Problem description

Fig. 1
figure 1

Event-triggered asynchronous control with packet dropouts

This work focuses on the design of the asynchronous dissipative controller for networked time-delay MJSs with packet dropouts under the event-triggered scheme, as shown in Fig. 1. Consider the networked time-delay MJSs as follows:

$$begin{aligned} left{ {begin{array}{*{20}{l}} {{x_{k + 1}} = {A_{{partial _k}}}{x_k} + {A_{d{partial _k}}}{x_{k – {d_k}}} + {B_{1{partial _k}}}{u_k} + {D_{1{partial _k}}}{w_k}}\ {{y_k} = {C_{{partial _k}}}{x_k} + {C_{d{partial _k}}}{x_{k – {d_k}}} + {B_{2{partial _k}}}{u_k} + {D_{2{partial _k}}}{w_k}}\ {{x_{{k_0}}} = {chi _{{k_0}}},{k_0} = – {d_2}, – {d_2} + 1, cdots , – 1,0} end{array}} right. end{aligned}$$

(1)

where ({x_k} in {{mathbb {R}}^{{n_x}}}), ({chi _{{k_0}}}), ({y_k} in {{mathbb {R}}^{{n_y}}}), ({u_k} in {{mathbb {R}}^{{n_u}}}), ({w_k} in {{mathbb {R}}^{{n_w}}})(({w_k} in l[0,infty ))) denote the system state, the system initial state, the controlled output, the control input and the disturbance input, respectively. There exists time delay ({d_k} in {{mathbb {N}}^ + }) in the system (1) with the upper bound ({d_1}) and the lower bound ({d_2}), ({d_1} < {d_2}). (({A_{{partial _k}}}), ({A_{d{partial _k}}}), ({B_{1{partial _k}}}), ({B_{2{partial _k}}}), ({C_{{partial _k}}}), ({C_{d{partial _k}}}), ({D_{1{partial _k}}}), ({D_{2{partial _k}}})) are the given real matrices with appropriate dimensions. The Markov jump process of system (1) is controlled by the mode parameter ({partial _k}) (({partial _k} in mathrm{{S}}), (mathrm {S} = { 1,2, cdots ,s})) and complies with the transition probability matrix (TPM) (Upsilon = [{pi _{pq}}]), in which the transition probability(TP) ({pi _{pq}}) is defined as follows:

$$begin{aligned} Pr { {partial _{k + 1}} = q|{partial _k} = p} = {pi _{pq}} end{aligned}$$

(2)

obviously, ({pi _{pq}} in [0,1]) and (sum nolimits _{q = 1}^s {{pi _{pq}} = 1}) for (forall p,q in mathrm{{S}}).

Considering the limited bandwidth and energy in the system, we will introduce an event-trigger to reduce the transmission rate of sampling signals and relieve the communication pressure. The event-triggered mechanism is as follows:

$$begin{aligned} {hat{x}_{ik}} = left{ {begin{array}{*{20}{c}} {{x_{ik}}}\ {{{hat{x}}_{i(k – 1)}}} end{array}} right. begin{array}{*{20}{c}} {}&{} end{array}begin{array}{*{20}{c}} {|{{hat{x}}_{i(k – 1)}} – {x_{ik}}| > {delta _i}|{x_{ik}}|}\ {|{{hat{x}}_{i(k – 1)}} – {x_{ik}}| le {delta _i}|{x_{ik}}|} end{array} end{aligned}$$

(3)

where ({delta _i} in left[ {0,1} right]) is error threshold. Denote ({H_k} = diag{ {Delta _{1k}},{Delta _{2k}}, cdots ,{Delta _{{n_x}k}}}), ({Delta _{ik}} in [ – {delta _i},{delta _i}],i = 1,2, cdots ,{n_x}). The sampling signal is transmitted to the controller only when the event-triggered condition is satisfied. Then according to (3), we can obtain

$$begin{aligned} {hat{x}_k} = (I + {H_k}){x_k} end{aligned}$$

(4)

Remark 1

Owing to the introduction of the event-triggered transmission scheme, the sampling signal does not need to be transmitted periodically, thus gaining the aim of reducing the data transmission frequency. In addition, we introduce a performance index of data transmission performance (DTP = {t_S}/{t_T} times 100%) to represent the communication performance [40], in which ({t_S}) and ({t_T}) indicate the transmission times of sampled data with and without the event-triggered mechanism, respectively.

Based on the output of the trigger (4), we will adopt the following asynchronous controller:

$$begin{aligned} {u_k} = {K_{{sigma _k}}}{hat{x}_k} end{aligned}$$

(5)

where ({K_{{sigma _k}}}) represents the controller gain, and ({sigma _k}) is the mode of the controller. The mode ({partial _k}) of system (1) affects the mode ({sigma _k}) of the controller through the conditional transition matrix(CPM) (Psi = [{theta _{pj}}]), and its conditional transition (CP) ({theta _{pj}}) is described as follows [41]:

$$begin{aligned} Pr { {sigma _k} = j|{partial _k} = p} = {theta _{pj}} end{aligned}$$

(6)

which denotes the controller works in mode j when the system (1) is in mode p, in which ({theta _{pj}} in [0,1]) and (sum nolimits _{j = 1}^s {{theta _{pj}}} = 1) for (forall p,j in S).

