Given the difficulty of obtaining or detecting the complete state in many systems, the following observers are considered:

$$begin{aligned} textstylebegin{cases} {{dot {hat{x}}_{i}}} ( t ) = A{{hat{x}}_{i}} ( t ) + B{u_{i}} ( t ) + G ( {{{hat{y}}_{i}} ( t ) – {y_{i}} ( t )} ), \ {{hat{y}}_{i}} ( t ) = C{{hat{x}}_{i}} ( t ), end{cases}displaystyle quad i = 1,2,ldots,N, end{aligned}$$

(3)

where ({hat{x}_{i}} ( t ) in {R^{n}}) and G represent the state and gain matrix of the observer, respectively, and ({hat{y}_{i}} ( t ) in {R^{p}}) is the output information of the observer-based system.

On the basis of the above observer (3), the event-triggered and leader-following consensus protocols are designed as follows:

$$begin{aligned} {u_{i}} ( t ) = {}& – K biggl( {sum_{j in {N_{i}}} {{a_{ij}} bigl( {{{hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {{ hat{x}}_{j}} bigl( {t_{k’}^{j}} bigr)} bigr) + {d_{i}} bigl( {{{ hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {x_{0}} bigl( {t_{k}^{i}} bigr)} bigr)} } biggr) \ &{}- Koperatorname{sig} { biggl( {sum_{j in {N_{i}}} {{a_{ij}} bigl( {{{ hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {{hat{x}}_{j}} bigl( {t_{k’}^{j}} bigr)} bigr) + {d_{i}} bigl( {{{ hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {x_{0}} bigl( {t_{k}^{i}} bigr)} bigr)} } biggr)^{alpha}}, end{aligned}$$

(4)

where (K = {B^{T}}P in {R^{m times n}}) is the control gain matrix, P is a positive definite matrix, (0 < alpha le 0.5), ({a_{ij}}) represents the ijth item of the adjacency matrix (mathcal{A}), (t_{k}^{i}) is the latest trigger moment of agent i, and (hat{x} ( {t_{k}^{i}} )) represents the latest broadcast state of agent i.

Remark 1

Controller (4) is designed in two parts. The first part aims to reduce the state error to near zero, and the second part ensures that the state error converges to zero within a finite time.

The state tracking error and observation error are defined as follows:

$$begin{aligned} {tilde{x}_{i}} ( t ) = {hat{x}_{i}} ( t ) – {x_{0}} ( t ), end{aligned}$$

(5)

$$begin{aligned} {h_{i}} ( t ) = {hat{x}_{i}} ( t ) – {x_{i}} ( t ). end{aligned}$$

(6)

Therefore, the following result is obtained:

$$begin{aligned} {{dot {tilde{x}}_{i}}} ( t ) ={}& A{tilde{x}_{i}} ( t ) – B{B^{T}}P biggl( {sum _{j in {N_{i}}} {{a_{ij}} bigl( {{{hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {{hat{x}}_{j}} bigl( {t_{k’}^{j}} bigr)} bigr) + {d_{i}} bigl( {{{ hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {x_{0}} bigl( {t_{k}^{i}} bigr)} bigr)} } biggr) \ & {}- B{B^{T}}Psig{ biggl( {sum_{j in {N_{i}}} {{a_{ij}} bigl( {{{hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {{hat{x}}_{j}} bigl( {t_{k’}^{j}} bigr)} bigr) + {d_{i}} bigl( {{{hat{x}}_{i}} bigl( {t_{k}^{i}} bigr) – {x_{0}} bigl( {t_{k}^{i}} bigr)} bigr)} } biggr)^{alpha}} \ &{}+ GC bigl( {{{hat{x}}_{i}} ( t ) – {x_{i}} ( t )} bigr). end{aligned}$$

(7)

The measurement error of the current state and the trigger state is defined as follows:

$$begin{aligned} {{{e_{i}} ( t ) = {tilde{x}_{i}} bigl( {t_{k}^{i}} bigr) – {tilde{x}_{i}} ( t ).}} end{aligned}$$

(8)

Substituting (8) into (7) obtains

$$begin{aligned} {{dot {tilde{x}}_{i}}} ( t ) ={}& A{{tilde{x}}_{i}} ( t ) – B{B^{T}}P ( {sum _{j in {N_{i}}} {{a_{ij}} bigl( { bigl( {{{tilde{x}}_{i}} ( t ) – {{tilde{x}}_{j}} ( t )} bigr) + bigl( {{e_{i}} ( t ) – {e_{j}} ( t )} bigr)} bigr)} } \ & {} + {d_{i}} bigl( {{{tilde{x}}_{i}} ( t ) + {e_{i}} ( t )} bigr) – B{B^{T}}Poperatorname{sig} biggl( {sum_{j in {N_{i}}} {{a_{ij}} bigl( { bigl( {{{ tilde{x}}_{i}} ( t ) – {{tilde{x}}_{j}} ( t )} bigr)} } } \ & + bigl( e_{i} ( t ) – e_{j} ( t ) bigr) bigr) + d_{i} bigl(tilde{x}_{i} ( t ) + e_{i} ( t ) bigr) biggr)^{alpha} + GC{h_{i}} ( t ). end{aligned}$$

(9)

The event-triggered mechanism can be applied by introducing a dynamic variable as follows:

$$begin{aligned} {dot{vartheta}_{i}} ( t ) = – {varepsilon _{i}} operatorname{sig} { bigl( {{vartheta _{i}} ( t )} bigr)^{gamma}}, end{aligned}$$

(10)

where ({vartheta _{i}}) is a non-zero real number, ({varepsilon _{i}} > 0), and (gamma in ( {0,1} )).

The event-triggered function of agent i is designed as follows:

$$begin{aligned} {f_{i}} bigl( {t,{e_{i}} ( t ),{{tilde{x}}_{i}} ( t ),{vartheta _{i}} ( t )} bigr) ={}& {eta _{1}} { biglVert {{e_{i}} ( t )} bigrVert ^{2}} + {eta _{2}} { biglVert {{e_{i}} ( t )} bigrVert ^{2alpha }} + {eta _{3}} { biglVert {{{ tilde{x}}_{i}} ( t )} bigrVert ^{2alpha }} \ &{}- rho {varepsilon _{i}}delta { biglvert {{vartheta _{i}} ( t )} bigrvert ^{2gamma }}, end{aligned}$$

(11)

where (rho in ( {0,1} )), (delta > 0). In addition,

$$begin{aligned}& {eta _{1}} ge {a_{1}} {lambda _{max }} bigl( {{H^{T}}H otimes PB{B^{T}}P} bigr), \& {eta _{2}} ge {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ ( {{N_{n}}} )^{1 – alpha }} {2^{alpha}} { biglVert { ( {H otimes {I_{N}}} )} bigrVert ^{2alpha }}, \& {eta _{3}} ge bigl( {{a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){{ ( {{N_{n}}} )}^{1 – alpha }} {2^{alpha}} + 1} bigr) times { biglVert { ( {H otimes {I_{N}}} )} bigrVert ^{2 alpha }}, end{aligned}$$

here ({a_{1}} > 0), ({a_{2}} > 0).

Under the proposed event-triggered consensus strategy, the agent i not only monitors its own state but also receives the broadcast state of its in-neighbors. The event will be triggered when ({f_{i}} ( {t,{e_{i}} ( t ),{{tilde{x}}_{i}} ( t ),{vartheta _{i}} ( t )} ) > 0). Then, the agent i updates its controller with its current state and broadcasts its current state to out-neighbors. At the same time, ({e_{i}} ( t )) is reset to zero. If the agent i receives the broadcast state of its in-neighbor, the controller will also be updated.

Remark 2

The threshold used in [15] and [18] is suitable for asymptotic consensus control; however, the threshold is incompatible for finite-time control because it cannot converge to 0 within a finite time. In contrast to the methods in [15] and [18], the threshold used in this study can ensure the convergence of the control to 0 within a finite time. This scheme also plays an important role in verifying the accuracy of the proposed MAS finite-time event-triggering algorithm.

Theorem 1

Consider systems (1) and (2) with observer (3) and control protocol (4). Suppose that Assumption 1holds. If there exists a positive definite matrix P and an appropriate positive scalar μ and β, such that the following Riccati inequality

$$begin{aligned} PA + {A^{T}}P – 2mu PB{B^{T}}P + beta {I_{n}} < 0 end{aligned}$$

(12)

holds, and the trigger function is given by (11), then the finitetime leaderfollowing consensus can be achieved for all initial conditions.

Proof

With the Kronecker product, (9) can be written in compact form as follows:

$$begin{aligned} dot {tilde{x}} ( t ) ={}& bigl( {{I_{N}} otimes A – H otimes B{B^{T}}P} bigr)tilde{x} ( t ) – bigl( {H otimes B{B^{T}}P} bigr)e ( t ) \ &{}- bigl( {{I_{N}} otimes B{B^{T}}P} bigr) operatorname{sig} { bigl( { ( {H otimes {I_{N}}} ) bigl( {tilde{x} ( t ) + e ( t )} bigr)} bigr)^{alpha}} \ &{} + ( {{I_{N}} otimes GC} )h ( t ), end{aligned}$$

(13)

where (tilde{x} ( t ) = { [ {tilde{x}_{1}^{T} ( t ),ldots,tilde{x}_{N}^{T} ( t )} ]^{T}}), (e ( t ) = { [ {e_{1}^{T} ( t ),ldots,e_{N}^{T} ( t )} ]^{T}}), (h ( t ) = { [ {h_{1}^{T} ( t ),ldots, h_{N}^{T} ( t )} ]^{T}}).

When Assumption 2 is satisfied, matrix H has N eigenvalues, and the real part of each eigenvalue is positive.

According to the observation error, ({dot{h}_{i}} ( t ) = ( {A + GC} ){h_{i}} ( t )). Thus,

$$begin{aligned} dot{h} ( t ) = bigl( {{I_{N}} otimes ( {A + GC} )} bigr)h ( t ). end{aligned}$$

(14)

If the observer feedback matrix G is designed such that (A + GC) is a Hurwitz matrix, then ({h_{i}} ( t )) will asymptotically approach zero. According to (13) and (14), the estimation error (h ( t )) is decoupled from the dynamics (tilde{x} ( t )), and the stability of (13) is equivalent to the stability of the following system:

$$begin{aligned} dot {tilde{x}} ( t ) ={}& bigl( {{I_{N}} otimes A – H otimes B{B^{T}}P} bigr)tilde{x} ( t ) – bigl( {H otimes B{B^{T}}P} bigr)e ( t ) \ &{}- bigl( {{I_{N}} otimes B{B^{T}}P} bigr) operatorname{sig} { bigl( { ( {H otimes {I_{N}}} ) bigl( {tilde{x} ( t ) + e ( t )} bigr)} bigr)^{alpha}}. end{aligned}$$

(15)

For system (15), the Lyapunov function is constructed as follows:

$$begin{aligned} V = {tilde{x}^{T}} ( {{I_{N}} otimes P} ) tilde{x} + sum_{i = 1}^{N} {frac{delta }{{1 + gamma }}} { vert {{ vartheta _{i}}} vert ^{1 + gamma }}. end{aligned}$$

(16)

Let ({V_{1}} = {tilde{x}^{T}} ( {{I_{N}} otimes P} )tilde{x}), ({V_{2}} = sum_{i = 1}^{N} {frac{delta }{{1 + gamma }}} { vert {{vartheta _{i}}} vert ^{1 + gamma }}). Take the derivative of ({V_{1}}) along the trajectory of system (15)

$$begin{aligned} {dot{V}_{1}} ={}& 2{tilde{x}^{T}} ( {{I_{N}} otimes P} ) dot {tilde{x}} \ ={}& 2{{tilde{x}}^{T}} ( {{I_{N}} otimes P} ) bigl[ { bigl( {{I_{N}} otimes A – H otimes B{B^{T}}P} bigr)tilde{x} – bigl( {H otimes B{B^{T}}P} bigr)e} \ &{} – bigl( {{I_{N}} otimes B{B^{T}}P} bigr) operatorname{sig} {{ bigl( { ( {H otimes {I_{N}}} ) ( {tilde{x} + e} )} bigr)}^{alpha}} bigr]. \ ={}& 2{{tilde{x}}^{T}} bigl( {{I_{N}} otimes P – H otimes PB{B^{T}}P} bigr)tilde{x} – 2{{tilde{x}}^{T}} bigl( {H otimes PB{B^{T}}P} bigr)e. \ &{} – 2{{tilde{x}}^{T}} bigl( {{I_{N}} otimes PB{B^{T}}P} bigr)operatorname{sig} { bigl( { ( {H otimes {I_{N}}} ) ( {tilde{x} + e} )} bigr)^{alpha}}. end{aligned}$$

(17)

After analysis, the first item of (17) obtains the following results:

$$begin{aligned} &2{tilde{x}^{T}} bigl( {{I_{N}} otimes P – H otimes PB{B^{T}}P} bigr)tilde{x} \ &quad = {tilde{x}^{T}} bigl( {{I_{N}} otimes bigl( {PA + {A^{T}}P} bigr) – 2H otimes PB{B^{T}}P} bigr)tilde{x} \ &quad le sum_{i = 1}^{N} {xi _{i}^{T}} bigl( { bigl( {PA + {A^{T}}P} bigr) – 2{lambda _{1}}PB{B^{T}}P} bigr){xi _{i}} \ &quad le sum_{i = 1}^{N} {xi _{i}^{T}} bigl( { bigl( {PA + {A^{T}}P} bigr) – 2mu PB{B^{T}}P} bigr){xi _{i}} \ &quad le – beta sum_{i = 1}^{N} {xi _{i}^{T}} {xi _{i }} \ &quad le – beta { Vert {tilde{x}} Vert ^{2}}. end{aligned}$$

(18)

According to Lemma 5, the second term in (17) is bounded as follows:

$$begin{aligned} – 2{tilde{x}^{T}} bigl( {H otimes PB{B^{T}}P} bigr)e ={}& – 2{ bigl( { bigl( {{I_{N}} otimes {B^{T}}P} bigr)tilde{x}} bigr)^{T}} bigl( {H otimes {B^{T}}P} bigr)e \ le{}& frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{1}}}}{ Vert {tilde{x}} Vert ^{2}} \ & {} + {a_{1}} {lambda _{max }} bigl( {{H^{T}}H otimes PB{B^{T}}P} bigr){ Vert e Vert ^{2}}. end{aligned}$$

(19)

According to Lemma 5, the last item in (17) is bounded as follows:

$$begin{aligned} &{}- 2{tilde{x}^{T}} bigl( {{I_{N}} otimes PB{B^{T}}P} bigr)operatorname{sig} { bigl( { ( {H otimes {I_{N}}} ) ( {tilde{x} + e} )} bigr)^{alpha}} \ &quad = – 2{ bigl( { bigl( {{I_{N}} otimes {B^{T}}P} bigr)tilde{x}} bigr)^{T}} bigl( {{I_{N}} otimes {B^{T}}P} bigr)operatorname{sig} { ( q )^{alpha}} \ &quad le frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{2}}}}{ Vert {tilde{x}} Vert ^{2}} + {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ bigl( {operatorname{sig} {{ ( q )}^{alpha}}} bigr)^{T}}operatorname{sig} { ( q )^{alpha}}, end{aligned}$$

(20)

where ({a_{1}} > 0), ({a_{2}} > 0), and (q = ( {H otimes {I_{N}}} ) ( {tilde{x} + e} )).

Substituting (18), (19), and (20) into (17) yields

$$begin{aligned} {dot{V}_{1}} ={}& 2{tilde{x}^{T}} ( {{I_{N}} otimes P} ) dot {tilde{x}} \ le{}& – beta { Vert {tilde{x}} Vert ^{2}} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{1}}}}{ Vert {tilde{x}} Vert ^{2}} + {a_{1}} {lambda _{max }} bigl( {{H^{T}}H otimes PB{B^{T}}P} bigr){ Vert e Vert ^{2}} \ & {} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{2}}}}{ Vert {tilde{x}} Vert ^{2}} + {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ bigl( {operatorname{sig} {{ ( q )}^{alpha}}} bigr)^{T}}operatorname{sig} { ( q )^{alpha}}. end{aligned}$$

(21)

