Static Output Feedback l2 – l∞ Asynchronous Control of Markov Jump Systems Under Dynamic Event-Triggered Scheme

This article studies the control problem of the discrete-time Markov jump systems (MJSs). First, the asynchronous switching between the plant modes and the controller modes is taken into consideration. To handle this issue, the estimated modes are utilized in the controller design. Second, in order to reduce the frequency of data transmission in the control system, a dynamic event-triggered communication scheme is introduced into the controller-to-actuator channel. Third, the exogenous disturbance is considered and the $l_{2}-l_{\infty }$ performance index is introduced. In addition, all the states are unavailable and only the output signals are used for the feedback design. The highlight of this study is that a more general case is considered, in which both the plant modes and the states are assumed to be unavailable. The fundamental issue is to determine the feedback gains and the event-triggering matrix simultaneously such that the closed-loop MJSs are stochastically stabilized with a certain level of $l_{2}-l_{\infty }$ performance. By constructing a mode-dependent Lyapunov function, a set of sufficient conditions is derived and the control algorithm is developed. At last, a numerical example and a DC-DC switched boost converter circuit are given to show the effectiveness and practicability of the proposed control technique.

tioning that most of these results are concerned with the static 66 ETM. Compared to the static ETM, the dynamic ETM can 67 further reduce the communication burden due to its introduc-68 tion of an additional dynamic variable [20], [21]. In [27], the 69 dynamic ETM was introduced into the sliding mode control  In response to the discussion above, it is imperative to 74 introduce the asynchronous control and event-triggered con-

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(2) Many studies on the asynchronous control of MJSs 89 such as [12], [13], [14], [15], [16] require that the states 90 are measurable. While in our work, a more general 91 case is considered, in which the states are assumed to 92 be unavailable. Then the static output feedback con-93 troller is designed for the MJSs. Moreover, based on 94 the feedback signal, the event-triggering condition is 95 constructed. 96 (3) In the literature on event-triggered asynchronous con-97 trol of MJSs, such as [22], [23], [24], [25], [26], little 98 attention is paid on the dynamic ETM. We will fill this 99 gap in this study. 100 The remainder of this study is organized as follows. 101 In Section 2, the dynamic ETM scheme together with some 102 preliminaries is introduced, based on which the asynchronous 103 control problem for MJSs is formulated. Section 3 presents 104 the main results of this work, in which the controller design is 105 derived in terms of a set of linear matrix inequalities (LMIs). 106 A numerical example is provided to verify the theoretical 107 findings in Section 4, and Section 5 summarizes this work.

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Notation: Throughout this work, · denotes the Euclidean 109 vector norm. l 2 [0, ∞); R n is the space of square-110 summable of n-dimensional vector functions over [0, ∞). 111 For w(k) ∈ l 2 [0, ∞); R n , its norm is defined as 112 Prob{X }, E{X } and E k {X } refer 116 to respectively the occurrence probability, the mathematical 117 expectation and the expectation conditional on the informa-118 tion available at time k. diag{·} represents a block-diagonal 119 matrix. λ max (·) and λ min (·) denote, respectively, the maximal 120 and minimal eigenvalue of a matrix. The symbol * means an 121 ellipsis for terms induced by symmetry.

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From η(0) ≥ 0 and ε 2 ≥ 1/ε 1 , it follows that, It is worth mentioning that (7) is an important relationship 202 used to derive the main results of this work.

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Remark 2: In this study, the ETM is only applied in the 204 C/A network. It is worth mentioning that the control tech-205 nique can also be extended to the case where the ETM is used 206 in S/C channel, or even in both S/C and C/A channels.

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Remark 3: In (5)-(6), e(k) denotes the difference between 208 the current control signalũ(k) and the oneũ(k t ) computed 209 at the last triggering instant k t . Once the triggering condition 210 in (6a) is satisfied a new event is triggered. Then the control 211 signal is transmitted to the actuator over the C/A network. 212 Otherwise, the transmission task is skipped and the actuator 213 continues to implement the control signal received at the last 214 triggered instant.

