Nonlinear Cooperative Output Regulation With Input Delay Compensation

Given the MASs consisting of the leader (1) and nonlinear followers (3)-(5), the controller in the form of (15) can solve Problem 1.

“Proposition 1.Given the MASs consisting of the leader (1) and nonlinear followers (3)-( 5), the controller in the form of (15) can solve Problem 1.”

• AssumptionThe authors state this as an assumption: The real parts of the eigenvalues of the leader system matrix S are zero.
• AssumptionThe authors state this as an assumption: The nonlinear function f_i satisfies |df_i(X_i)/dX_i| ≤ ℓ for a positive constant ℓ.
• AssumptionThe authors state this as an assumption: The input delay D_i is a positive constant, can be different for every follower, and can be any large positive constant.

Under the transformation (21), the system (3)-(5) with the periodic event-triggered filter (31)-(32) is mapped into the target system (35)-(38).

“Proposition 4.Under the transformation (21), the system (3)-( 5) with the periodic event-triggered filter ( 31)-(32) is mapped into the following target system:”

• AssumptionThe authors state this as an assumption: The nonlinear function f_i satisfies |df_i(X_i)/dX_i| ≤ ℓ for a positive constant ℓ.
• AssumptionThe authors state this as an assumption: In Section V, there is no delay mismatch, i.e., D_i = D̂_i.

For the MASs (1)-(5) with the distributed observer (6)-(7), there exists a positive constant M such that for any κT ≤ M with κ = max{κ1, κ2} and T = max_{i,j} {Tij}, the estimation errors S̃i = Ŝi - S and ṽi = v̂i - v tend to the origin exponentially.

“Theorem 1.Consider the MASs (1)-( 5) with the distributed observer ( 6)-( 7), there exists a positive constant M such that for any κT ≤ M with κ ≜ max{κ 1 , κ 2 } and T ≜ max i,j∈{1,2,...,N } {T ij }, the estimation errors Si ≜ Ŝi -S and ṽi ≜ vi -v(i = 1, ..., N ) will tend to the origin exponentially.”

• AssumptionThe authors state this as an assumption: The real parts of the eigenvalues of the leader system matrix S are zero.
• AssumptionThe authors state this as an assumption: The communication graph G is directed and at least has a directed spanning tree.
• AssumptionThe authors state this as an assumption: Only a small portion of the followers can access the leader; the system matrix S is known only by a small portion of followers.
• AssumptionThe authors state this as an assumption: The network of multi-agent systems is fully asynchronous: agents have various sampling periods and event-triggered time instants.

Under the coordinate transformation (21), the system (3)-(5) is mapped into the target system (22)-(28).

“Proposition 2. Under the coordinate transformation (21), the system (3)-( 5) is mapped into the following target system: X-dynamics:”

• AssumptionThe authors state this as an assumption: The nonlinear function f_i satisfies |df_i(X_i)/dX_i| ≤ ℓ for a positive constant ℓ.

There exists a positive constant D* and small enough λj (j=2,3,4) such that for all |D̃_i| ≤ D*, V_i ≤ -a V_i + ε_i(t), where ε_i(t) ∈ E(D̃_i).

“Proposition 3.There exists a positive constant D * and small enough λ j (j = 2, 3, 4) such that for ∀| D i | ≤ D * , Vi ≤ -aV i + ε i (t),(30) where i = 1, 2, ..., N , a > 0 is a constant, ε i (t) ∈ E( D i ).”

• AssumptionThe authors state this as an assumption: The nonlinear function f_i satisfies |df_i(X_i)/dX_i| ≤ ℓ for a positive constant ℓ.
• AssumptionThe authors state this as an assumption: The input delay D_i is a positive constant, can be different for every follower, and can be any large positive constant.

Given the MASs with leader (1)-(2) and nonlinear followers (3)-(5), there exists a control law (20) with a distributed observer (6)-(7) and a filter (31)-(32) such that Problem 1 is solvable and the output regulation error |e_i(t)| ∈ E(0).