Remark 2

The controller and the physical plant (1) are mode asynchronous, and the asynchronous relationship is described by an HMM, which can well describe the asynchronization and connection between the controller and the plant through CPM. The controller’s mode is influenced by the plant’s mode, and their asynchronous level is reflected by the CPs. In addition, the asynchronous controller under the HMM scheme is more general, which covers both synchronous (i.e., (Psi mathrm{{ = }}I)) and mode-independent (i.e., ({sigma _k} in { 1})) cases [42].

Considering that there exists network between the controller and the actuator, packet dropouts are inevitable. A Bernoulli stochastic process is used in this work to describe the packet dropout process:

$$begin{aligned} {hat{u}_k} = {beta _k}{u_k} end{aligned}$$

(7)

where ({beta _k}) denotes the Bernoulli process with

$$begin{aligned} Pr { {beta _k} = 1} = beta , Pr { {beta _k} = 0} = 1 – beta end{aligned}$$

(8)

and satisfies

$$begin{aligned} mathrm{E}{ {beta _k}} = beta , mathrm{E}{ beta _k^2} = beta end{aligned}$$

(9)

Furthermore, to facilitate subsequent derivations. Defining ({{bar{beta }} _k} = {beta _k} – beta), we obtain

$$begin{aligned} mathrm{E}{ {{bar{beta }} _k}} = 0, mathrm{E}{ {bar{beta }} _k^2} = {{bar{beta }} ^2} end{aligned}$$

(10)

where ({bar{beta }} = sqrt{beta – {beta ^2}}).

For the convenience of expression, let ({partial _k} = p), ({partial _{k + 1}} = q) and ({sigma _k} = j). According to (1), (4) and (7), we can obtain the following closed-loop dynamic system:

$$begin{aligned} left{ {begin{array}{*{20}{l}} {{x_{k + 1}} = {{bar{A}}_{pjk}}{x_k} + {A_{dp}}{x_{k – d_{k}}} + {D_{1p}}{w_k}}\ {{y_k} = {{bar{C}}_{pjk}}{x_k} + {C_{dp}}{x_{k – d _{k}}} + {D_{2p}}{w_k}} end{array}} right. end{aligned}$$

(11)

where ({bar{A}_{pjk}} = {A_p} + {beta _k}{B_{1p}}{K_j}(I+{H_k})), ({bar{C}_{pjk}} = {C_p} + {beta _k}{B_{2p}}{K_j}(I+{H_k})).

Next, some important lemmas and definitions are introduced to facilitate the work of this paper.

Definition 2.1

[43] The system (11) is stochastically stable, if ({w_k} equiv 0) and for any initial condition (({x_0},{partial _0})), satisfying

$$begin{aligned} mathrm{E} left{ sum limits _{k = 0}^infty ||x_k||^2|x_0,partial _0 right} < infty end{aligned}$$

(12)

Definition 2.2

[43] Given a constant (gamma > 0), matrices (mu le 0), (vartheta) and symmetric (upsilon), the closed-loop system (11) is considered to be strictly ((mu ,vartheta ,upsilon ) – gamma -)dissipative, for any positive integer N, when ({w_k} in l[0,infty )) and under 0 initial condition, satisfying

$$begin{aligned} sum limits _{k = 0}^N {mathrm{E}{ F({w_k},{y_k})} ge gamma sum limits _{k = 0}^N {w_k^{mathrm{{T}}}{w_k}} } end{aligned}$$

(13)

where (F({w_k},{y_k}) = y_k^{mathrm{{T}}}mu {y_k} + 2y_k^{mathrm{{T}}}vartheta {w_k} + w_k^{mathrm{{T}}}upsilon {w_k}), and (mu buildrel Delta over = – U_1^{mathrm{{T}}}{U_1}) is negative semi-definite.

Lemma 2.1

[44] Given the matrices A, B, C and ({A^{mathrm{{T}}}} = A), then

$$begin{aligned} A + CB + {C^{mathrm{{T}}}}{B^{mathrm{{T}}}} < 0 end{aligned}$$

(14)

holds if there exists a matrix (D > 0), satisfying

$$begin{aligned} A + C{D^{ – 1}}{C^{mathrm{{T}}}} + {B^{mathrm{{T}}}}DB < 0 end{aligned}$$

(15)

Stability and dissipativity analysis of the system

This section will derive a sufficient condition to ensure that the system (11) is stochastically stable and strictly ((mu ,vartheta ,upsilon ) – gamma -)dissipative.

Theorem 2.1

The system (11) is stochastically stable and strictly ((mu ,vartheta ,upsilon ) – gamma -)dissipative, if there exist a matrix ({K_j} in {{mathbb {R}}^{{n_u} times {n_x}}}), positive matrices ({P_p} in {{mathbb {R}}^{{n_x} times {n_x}}}), ({Q} in {{mathbb {R}}^{{n_x} times {n_x}}}), ({R_{pj}} in {{mathbb {R}}^{{n_x} times {n_x}}}) and a positive diagonal matrix ({G_{pj}} in {{mathbb {R}}^{{n_u} times {n_u}}}), for (forall p,j in mathrm{{S}}), satisfying

$$begin{aligned} sum limits _{j = 1}^s {{theta _{pj}}{R_{pj}} < {P_p}} end{aligned}$$

(16)

$$begin{aligned} left[ {begin{array}{*{20}{c}} {{Pi _{pj}}}&{}{{N_{pj}}}&{}{L^{mathrm{{T}}}Lambda {G_{pj}}}\ *&{}{ – {G_{pj}}}&{}0\ *&{}*&{}{ – {G_{pj}}} end{array}} right] < 0 end{aligned}$$

(17)

where

({Pi _{pj}} = left[ {begin{array}{*{20}{c}} { – bar{P}_p^{ – 1}}&{}0&{}0&{}0&{}{bar{A}_{pj}^*}&{}{{A_{dp}}}&{}{{D_{1p}}}\ *&{}{ – bar{P}_p^{ – 1}}&{}0&{}0&{}{{bar{beta }} {{bar{B}}_{1pj}}}&{}0&{}0\ *&{}*&{}{ – I}&{}0&{}{{U_1}bar{C}_{pj}^*}&{}{{U_1}{C_{dp}}}&{}{{U_1}{D_{2p}}}\ *&{}*&{}*&{}{ – I}&{}{{bar{beta }} {U_1}{{bar{B}}_{2pj}}}&{}0&{}0\ *&{}*&{}*&{}*&{}{dQ – {R_{pj}}}&{}0&{}{ – bar{C}_{pj}^{*mathrm{{T}}}vartheta }\ *&{}*&{}*&{}*&{}*&{}{ – Q}&{}{ – C_{dp}^{mathrm{{T}}}vartheta } \ *&{}*&{}*&{}*&{}*&{}*&{}M_p end{array}} right]),

({N_{pj}} = {left[ {begin{array}{*{20}{c}} {beta bar{B}_{1pj}^{mathrm{{T}}}}&{{bar{beta }} bar{B}_{1pj}^{mathrm{{T}}}}&{beta bar{B}_{2pj}^{mathrm{{T}}}}&{{bar{beta }} bar{B}_{2pj}^{mathrm{{T}}}}&0&0&{beta {{({vartheta ^{mathrm{{T}}}}{B_{2pj}}{K_j})}^{mathrm{{T}}}}} end{array}} right] ^{mathrm{{T}}}}),

({L} = left[ {begin{array}{*{20}{c}} 0&0&0&0&I&0&0 end{array}} right]), (bar{A}_{pj}^* = {A_p} + beta {B_{1p}}{K_j}), ({bar{B}_{1pj}} = {B_{1p}}{K_j}),

(bar{C}_{pj}^* = {C_p} + beta {B_{2p}}{K_j}), ({bar{B}_{2pj}} = {B_{2p}}{K_j}), (Lambda = diag{ {delta _1},{delta _2}, cdots ,{delta _{{n_x}}}}),

(M_p = – D_{2p}^{mathrm{{T}}}vartheta – {vartheta ^{mathrm{{T}}}}{D_{2p}} + gamma I – upsilon), ({bar{P}_p} = sum limits _{q = 1}^s {{pi _{pq}}{P_q}}), (d = {d_2} – {d_1} + 1).

Proof

First, using Schur complement to (17), we obtain that

$$begin{aligned} {Pi _{pj}} + N_{pj}^{mathrm{{T}}}G_{pj}^{ – 1}{N_{pj}} + {L}Lambda {G_{pj}}Lambda L^{mathrm{{T}}} < 0 end{aligned}$$

(18)

By Lemma 2.1, we have

$$begin{aligned} {{bar{Pi }} _{pjk}} buildrel Delta over = {Pi _{pj}} + N_{pj}^{mathrm{{T}}}{H_k}L^{mathrm{{T}}} + {L}{H_k}{N_{pj}} < 0 end{aligned}$$