On the basis of equation (10), the time derivative of ({V_{2}}) satisfies

$$begin{aligned} {dot{V}_{2}} = sum_{i = 1}^{N} delta operatorname{sig} { ( {{ vartheta _{i}}} )^{gamma}} { dot{vartheta}_{i}} = – sum_{i = 1}^{N} {delta {varepsilon _{i}} {{ vert {{vartheta _{i}}} vert }^{2gamma }}}. end{aligned}$$

(22)

Consequently, combining (21) and (22), we obtain

$$begin{aligned} dot{V} ={}& {dot{V}_{1}} + {dot{V}_{2}} \ le{}& – beta { Vert {tilde{x}} Vert ^{2}} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{1}}}}{ Vert {tilde{x}} Vert ^{2}} + {a_{1}} {lambda _{max }} bigl( {{H^{T}}H otimes PB{B^{T}}P} bigr){ Vert e Vert ^{2}} \ &{} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{2}}}}{ Vert {tilde{x}} Vert ^{2}} + {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ bigl( {operatorname{sig} {{ ( q )}^{alpha}}} bigr)^{T}}operatorname{sig} { ( q )^{alpha}} \ & {} – sum_{i = 1}^{N} {delta {varepsilon _{i}} {{ vert {{ vartheta _{i}}} vert }^{2gamma }}} \ ={}& biggl( { – beta + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{1}}}} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{2}}}}} biggr){ Vert {tilde{x}} Vert ^{2}} \ & {} + {a_{1}} {lambda _{max }} bigl( {{H^{T}}H otimes PB{B^{T}}P} bigr){ Vert e Vert ^{2}} \ & {} + {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ bigl( {operatorname{sig} {{ ( q )}^{alpha}}} bigr)^{T}}operatorname{sig} { ( q )^{alpha}} – sum _{i = 1}^{N} {delta {varepsilon _{i}} {{ vert {{vartheta _{i}}} vert }^{2gamma }}}. end{aligned}$$

(23)

According to Lemma 2, for ({a_{2}}{lambda _{max }} ( {PB{B^{T}}P} ){ ( {operatorname{sig}{{ ( q )}^{alpha}}} )^{T}}operatorname{sig}{ ( q )^{alpha}}), the following results can be attained:

$$begin{aligned} &{a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ bigl( {operatorname{sig} {{ ( q )}^{alpha}}} bigr)^{T}}operatorname{sig} { ( q )^{alpha}} \ &quad = {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr) biglVert { ( {H otimes {I_{N}}} ) ( {tilde{x} + e} )} bigrVert _{2alpha }^{2alpha } \ &quad le {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ ( {{N_{n}}} )^{1 – alpha }} { biglVert { ( {H otimes {I_{N}}} ) tilde{x} + ( {H otimes {I_{N}}} )e} bigrVert ^{2alpha }} \ & quad le {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ ( {{N_{n}}} )^{1 – alpha }} { bigl( {2{{ biglVert { ( {H otimes {I_{N}}} )tilde{x}} bigrVert }^{2}} + 2{{ biglVert { ( {H otimes {I_{N}}} )e} bigrVert }^{2}}} bigr)^{alpha}} \ &quad le {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ ( {{N_{n}}} )^{1 – alpha }} {2^{alpha}} bigl( {{{ biglVert { ( {H otimes {I_{N}}} ) tilde{x}} bigrVert }^{2alpha }} + {{ biglVert { ( {H otimes {I_{N}}} )e} bigrVert }^{2alpha }}} bigr) \ &quad = bigl( {{a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){{ ( {{N_{n}}} )}^{1 – alpha }} {2^{alpha}}+ 1} bigr){ biglVert { ( {H otimes {I_{N}}} ) tilde{x}} bigrVert ^{2alpha }} – { biglVert { ( {H otimes {I_{N}}} )tilde{x}} bigrVert ^{2alpha }} \ & qquad {} + {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ ( {{N_{n}}} )^{1 – alpha }} {2^{alpha}} { biglVert { ( {H otimes {I_{N}}} )e} bigrVert ^{2alpha }} \ &quad le bigl( {{a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){{ ( {{N_{n}}} )}^{1 – alpha }} {2^{alpha}}+ 1} bigr){ biglVert { ( {H otimes {I_{N}}} )} bigrVert ^{2 alpha }} { Vert {tilde{x}} Vert ^{2alpha }} – { biglVert { ( {H otimes {I_{N}}} )tilde{x}} bigrVert ^{2alpha }} \ & qquad {} + {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ ( {{N_{n}}} )^{1 – alpha }} {2^{alpha}} { biglVert { ( {H otimes {I_{N}}} )} bigrVert ^{2alpha }} { Vert e Vert ^{2 alpha }}. end{aligned}$$