“Theorem 3. Given the MASs with leader (1)-(2) and nonlinear followers (3)-(5).There exists a control law (20) with a distributed observer (6)-(7) and a filter (31)-(32) such that Problem 1 is solvable and the output regulation error |e i (t)| ∈ E(0).”

• AssumptionThe authors state this as an assumption: The real parts of the eigenvalues of the leader system matrix S are zero.
• AssumptionThe authors state this as an assumption: The communication graph G is directed and at least has a directed spanning tree.
• AssumptionThe authors state this as an assumption: Only a small portion of the followers can access the leader; the system matrix S is known only by a small portion of followers.
• AssumptionThe authors state this as an assumption: The nonlinear function f_i satisfies |df_i(X_i)/dX_i| ≤ ℓ for a positive constant ℓ.
• AssumptionThe authors state this as an assumption: The input delay D_i is a positive constant, can be different for every follower, and can be any large positive constant.
• AssumptionThe authors state this as an assumption: The network of multi-agent systems is fully asynchronous: agents have various sampling periods and event-triggered time instants.
• AssumptionThe authors state this as an assumption: In Section V, there is no delay mismatch, i.e., D_i = D̂_i.

∆_i2(x, t) ∈ E(0) for all x ∈ [0, 1].

“From Theorem 1, we can conclude that ∆ i2 (x, t) ∈ E(0) for ∀x ∈ [0, 1].”

There exist small enough T_i and λ_j (j = 2, ..., 5) such that V_i ≤ -aV_i + ε_i.

“We can see that there exist small enough T i and λ j (j = 2, ..., 5) such that Vi ≤ -aV i + ε i .The proof is completed.”

When D_i = D̂_i = 0.15 s, the output regulation errors e_i (i = 1, 2, 3, 4) all converge to zero.

“This demonstrates that e i (i = 1, 2, 3, 4) all converge to zero.”

When PETM-B is invoked, the output regulation errors converge to zero exponentially.

“We can see that the output regulation errors also converge to zero exponentially.”

V(t0) is bounded.

“Since (43) is a linear system, V does not exhibit finite-time escape on [0, t 0 ).Therefore, V (t 0 ) is bounded.”

In the simulation, the estimation values v_i of the four followers quickly track the true value v.

“We can see that the estimation values vi of the 4 followers quickly track the true value v.”

The transmission times of the proposed method are less than 10% of the sampled data control with a 10 ms sampling period.

“According to the simulation results, the transmission times of the proposed method is less than 10% of the sampled data control with 10ms sampling period.”

When the estimated delay equals the actual delay (D_i = D̂_i = 0.15 s), the outputs of the four followers quickly track the leader's output.

“Fig. 4(a) demonstrates the control performance of the proposed control scheme when the estimated delay is selected as Di = D i = 0.15s.We can see that the outputs of the followers quickly track the output of the leader.”

For the high-order nonlinear system (69), there exists a control law (77) with periodic event-triggered adaptive distributed observer (6)-(7) such that Problem 1 is solvable.

“There exists a control law (77) with periodic event-triggered adaptive distributed observer (6)-( 7) such that Problem 1 is solvable.”

• AssumptionThe authors state this as an assumption: The nonlinear function f_i(X_i1, X_i2, ..., X_in) in the high-order system (69) satisfies the global Lipschitz condition.

ζ_i0(t) ∈ E(0).

“By Theorem 1, we know ζ i0 (t) ∈ E(0). (23) is proved.”

For all i = 1, 2, ..., N, ||ṽ_i|| ∈ E(0) if ||S_i|| ∈ E(0).

“We will show for ∀i = 1, 2, ..., N , ||ṽ i || ∈ E(0) if || Si || ∈ E(0).”