(19)

and thus obtain

$$begin{aligned} {{bar{Pi }} _{pjk}} = left[ {begin{array}{*{20}{c}} { – bar{P}_p^{ – 1}}&{}0&{}0&{}0&{}{{{tilde{A}}_{pjk}}}&{}{{A_{dp}}}&{}{{D_{1p}}}\ *&{}{ – bar{P}_p^{ – 1}}&{}0&{}0&{}{{bar{beta }} {{tilde{B}}_{1pjk}}}&{}0&{}0\ *&{}*&{}{ – I}&{}0&{}{{U_1}{{tilde{C}}_{pjk}}}&{}{{U_1}{C_{dp}}}&{}{{U_1}{D_{2p}}}\ *&{}*&{}*&{}{ – I}&{}{{bar{beta }} {U_1}{{tilde{B}}_{2pjk}}}&{}0&{}0\ *&{}*&{}*&{}*&{}{dQ – {R_{pj}}}&{}0&{}{ – tilde{C}_{pjk}^{mathrm{{T}}}vartheta }\ *&{}*&{}*&{}*&{}*&{}{ – Q}&{}{ – C_{dp}^{mathrm{{T}}}vartheta }\ *&{}*&{}*&{}*&{}*&{}*&{}M_p end{array}} right] < 0 end{aligned}$$

(20)

where

({tilde{A}_{pjk}} = {A_p} + beta {B_{1p}}{K_j}(I + {H_k})), ({tilde{B}_{1pjk}} = {B_{1p}}{K_j}(I + {H_k})),

({tilde{C}_{pjk}} = {C_p} + beta {B_{2p}}{K_j}(I + {H_k})), ({tilde{B}_{2pjk}} = {B_{2p}}{K_j}(I + {H_k})).

which implies

$$begin{aligned} left[ {begin{array}{*{20}{c}} { – bar{P}_p^{ – 1}}&{}0&{}{{{tilde{A}}_{pjk}}}&{}{{A_{dp}}}\ *&{}{ – bar{P}_p^{ – 1}}&{}{{bar{beta }} {{tilde{B}}_{1pjk}}}&{}0\ *&{}*&{}{dQ – {R_{pj}}}&{}0\ *&{}*&{}*&{}{ – Q} end{array}} right] < 0 end{aligned}$$

(21)

Then, using Schur complement to (21) and (20) again, we have

(22)

where

figure a

({{bar{Gamma }}_p ^2} = diag{ – bar{P}_p, – bar{P}_p, – I, – I}), ({Gamma ^1} = left[ {begin{array}{*{20}{c}} {dQ}&{}0\ *&{}{ – Q} end{array}} right]), ({phi _{pjk}} = left[ {begin{array}{*{20}{c}} {{{tilde{A}}_{pjk}}}&{}{{A_{dp}}}\ {{bar{beta }} {{tilde{B}}_{1pjk}}}&{}0 end{array}} right]),

(Gamma _{pjk}^2 = left[ {begin{array}{*{20}{c}} {dQ}&{}0&{}{ – tilde{C}_{pjk}^{mathrm{{T}}}vartheta }\ *&{}Q&{}{ – C_{dp}^{mathrm{{T}}}vartheta }\ *&{}*&{}{{M_p}} end{array}} right]), ({varphi _{pjk}} = left[ {begin{array}{*{20}{c}} {{{tilde{A}}_{pjk}}}&{}{{A_{dp}}}&{}{{D_{1p}}}\ {{bar{beta }} {{tilde{B}}_{1pjk}}}&{}0&{}0\ {{U_1}{{tilde{C}}_{pjk}}}&{}{{U_1}{C_{dp}}}&{}{{U_1}{D_{2p}}}\ {{bar{beta }} {U_1}{{tilde{B}}_{2pjk}}}&{}0&{}0 end{array}} right]).

Next, choose the mode-dependent Lyapunov–Krasovskii function as follows:

$$begin{aligned} {V_k} = sum limits _{l = 1}^2 {{V_{lk}}} end{aligned}$$

(23)

where ({V_{1k}} = x_k^{mathrm{{T}}}{P_{{partial _k}}}{x_k}), ({V_{2k}} = sum limits _{b = – {d_2} + 1}^{ – {d_1} + 1} {sum limits _{a = k – 1 + b}^{k – 1} {x_a^{mathrm{{T}}}} } Q{x_a}). We introduce ({xi _{1k}} = {left[ {begin{array}{*{20}{c}} {x_k^{mathrm{{T}}}}&{x_{k – d_k}^{mathrm{{T}}}} end{array}} right] ^{mathrm{{T}}}}) and ({xi _k} = {left[ {begin{array}{*{20}{c}} {xi _{1k}^{mathrm{{T}}}}&{w_k^{mathrm{{T}}}} end{array}} right] ^{mathrm{{T}}}}). Denoting (nabla {V_k}) as the forward differential of ({V_k}),

$$begin{aligned} mathrm{E}{ nabla {V_k}} = mathrm{E}{ nabla {V_{1k}}} + mathrm{E}{ nabla {V_{2k}}} end{aligned}$$

(24)

