Distributed Matrix Pencil Formulations for Prescribed-Time Leader-Following Consensus of MASs with Unknown Sensor Sensitivity
Hefu Ye Changyun Wen Yongduan Song




Distributed Matrix Pencil Formulations for Prescribed-Time Leader-Following Consensus of MASs with Unknown Sensor Sensitivity
    Hefu Ye, Changyun Wen, Fellow, IEEE, and Yongduan Song^*, Fellow, IEEE 

Hefu  Ye  and Yongduan Song are   with  the School of Automation, Chongqing University, 400044, China. Engineering, Nanyang Technological University,  639798, