −2 −3 , K 2m = −12 −6 , λ m = 2, a im = 0.1, (i = 1, 2; m = 1, 2, 3), ρ = 0.1, G = 1 0 and γ = 3.5. By solving the condition in (22), it obtains the following feasible solutions:    Note that although the issue about sliding mode control based on observer for T-S model-based Markovian jump systems has been investigated in this paper, it still leaves much space for improvements. A future study should tackle new problems such as time delay and packet dropout.
Therefore, the design of state observer in this paper is implemented in the following way: First, select appropriate gain matrices Last, design the sliding mode controller proposed in (33). The diagram of overall implementation is presented in the following Figure 5.   Note that although the issue about sliding mode control based on observer for T-S model-based Markovian jump systems has been investigated in this paper, it still leaves much space for improvements. A future study should tackle new problems such as time delay and packet dropout.
Therefore, the design of state observer in this paper is implemented in the following way: First, select appropriate gain matrices Last, design the sliding mode controller proposed in (33). The diagram of overall implementation is presented in the following Figure 5.   Note that although the issue about sliding mode control based on observer for T-S model-based Markovian jump systems has been investigated in this paper, it still leaves much space for improvements. A future study should tackle new problems such as time delay and packet dropout.
Therefore, the design of state observer in this paper is implemented in the following way: First, select appropriate gain matrices Last, design the sliding mode controller proposed in (33). The diagram of overall implementation is presented in the following Figure 5. Note that although the issue about sliding mode control based on observer for T-S model-based Markovian jump systems has been investigated in this paper, it still leaves much space for improvements. A future study should tackle new problems such as time delay and packet dropout.
Therefore, the design of state observer in this paper is implemented in the following way: First, select appropriate gain matrices K i,m such that A i,m + B m K i,m is Hurwitz. Second, obtain observer gain matrices L i,m by solving the inequalities in Theorem 1 or Theorem 2. Third, set an event-generator based on the parameter obtained in the second step. Last, design the sliding mode controller proposed in (33). The diagram of overall implementation is presented in the following Figure 5. Finally, in order to verify the advantage of the proposed method, a simulation study is conducted by comparing the system performance on original system based on the state observer without output time-delay, i.e., t . Taking the same parameters above, the simulation result is presented in Figure 6, from which it is seen that, compared with the system performance in Figure 2, much longer time is needed for the system to reach its steady state and the system stability is also affected to some extent. Therefore, proposing a time-delay Luenberger observer is a benefit for the system performance.

Conclusions
In this paper, the issue about sliding mode control based on observer for T-S modelbased Markovian jump systems was investigated. Firstly, it involved designing an eventtriggered based time-delay sliding mode observer, which can suppress the error and obtain good stability. On this basis, a novel integral sliding surface was proposed and the observer gain matrices can be computed in the design process. Then, according to stochastic stability theory, the H∞ performance of the sliding mode dynamics and the error dynamics were ensured in terms of LMI conditions. In addition, a fuzzy sliding mode controller was constructed to guarantee the finite-time reachability of the predefined sliding surface. Finally, numerical examples based on robotics were presented to verify the effectiveness of the proposed method.  Finally, in order to verify the advantage of the proposed method, a simulation study is conducted by comparing the system performance on original system based on the state observer without output time-delay, i.e.,ŷ(t − τ(t)) is changed byŷ(t). Taking the same parameters above, the simulation result is presented in Figure 6, from which it is seen that, compared with the system performance in Figure 2, much longer time is needed for the system to reach its steady state and the system stability is also affected to some extent. Therefore, proposing a time-delay Luenberger observer is a benefit for the system performance. Finally, in order to verify the advantage of the proposed method, a simulation study is conducted by comparing the system performance on original system based on the state observer without output time-delay, i.e., t . Taking the same parameters above, the simulation result is presented in Figure 6, from which it is seen that, compared with the system performance in Figure 2, much longer time is needed for the system to reach its steady state and the system stability is also affected to some extent. Therefore, proposing a time-delay Luenberger observer is a benefit for the system performance.

Conclusions
In this paper, the issue about sliding mode control based on observer for T-S modelbased Markovian jump systems was investigated. Firstly, it involved designing an eventtriggered based time-delay sliding mode observer, which can suppress the error and obtain good stability. On this basis, a novel integral sliding surface was proposed and the observer gain matrices can be computed in the design process. Then, according to stochastic stability theory, the H∞ performance of the sliding mode dynamics and the error dynamics were ensured in terms of LMI conditions. In addition, a fuzzy sliding mode controller was constructed to guarantee the finite-time reachability of the predefined sliding surface. Finally, numerical examples based on robotics were presented to verify the effectiveness of the proposed method.

Conclusions
In this paper, the issue about sliding mode control based on observer for T-S modelbased Markovian jump systems was investigated. Firstly, it involved designing an eventtriggered based time-delay sliding mode observer, which can suppress the error and obtain good stability. On this basis, a novel integral sliding surface was proposed and the observer gain matrices can be computed in the design process. Then, according to stochastic stability theory, the H ∞ performance of the sliding mode dynamics and the error dynamics were ensured in terms of LMI conditions. In addition, a fuzzy sliding mode controller was constructed to guarantee the finite-time reachability of the predefined sliding surface.