Then, figure out (mathrm{E}{ nabla {V_{1k}}}) and (mathrm{E}{ nabla {V_{2k}}})

$$begin{aligned} begin{array}{l} mathrm{E}{ nabla {V_{1k}}} = left{ {{V_{1(k + 1)}} – {V_{1k}}|{x_k},{partial _k} = p} right} \ qquad = mathrm{E}{ x_{k + 1}^{mathrm{{T}}}{P_q}{x_{k + 1}} – x_k^{mathrm{{T}}}{P_p}{x_k}} \ qquad = mathrm{E}{ sum limits _{j = 1}^s {sum limits _{q = 1}^s {{theta _{pj}}{pi _{pq}}x_{k + 1}^{mathrm{{T}}}{P_q}{x_{k + 1}} – x_k^{mathrm{{T}}}{P_p}{x_k}} } } \ qquad = mathrm{E}{ sum limits _{j = 1}^s {{theta _{pj}}{xi _k^{mathrm{{T}}}}left[ {begin{array}{*{20}{c}} {phi _{pjk}^{mathrm{{T}}}}\ {D_{1p}^{mathrm{{T}}}} end{array}} right] {{bar{P}}_p}left[ {begin{array}{*{20}{c}} {{phi _{pjk}}}&{{D_{1p}}} end{array}} right] {xi _k} – x_k^{mathrm{{T}}}{P_p}{x_k}} } end{array} end{aligned}$$

(25)

$$begin{aligned} begin{array}{l} mathrm{E}{ nabla {V_{2k}}} = mathrm{E}{ {V_{2(k + 1)}} – {V_{2k}}} \ qquad = mathrm{E}{ sum limits _{b = – {d_2} + 1}^{ – {d_1} + 1} {sum limits _{a = k + b}^k {x_a^{mathrm{{T}}}} } Q{x_a} – sum limits _{b = – {d_2} + 1}^{ – {d_1} + 1} {sum limits _{a = k – 1 + b}^{k – 1} {x_a^{mathrm{{T}}}} } Q{x_a}} \ qquad = mathrm{E}{ sum limits _{b = – {d_2} + 1}^{ – {d_1} + 1} {{ x_k^{mathrm{{T}}}Q{x_k} – x_{k – 1 + b}^{mathrm{{T}}}Q{x_{k – 1 + b}}} } } \ le mathrm{E}{ x_k^{mathrm{{T}}}dQ{x_k} – x_{k – d_{k}}^{mathrm{{T}}}Q{x_{k – d_{k}}}} \ qquad = mathrm{E}{ xi _{1k}^{mathrm{{T}}}{Gamma ^1}{xi _{1k}}} end{array} end{aligned}$$

(26)

where “(le)” is obtained from (27) and (28).

$$begin{aligned}&sum limits _{b = – {d_2} + 1}^{ – {d_1} + 1} {x_k^{mathrm{{T}}}Q{x_k}} = x_k^{mathrm{{T}}}dQ{x_k} end{aligned}$$

(27)

$$begin{aligned}&sum limits _{b = – {d_2} + 1}^{ – {d_1} + 1} {x_{k – 1 + b}^{mathrm{{T}}}Q{x_{k – 1 + b}}} = sum limits _{b = k – {d_2}}^{k – {d_1}} {x_b^{mathrm{{T}}}Q{x_b}} ge x_{k – d_{k}}^{mathrm{{T}}}Q{x_{k – d_{k}}} end{aligned}$$

(28)

Noting that ({w_k} equiv 0) in the definition of stochastic stability, and combining (24), (25) and (26), we can get

$$begin{aligned} begin{array}{l} mathrm{E}{ nabla {V_k}} = mathrm{E}{ nabla {V_{1k}}} + mathrm{E}{ nabla {V_{2k}}} \ le mathrm{E}{ sum limits _{j = 1}^s {{theta _{pj}}xi _{1k}^{mathrm{{T}}}{Omega _{1pjk}}{xi _{1k}} – x_k^{mathrm{{T}}}{P_p}{x_k}} } \ < mathrm{E}{ x_k^{mathrm{{T}}}(sum limits _{j = 1}^s {{theta _{pj}}{R_{pj}} – {P_p})} {x_k}} \ le varpi mathrm{E}{ x_k^{mathrm{{T}}}{x_k}} end{array} end{aligned}$$

(29)

where “<” is obtained from (22), and (varpi = {lambda _{max }}(sum limits _{j = 1}^s {{theta _{pj}}{R_{pj}} – {P_p}} )).

Accordingly,

$$begin{aligned} mathrm{E}{ sum limits _0^infty {nabla {V_k}} } = mathrm{E}{ {V_infty } – {V_0}} le varpi mathrm{E}{ sum limits _0^infty {x_k^{mathrm{{T}}}{x_k}} } end{aligned}$$

(30)

We know that (varpi < 0) from (16); hence,

$$begin{aligned} mathrm{E}left{ sum limits _0^infty {x_k^{mathrm{{T}}}{x_k}} right} < infty end{aligned}$$

(31)

which conforms to definition 2.1; namely, the stochastic stability of the system (11) is proved.