When each agent has different input delays (D1 = 0.01 s, D2 = 0.03 s, D3 = 0.15 s, D4 = 0.30 s), the actual outputs of the agents track the leader very well.

“Fig. 5 shows the control performance when all the agents have different input delays.We can see that the actual outputs of the agents can track the leader very well.”

For the high-order system, there exists a sufficiently small positive constant D* such that for all |D_i - D̂_i| ≤ D* (i = 1, 2, ..., N), the output regulation error |e_i(t)| ∈ E(D_i).

“Specifically, there exists a sufficiently small positive constant D * such that for ∀| Di -D i | ≤ D * (i = 1, 2, .., N ), the output regulation error |e i (t)| ∈ E( D i ).”

• AssumptionThe authors state this as an assumption: The nonlinear function f_i(X_i1, X_i2, ..., X_in) in the high-order system (69) satisfies the global Lipschitz condition.

When PETM-B is invoked, communication resource has been saved.

“We can see that the communication resource has been saved.”

For the revised follower models with unknown external disturbance, the output regulation errors of the four followers all converge to the neighborhood of the origin asymptotically.

“It can be seen that the output regulation errors of the four followers can all converge to the neighborhood of the origin asymptotically.”

When a delay mismatch exists (D_i = 0.2 s ≠ D̂_i = 0.15 s), the output regulation errors e_i (i = 1, 2, 3, 4) converge to a small set.

“If we take Di = 0.2s ̸ = D i = 0.15s implying that a delay mismatch exists, the tracking performance of the MASs is illustrated in Fig. 4(b).We can see that e i (i = 1, 2, 3, 4) converge to a small set.”

The maximally allowable sampling periods Tij can be made large by decreasing the observer gain κ, with a sacrifice of convergence rate.

“This implies that the maximally allowable sampling periods T ij can be made large by decreasing the observer gain κ with a sacrifice of convergence rate.”

A long input delay is easy to cause a loss of stability in multi-agent systems.

“A long input delay is easy to cause a loss of stability [17], [18].”

• cites [17], [18]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If there exists no delay mismatch, then the output regulation will converge to the origin exponentially.

“If there exists no delay mismatch, then the output regulation will converge to the origin exponentially.”

The proposed method deals with the cooperative output regulation problem for nonlinear multi-agent systems, which is more general than the traditional consensus problem for linear multi-agent systems.

“Our method deals with the COR (cooperative output regulation) problem for nonlinear MASs.This is more general than the traditional consensus problem for linear MASs [23], [24].”

• cites [23], [24]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The proposed periodic event-triggered adaptive distributed observer can deal with fully asynchronous periodic event-triggered transmission data, where every agent pair has a different sampling period.

“Different from the existing works [14], [28], the proposed distributed observer can deal with fully asynchronous periodic event-triggered transmission data, i.e., every agent pair has a different sampling period.”

• cites [14], [28]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Under the delay mismatch, practical cooperative output regulation (COR) is achieved.

“Hence, the practical COR is achieved.”

If semiglobal stability of the closed-loop system is considered, the global Lipschitz condition can be removed.

“Moreover, if we consider semiglobal stability of the closed-loop system, the global Lipschitz condition can also be removed.”

The proposed results are effective.

“This indicates that the proposed results are effective.”

The proposed periodic event-triggered condition can reduce the transmission times considerably.

“This implies that the proposed periodic event-triggered condition can reduce the transmission times considerably.”

A new predictor feedback controller is proposed to handle the long input delay in the nonlinear MASs.

“Based on fully asynchronous sampled data, a new predictor feedback controller is proposed to handle the long input delay in the nonlinear MASs.”

The proposed method can handle more complex nonlinear systems.

“This indicates that the proposed method can handle more complex nonlinear systems, and is robust to external disturbance.”

The proposed method is robust to external disturbance.

“This indicates that the proposed method can handle more complex nonlinear systems, and is robust to external disturbance.”