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Remark 4: In most of the existing results on ETM-based 216 control of MJSs, it is assumed that the states can be mea-217 surable [22], [23], [24], [25], [26], [27]. Then the event-218 triggering condition is constructed based on the system states. 219 It is not practical due to the unavailability of the states in 220 many cases. In this study, a more general case is considered, 221 in which the event-triggering condition is constructed based 222 on the output signal. Then the design of SOF is investigated. 223 Remark 5: Note that the triggering condition (6a) depends 224 not only on the current feedback signalsũ(k) and the one 225 obtained at the last triggered instant but also on an internal 226 dynamic state η(k), which is called dynamic ETM in the 227 literature [20], [21]. When ε 1 → ∞, the triggering condition 228 (6a) reduces to which is called static ETM [17], [18]. Compared with static 231 ETM, dynamic ETM has a lower triggering rate and thus can 232 further save the network resources [20], [21].

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Under dynamic ETM, only the feedback signalũ(k t ) com-234 puted at triggered instant is available for the actuator.ũ(k t ) 235 will be held until next event happens at instant k t+1 . To keep 236 the control input signal continuous, a ZOH is embedded at 237 the actuator side. The real control signal implemented to the 238 plant is Based on (5), the controller (3) can be rewritten as The closed-loop system is then obtained, In the following, the matrices with argument θ (k) = i 248 are denoted by their subscript for simplicity. For instance, In this work, the aim is to determine the mode-dependent initial state x(0) = x 0 and initial mode θ (0) = θ 0 , the closed- Given a scalar γ > 0, system (10) is said to be MSS with an for any w(k) ∈ l 2 [0, ∞); R n and some nonnegative 265 definite function ς(·) satisfying ς (0, 0) = 0.

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In this section, we first analyze the MSS with a certain level R n x ×n x , ∈ R n u ×n u satisfying the following conditions, 283 then the closed-loop system (10) is MSS with an l 2 − l ∞ 284 performance γ .

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Proof: Define a mode-dependent Lypunov function as 286 follows where P(θ (k)) ∈ R n x ×n x is a positive matrix to be determined. 290 For θ (k) = i, taking the conditional expectation of the differ-291 ence of V (k) along the trajectories of closed-loop system (10), 292 one has One can obtain the 296 relations as follows, Note that ε 2 < 1 and η(k) ≥ 0. It leads to (ε 2 −1)η(k) ≤ 0 and 303 Combining (18) and (19), one has where the matrices 2 , 4 are given in (16), and From (14), it follows 1 < M j=1 τ ij 1 . This together with

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Taking the expected values conditional on the information 333 available at time 0 further results in In the absence of the external distrubance, i.e., w(k) = 0, one 336 has that ξ (k) = ξ (k) withξ (k) = x T (k) e T (k) T .

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Note that when = 1, one has that

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On the other hand, from (25) it infers to

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In Theorem 1, a set of conditions is obtained to ensure the 377 MSS with an l 2 − l ∞ performance γ for the closed-loop sys-378 tem. However, it is difficult to directly solve the inequalities in 379 Theorem 1 due to the coupling terms in (13)- (15). By taking 380 into account the solvability, a set of sufficient conditions 381 for (13)- (15) is derived in the below.

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Theorem 2: Consider the MJS (1). Given a scalar γ > 383 0 and ETM constants δ, ε 1 , ε 2 satisfying (6c), if there exist 384 positive definite matrices X i ∈ R n x ×n x , H ij ∈ R n x ×n x , 385 Z ∈ R n u ×n u , matrices G i ∈ R n x ×n x , U j ∈ R n u ×n y , V j ∈ 386 R n y ×n y such that the following LMIs are feasible, Then (42) can be rearranged as Then, a sufficient condition for (44) is obtained as To see this, Pre-and post-multiplying I (V −1 j 8 ) T and its 440 transpose on both sides of (45), the condition in (44) is then 441 obtained. (45) can be rewritten as Introducing the variable definition  Pre-and post-multiplying diag{P −1 i , I } and its transpose on 455 both sides of (15), it immediately results in the LMIs in (39).