(24)

Substituting (24) into (23) obtains the following:

$$begin{aligned} dot{V}le{}& biggl( { – beta + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{1}}}} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{2}}}}} biggr){ Vert {tilde{x}} Vert ^{2}} \ & {} + {a_{1}} {lambda _{max }} bigl( {{H^{T}}H otimes PB{B^{T}}P} bigr){ Vert e Vert ^{2}} \ & {} + bigl( {{a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){{ ( {{N_{n}}} )}^{1 – alpha }} {2^{alpha}}+ 1} bigr){ biglVert { ( {H otimes {I_{N}}} )} bigrVert ^{2 alpha }} { Vert {tilde{x}} Vert ^{2alpha }} \ &{} + {a_{2}} {lambda _{max }} bigl( {PB{B^{T}}P} bigr){ ( {{N_{n}}} )^{1 – alpha }} {2^{alpha}} { biglVert { ( {H otimes {I_{N}}} )} bigrVert ^{2alpha }} { Vert e Vert ^{2alpha }} \ & {} – { biglVert { ( {H otimes {I_{N}}} )tilde{x}} bigrVert ^{2alpha }} – sum_{i = 1}^{N} {delta { varepsilon _{i}} {{ vert {{vartheta _{i}}} vert }^{2gamma }}}. end{aligned}$$

(25)

Using the triggering functions (11) and (25), can be rewritten as

$$begin{aligned} dot{V} le{}& biggl( { – beta + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{1}}}} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{2}}}}} biggr){ Vert {tilde{x}} Vert ^{2}} \ & {} – { biglVert { ( {H otimes {I_{N}}} )tilde{x}} bigrVert ^{2alpha }} – ( {1 – rho } )delta sum_{i = 1}^{N} {{varepsilon _{i}} {{ vert {{vartheta _{i}}} vert }^{2gamma }}}. end{aligned}$$

(26)

Let (beta > frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{1}}}} + frac{{{lambda _{max }} ( {PB{B^{T}}P} )}}{{{a_{2}}}}). Then,

$$begin{aligned} dot{V} le – { biglVert { ( {H otimes {I_{N}}} )tilde{x}} bigrVert ^{2alpha }} – ( {1 – rho } )delta sum _{i = 1}^{N} {{varepsilon _{i}} {{ vert {{vartheta _{i}}} vert }^{2gamma }}}. end{aligned}$$

(27)

After analysis, the first part of (27) obtains the following results:

$$begin{aligned} – { biglVert { ( {H otimes {I_{N}}} )tilde{x}} bigrVert ^{2 alpha }} = {}& – { biglVert {{{tilde{x}}^{T}} bigl( {{H^{T}} otimes {I_{N}}} bigr) ( {H otimes {I_{N}}} )tilde{x}} bigrVert ^{alpha}} \ = {}& – { biglVert {{{tilde{x}}^{T}} bigl( {{H^{T}}H otimes {I_{N}}} bigr)tilde{x}} bigrVert ^{alpha}} \ = {}& – { biggl( { frac{{{{tilde{x}}^{T}} ( {{H^{T}}H otimes {I_{N}}} )tilde{x}}}{{{{tilde{x}}^{T}} ( {{I_{N}} otimes P} )tilde{x}}}{{ tilde{x}}^{T}} ( {{I_{N}} otimes P} )tilde{x}} biggr)^{alpha}} \ ={}& – { biggl( { frac{{{{tilde{x}}^{T}} ( {{H^{T}}H otimes {I_{N}}} )tilde{x}}}{{{{tilde{x}}^{T}} ( {{I_{N}} otimes P} )tilde{x}}}} biggr)^{alpha}}V_{1}^{alpha} \ le{} & – { biggl( { frac{{{lambda _{min }} ( {{H^{T}}H} )}}{{{lambda _{max }} ( P )}}} biggr)^{alpha}}V_{1}^{alpha} \ = {}& – {c_{1}}V_{1}^{alpha}, end{aligned}$$

(28)

where ({c_{1}} = { ( { frac{{{lambda _{min }} ( {{H^{T}}H} )}}{{{lambda _{max }} ( P )}}} )^{alpha}} > 0).