Next, we will show that the system (11) is strictly ((mu ,vartheta ,upsilon ) – gamma -)dissipative. Define the performance index as

$$begin{aligned} begin{array}{l} J = sum limits _{k = 0}^infty {mathrm{E}{ w_k^{mathrm{{T}}}(gamma I – upsilon ){y_k} – y_k^{mathrm{{T}}}mu {y_k} – 2y_k^{mathrm{{T}}}vartheta {w_k}} } \ le sum limits _{k = 0}^infty {mathrm{E}{ } w_k^{mathrm{{T}}}(gamma I – upsilon ){y_k} – y_k^{mathrm{{T}}}mu {y_k} – 2y_k^{mathrm{{T}}}vartheta {w_k} + nabla {V_k}} end{array} end{aligned}$$

(32)

Then, calculate (mathrm{E}{ nabla {V_{1k}}}) and (mathrm{E}{ nabla {V_{2k}}}), respectively,

$$begin{aligned} mathrm{E}left{ {nabla {V_{1k}}} right}= & {} mathrm{E}left{ {sum limits _{j = 1}^s {{theta _{pj}}x_{k + 1}^{mathrm{{T}}}{{bar{P}}_p}{x_{k + 1}} – x_k^{mathrm{{T}}}{P_p}{x_k}} } right} end{aligned}$$

(33)

$$begin{aligned} mathrm{E}left{ {nabla {V_{2k}}} right}= & {} mathrm{E}left{ {sum limits _{j = 1}^s {{theta _{pj}}{xi _{k} ^{mathrm{{T}}}}diag{ {Gamma ^1},0} xi _{k} } } right} end{aligned}$$

(34)

Combining (32), (33) and (34), we can obtain

$$begin{aligned} begin{array}{l} J le sum limits _{k = 0}^infty {mathrm{E}{ sum limits _{j = 1}^s {{theta _{pj}}{xi _k ^{mathrm{{T}}}}{Omega _{2pjk}}xi _k – x_k^{mathrm{{T}}}{P_p}{x_k}} } } \< sum limits _{k = 0}^infty {mathrm{E}{ x_k^{mathrm{{T}}}(sum limits _{j = 1}^s {{theta _{pj}}{R_{pj}} – {P_p}} )} {x_k}} \ < 0 end{array} end{aligned}$$

(35)

where the two “<” are obtained from (22) and (16), respectively. By definition 2.2, we can know that the system (11) is strictly ((mu ,vartheta ,upsilon ) – gamma -)dissipative. Thus, the proof is completed. (square)

Remark 3

A sufficient condition for stochastic stability and strict dissipativity is derived for the system (11) in Theorem 2.1. However, in terms of existing nonlinear term in Theorem 2.1, we cannot parameterize the controller gain directly by conditions (16) and (17); therefore, further linearization is required.

Design of the event-triggered asynchronous controller

This section will provide a design method for an event-triggered asynchronous controller and further determine the controller gains.

Theorem 2.2

The system (11) is stochastically stable and strictly ((mu ,vartheta ,upsilon ) – gamma -)dissipative, if there exist matrices ({bar{K}_j} in {{mathbb {R}}^{{n_u} times {n_x}}}), (V in {{mathbb {R}}^{{n_x} times {n_x}}}), positive matrices ({bar{P}_p} in {{mathbb {R}}^{{n_x} times {n_x}}}), ({bar{R}_{pj}} in {{mathbb {R}}^{{n_x} times {n_x}}}), (bar{Q} in {{mathbb {R}}^{{n_x} times {n_x}}}) and a positive diagonal matrix ({bar{G}_{pj}} in {{mathbb {R}}^{{n_u} times {n_u}}}), for (forall p,j in mathrm{{S}}), satisfying

$$begin{aligned} left[ {begin{array}{*{20}{c}} { – {{bar{P}}_p}}&{}{{J_p}}\ *&{}{{{hat{R}}_p}} end{array}} right] < 0 end{aligned}$$

(36)

$$begin{aligned} left[ {begin{array}{*{20}{c}} {{X_{pj}}}&{}{{Y_{pj}}}&{}{{Z_{pj}}}\ *&{}{ – I}&{}0\ *&{}*&{}{{{hat{P}}}} end{array}} right] < 0 end{aligned}$$

(37)

where

({J_p} = [sqrt{{theta _{p1}}} {bar{P}_p} cdots sqrt{{theta _{pj}}} {bar{P}_p} cdots sqrt{{theta _{ps}}} {bar{P}_p}]), ({hat{R}_p} = diag{ – {bar{R}_{p1}}, cdots , – {bar{R}_{pj}}, cdots , – {bar{R}_{ps}}}),

({X_{pj}} = left[ {begin{array}{*{20}{c}} {dbar{Q} + {{bar{R}}_{pj}} – {V^{mathrm{{T}}}} – V}&{}0&{}{ – tilde{C}_{pj}^{*mathrm{{T}}}vartheta }&{}0&{}{Lambda {{bar{G}}_{pj}}}\ *&{}{ – bar{Q}}&{}{ – bar{C}_{dp}^{mathrm{{T}}}vartheta }&{}0&{}0\ *&{}*&{}M_p&{}{beta {vartheta ^{mathrm{{T}}}}{B_{2p}}{{bar{K}}_j}}&{}0\ *&{}*&{}*&{}{ – {{bar{G}}_{pj}}}&{}0\ *&{}*&{}*&{}*&{}{ – {{bar{G}}_{pj}}} end{array}} right]),