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The proof is completed.

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Remark 6: In Theorem 1, the disturbance attenuation level 458 γ is a prescribed given scalar. In many practical applications, 459 it is necessary to obtain the optimal γ . Replacing the term

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(2) At time k ≥ 0, sample the outputs y(k), obtain the mode 478 estimation σ (k) and determine the triggering instant k t .

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If k = 0 or an event is triggered, go to step 3; otherwise 480 go to step 4.

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(4) Implement the SOF control signal received at the latest 483 triggered instant.

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In order to show the effectiveness and practicability of the Let the scalars in (6c) be δ = 0.5, ε 1 = 2.5 and ε 2 = 0.5. 501 Select the l 2 − l ∞ performance γ as γ = 5. The initial 502 conditions for (1)  By 1000 repeated simulations using the same initial con-510 ditions given above, the state trajectories (in gray lines) and 511 their mean values (in blue lines) are plotted in Fig.2. The 512 corresponding control inputs (in gray lines) and their mean 513 values (in blue lines) are shown in Fig.3. It is clear that the 514 state trajectories converge to zero. In one of these 1000 sim-515 ulations, the evolution of system mode θ(k) and controller 516 mode σ (k) are given in Fig.4. It can be seen that the controller 517 mode is asynchronous with the system mode. From these 518 figures, it is shown that the stability is guaranteed under the 519 asynchronous switching.

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Next, we will show the transmission rate measured by 521 T r = The number of triggered instants System runtime .

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By 1000 calculations, the mean value of transmission rate 523 is T r = 0.4876. This means that only 48.76% packets 524 are released and transmitted over the C/A channel. In other 525 words, 51.24% network bandwidth is saved during the sim-526 ulation. For comparison, the mean value of T r of the static 527 ETM (8) is obtained as T r = 0.5376. One control trajectory 528 of these 1000 simulations and the corresponding triggered 529 instants are shown in Fig.5, where 1 means that an event is 530 triggered and 0 otherwise. From these results, it is clear that 531 the event-triggered communication scheme can really reduce 532 the frequency of data transmission. Compared to the static 533 ETM, the dynamic ETM can further reduce the communica-534 tion burden.

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At last, let us show the level of l 2 − l ∞ performance. 536 Without of generality, the initial conditions are reset as x(0) = 537 0 0 T and η(0) = 0. By 1000 calculations the ration of 538 z(k) l ∞ / w(k) l 2 (in gray lines) and their mean values (in 539 blue lines) are given in Fig.6, from which it is clear that the 540 ration is always less than 5. This implies that the predefined 541 level of l 2 − l ∞ performance γ = 5 is guaranteed. From 542 the aforementioned simulation results, the effectiveness of the 543 control technique presented in this work has been confirmed. 544 Example 2: (DC-DC switched boost converter circuit) 545 Consider a DC-DC switched boost converter circuit [7]. 546 As shown in Fig.7, the DC-DC switched boost converter 547 circuit is composed of a power source V in (t), an inductor 548 Y. Lin, T. Shi: Static Output Feedback l 2 − l ∞ Asynchronous Control of MJSs Under Dynamic Event-Triggered Scheme Model 2: Model 2: Other system matrices are chosen as,   values (in blue lines) are shown in Fig.9. In one of these 589 1000 realizations, the evolution of system mode θ (k) and 590 controller mode σ (k) are given in Fig.10. It can be seen that 591 the controller mode is asynchronous with the system mode. 592 From these figures, it is clear that the stability is ensured by 593 the obtained asynchronous static output feedback controller. 594 By 1000 calculations, the mean value of transmission 595 rate is T r = 0.2282. One control trajectory of these 596 shown in Fig.11, where 1 means that an event is triggered 598 and 0 otherwise. From these results, it is clear that the circuit 599 works well with reduced communication burden.