Meanwhile, the second part of (27) indicates that the term can be bounded as follows:

$$begin{aligned} – ( {1 – rho } )delta sum_{i = 1}^{N} {{ varepsilon _{i}} {{ vert {{vartheta _{i}}} vert }^{2gamma }}} ={} &- ( {1 – rho } ){varepsilon _{min }}sum _{i = 1}^{N} { ( {1 + gamma } ) frac{delta }{{1 + gamma }}{{ vert {{vartheta _{i}}} vert }^{ ( {1 + gamma } )frac{{2gamma }}{{1 + gamma }}}}} \ le {}& – ( {1 – rho } ){varepsilon _{min }} frac{{{{ ( {1 + gamma } )}^{frac{{2gamma }}{{1 + gamma }}}}}}{{{delta ^{frac{{gamma – 1}}{{1 + gamma }}}}}}{ Biggl( {sum_{i = 1}^{N} {frac{delta }{{1 + gamma }}{{ vert {{vartheta _{i}}} vert }^{ ( {1 + gamma } )}}} } Biggr)^{frac{{2gamma }}{{1 + gamma }}}} \ = {}& – {c_{2}}V_{2}^{frac{{2gamma }}{{1 + gamma }}}, end{aligned}$$

(29)

where ({c_{2}} = ( {1 – rho } ){varepsilon _{min }} frac{{{{ ( {1 + gamma } )}^{frac{{2gamma }}{{1 + gamma }}}}}}{{{delta ^{frac{{gamma – 1}}{{1 + gamma }}}}}} > 0), and ({varepsilon _{min }}=min { {{varepsilon _{1}},ldots,{ varepsilon _{N}}} }).

According to (28) and (29), (27) can further obtain the following results:

$$begin{aligned} dot{V} le – {c_{1}}V_{1}^{alpha}- {c_{2}}V_{2}^{ frac{{2gamma }}{{1 + gamma }}}. end{aligned}$$

(30)

According to (30), V will converge to (V le 1) within a finite time. This scenario indicates that ({V_{1}} le 1) and ({V_{2}} le 1) hold within a finite time. Furthermore, with (gamma in ( {0,1} )) in (10), one has (alpha < frac{{2alpha }}{{1 + alpha }}) and (frac{{2gamma }}{{1 + gamma }} < frac{{2alpha }}{{1 + alpha }}), then (V_{1}^{alpha}> V_{1}^{frac{{2alpha }}{{1 + alpha }}}) and (V_{2}^{frac{{2gamma }}{{1 + gamma }}} > V_{2}^{ frac{{2alpha }}{{1 + alpha }}}). Then, we have

$$begin{aligned} dot{V} &le – {c_{1}}V_{1}^{frac{{2alpha }}{{1 + alpha }}} – {c_{2}}V_{2}^{ frac{{2alpha }}{{1 + alpha }}} \ &< – min ( {{c_{1}},{c_{2}}} ) bigl( {V_{1}^{ frac{{2alpha }}{{1+ alpha }}}+ V_{2}^{ frac{{2alpha }}{{1+ alpha }}}} bigr) \ &< – min ( {{c_{1}},{c_{2}}} ){ ( {{V_{1}}+ {V_{2}}} )^{frac{{2alpha }}{{1+ alpha }}}} \ &< – c{V^{eta}}, end{aligned}$$

(31)

where (c = min ( {{c_{1}},{c_{2}}} )) and (eta = frac{{2alpha }}{{1+ alpha }}).