({Y_{pj}}={left[ {begin{array}{*{20}{c}} {{U_1}({C_p}V + beta {B_{2p}}{{bar{K}}_j})}&{}{{U_1}{C_{dp}}V}&{}{{U_1}{D_{2p}}}&{}{beta {U_1}{B_{2p}}{{bar{K}}_j}}&{}0\ {{bar{beta }} {U_1}{B_{2p}}{{bar{K}}_j}}&{}0&{}0&{}{{bar{beta }} {U_1}{B_{2p}}{{bar{K}}_j}}&{}0 end{array}} right] ^{mathrm{{T}}}}),

({Z_{pj}} = left[ {begin{array}{*{20}{c}} {sqrt{{pi _{p1}}} W_{pj}^{mathrm{{T}}}}&{sqrt{{pi _{p2}}} W_{pj}^{mathrm{{T}}}}&cdots&{sqrt{{pi _{ps}}} W_{pj}^{mathrm{{T}}}} end{array}} right]),

({W_{pj}} = {left[ {begin{array}{*{20}{c}} {{A_p}V + beta {B_{1p}}{{bar{K}}_j}}&{}{{A_{dp}}V}&{}{{D_{1p}}}&{}{beta {B_{1p}}{{bar{K}}_j}}&{}0\ {{bar{beta }} {B_{1p}}{{bar{K}}_j}}&{}0&{}0&{}{{bar{beta }} {B_{1p}}{{bar{K}}_j}}&{}0 end{array}} right] }),

(tilde{C}_{pj}^* = {C_p}V + beta {B_{2p}}{bar{K}_j}), ({bar{C}_{dp}} = {C_{dp}}V), ({hat{P}} = diag{ – {bar{P}_1}, – {bar{P}_2}, cdots , – {bar{P}_s}}).

and the controller gain ({K_j}) can be determined by

$$begin{aligned} {K_j} = {bar{K} _j}{V^{ – 1}} end{aligned}$$

(38)

Proof

First, we define

$$begin{aligned} begin{array}{c} {{bar{P}}_p} = P_p^{ – 1},{{bar{R}}_{pj}} = R_{pj}^{ – 1},{{bar{K}}_j} = {K_j}V, bar{Q} = {V^{mathrm{{T}}}}QV,{{bar{G}}_{pj}} = {V^{mathrm{{T}}}}{{ G}_{pj}}V end{array} end{aligned}$$

(39)

where V is an invertible slack matrix. Applying a congruence conversion to (36) by (diag{ {P_p},I, cdots ,I}), one has

$$begin{aligned} left[ {begin{array}{*{20}{c}} { – {P_p}}&{}{{{bar{J}}_p}}\ *&{}{{{hat{R}}_p}} end{array}} right] < 0 end{aligned}$$

(40)

where ({bar{J}_p} = [sqrt{{theta _{p1}}} I cdots sqrt{{theta _{pj}}} I cdots sqrt{{theta _{ps}}} I]). Then, (40) is equivalent to (16).

Moreover, due to the fact that

$$begin{aligned} {({bar{R}_{pj}} – V)^{mathrm{{T}}}}bar{R}_{pj}^{ – 1}({bar{R}_{pj}} – V) ge 0 end{aligned}$$

(41)

namely

$$begin{aligned} – {V^{mathrm{{T}}}}bar{R}_{pj}^{ – 1}V le {bar{R}_{pj}} – {V^{mathrm{{T}}}} – V end{aligned}$$

(42)

Then, (37) implies

$$begin{aligned} left[ {begin{array}{*{20}{c}} {{{bar{X}}_{pj}}}&{}{{Y_{pj}}}&{}{{Z_{pj}}}\ *&{}{ – I}&{}0\ *&{}*&{}{{{hat{P}}}} end{array}} right] < 0 end{aligned}$$

(43)

where

({bar{X}_{pj}} = left[ {begin{array}{*{20}{c}} {dbar{Q} + {V^{mathrm{{T}}}}{{bar{R}}_{pj}}V}&{}0&{}{ – tilde{C}_{pj}^{*mathrm{{T}}}vartheta }&{}0&{}{Lambda {{bar{G}}_{pj}}}\ *&{}{ – bar{Q}}&{}{ – bar{C}_{dp}^{mathrm{{T}}}vartheta }&{}0&{}0\ *&{}*&{}M_p&{}{beta {vartheta ^{mathrm{{T}}}}{B_{2p}}{{bar{K}}_j}}&{}0\ *&{}*&{}*&{}{ – {{bar{G}}_{pj}}}&{}0\ *&{}*&{}*&{}*&{}{ – {{bar{G}}_{pj}}} end{array}} right]).

Denoting (Theta = diag{ {({V^{mathrm{{T}}}})^{ – 1}},{({V^{mathrm{{T}}}})^{ – 1}},I,{({V^{mathrm{{T}}}})^{ – 1}},{({V^{mathrm{{T}}}})^{ – 1}},I, cdots ,I}), and applying a congruence conversion to (43) by (Theta) , we can get

$$begin{aligned} left[ {begin{array}{*{20}{c}} {{{tilde{X}}_{pj}}}&{}{{{tilde{Y}}_{pj}}}&{}{{{tilde{Z}}_{pj}}}\ *&{}{ – I}&{}0\ *&{}*&{}{{{hat{P}}}} end{array}} right] < 0 end{aligned}$$

(44)

where

({X_{pj}} = left[ {begin{array}{*{20}{c}} {dQ – {R_{pj}}}&{}0&{}{ – bar{C}_{pj}^{*mathrm{{T}}}vartheta }&{}0&{}{Lambda {G_{pj}}}\ *&{}{ – Q}&{}{ – C_{dp}^{mathrm{{T}}}vartheta }&{}0&{}0\ *&{}*&{}M_p&{}{beta {vartheta ^{mathrm{{T}}}}{B_{2p}}{{ K}_j}}&{}0\ *&{}*&{}*&{}{ – {G_{pj}}}&{}0\ *&{}*&{}*&{}*&{}{ – {G_{pj}}} end{array}} right]),

({tilde{Y}_{pj}} = {left[ {begin{array}{*{20}{c}} {{U_1}({C_p} + beta {B_{2p}}{K_j})}&{}{{U_1}{C_{dp}}}&{}{{U_1}{D_{2p}}}&{}{beta {U_1}{B_{2p}}{K_j}}&{}0\ {{bar{beta }} {U_1}{B_{2p}}{K_j}}&{}0&{}0&{}{{bar{beta }} {U_1}{B_{2p}}{K_j}}&{}0 end{array}} right] ^{mathrm{{T}}}}),

({tilde{Z}_{pj}} = left[ {begin{array}{*{20}{c}} {sqrt{{pi _{p1}}} tilde{W}_{pj}^{mathrm{{T}}}}&{sqrt{{pi _{p2}}} tilde{W}_{pj}^{mathrm{{T}}}}&cdots&{sqrt{{pi _{ps}}} tilde{W}_{pj}^{mathrm{{T}}}} end{array}} right]),

({tilde{W}_{pj}} = {left[ {begin{array}{*{20}{c}} {{A_p} + beta {B_{1p}}{K_j}}&{}{{A_{dp}}}&{}{{D_{1p}}}&{}{beta {B_{1p}}{K_j}}&{}0\ {{bar{beta }} {B_{1p}}{K_j}}&{}0&{}0&{}{{bar{beta }} {B_{1p}}{K_j}}&{}0 end{array}} right] }).

By employing Schur complement to (44), we get (17), and the proof is accomplished. (square)

Remark 4

In Theorem 2.1, it is difficult to compute the controller gain due to the nonlinear term. Thus, we introduce the slack matrix V in Theorem 2.2 and transform the nonlinear problem into LMIs by using matrix scaling and slack matrix techniques.

Remark 5

Based on dissipative theory, the larger (gamma) implies, the better dissipative performance. We can obtain the optimal performance ({gamma ^*}) by solving a convex optimization problem as follows:

$$begin{aligned} left{ {begin{array}{*{20}{c}} {begin{array}{*{20}{c}} {min }\ {s.t.} end{array}}&{}{begin{array}{*{20}{c}} { – gamma }\ {(36),(37)} end{array}} end{array}} right. end{aligned}$$

(45)

Moreover, it is noted that the dissipative performance also includes two special performances:

(1) ({H_infty }): let (mu = – I,vartheta = 0,upsilon = ({gamma ^2} + gamma )I) in (37).

(2) Passivity: when ({{mathbb {R}}^{{n_y}}} = {{mathbb {R}}^{{n_w}}}), let (mu = 0,vartheta = I,upsilon = 2gamma I) in (37).

Remark 6

Compared with the literature [26], although the same asynchronization method is introduced in this work, there are great differences in the issues of interest, the data transmission mechanism and the system performance. The work [26] mainly considered the ({H_infty }) control of MJSs with time delay and quantization. However, we consider not only the time delay, but also the packet dropouts between the controller and the actuator. Considering the limited communication resources, we also introduce an event-triggered mechanism to reduce communication consumption. In addition, the dissipative control we consider is more general, which covers both ({H_infty }) control and passive control. In particular, due to the introduction of event-triggered mechanism and packet dropouts, the technical derivation of the design method in this work is more complex.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Disclaimer:

This article is autogenerated using RSS feeds and has not been created or edited by OA JF.

Click here for Source link (https://www.springeropen.com/)