According to Lemma 3, we can derive (V ( t ) to 0) within a finite time T. At this phase, the proof is completed. □

Remark 3

Consider the event-triggered mechanism and finite-time consensus [19, 20, 22, 25, 27] and the event-triggered mechanism and the problem of unmeasurable state [28, 29, 31, 32, 34] in the literature. The problems of unmeasurable state and convergence within finite time are also considered in [35]. However, the aforementioned studies have not considered simultaneously the problems of unmeasurable state, event-triggered mechanism, and finite-time consensus. These problems are jointly investigated in the current research.

Remark 4

An observer-based event-triggered strategy is proposed, and the event-triggered condition (11) is distributed, and the trigger time of each agent is independent. Under the finite-time event-triggered consensus protocol, when the state-based measurement error of agent i exceeds a given threshold, an event will be triggered for it, the controller will be updated with the current state, and its current state will be broadcast to external neighbors. At the same time, the state-based agent i measurement error is reset to zero. If the state-based measurement error is less than the given threshold, it will not trigger, and no communication is required until the next event is triggered.

Theorem 2

Consider the leaderfollower MAS (1) and (2). If the eventtrigger condition (11) is satisfied, then the Zeno behavior can be avoided under the effect of consensus control protocol (4).

Proof

Assuming that the current trigger time is (t_{k}^{i}), the next trigger time (t_{k+ 1}^{i}) is determined by event-trigger condition (11). Consider the time interval (t in [ {t_{k}^{i},t_{k + 1}^{i}} )), and let the event interval time be (tau = t_{k + 1}^{i} – t_{k}^{i}). From the previous analysis, we know that ({tilde{x}_{i}}) and ({e_{i}}) are convergent, and (Vert {{{tilde{x}}_{i}}} Vert ) and (Vert {{e_{i}}} Vert ) are bounded. Let the upper bounds of (Vert {{{tilde{x}}_{i}}} Vert ) and (Vert {{e_{i}}} Vert ) be ({b_{1}}tau ) and ({b_{2}}tau ), respectively, where ({b_{1}}) and ({b_{2}}) are positive constants. Then, we can derive the following results:

$$begin{aligned} {eta _{1}} { Vert {{e_{i}}} Vert ^{2}} + {eta _{2}} { Vert {{e_{i}}} Vert ^{2alpha }} + {eta _{3}} { Vert {tilde{x}} Vert ^{2 alpha }} le {eta _{1}} { ( {{b_{1}}tau } )^{2}} + { eta _{2}} { ( {{b_{1}}tau } )^{2alpha }}+ { eta _{3}} { ( {{b_{2}}tau } )^{2alpha }} stackrel{ Delta}{=} {mathrm{{q}}} ( tau ). end{aligned}$$

(32)

The lower bound of the time interval can be determined using the solution to (32).

$$begin{aligned} rho {varepsilon _{i}}delta { biglvert {{vartheta _{i}} ( t )} bigrvert ^{2gamma }}={}&{eta _{1}} { Vert {{e_{i}}} Vert ^{2}} + {eta _{2}} { Vert {{e_{i}}} Vert ^{2alpha }} + {eta _{3}} { Vert {tilde{x}} Vert ^{2alpha }} \ = {}&{eta _{1}} { ( {{b_{1}} {tau _{1}}} )^{2}} + {eta _{2}} { ( {{b_{1}} {tau _{1}}} )^{2alpha }}+ {eta _{3}} { ( {{b_{2}} {tau _{1}}} )^{2alpha }}. end{aligned}$$

(33)

According to equation (33), if ({vartheta _{i}} ( t ) ne 0), then (tau ge {tau _{1}} > 0). As the proposed dynamic threshold will converge to 0 within a finite time, the system cannot guarantee that the lower bound of the time interval between events will be strictly greater than zero. However, if the appropriate parameters are selected such that the time of system consensus is less than the time ({vartheta _{i}} ( t )) converges to 0 (i.e., the finite-time consensus is achieved before the dynamic threshold of each agent converges to 0), then the Zeno behavior will not occur. □

Remark 5

The research results of this study provide ideas for solving the problem of finite-time output consensus in general linear MAS using event-triggered mechanisms. In particular, this study can help to ensure that the Zeno behavior will not occur when appropriate parameters are selected. Our future research will focus on the event-triggered mechanism that will not have Zeno behavior at all.

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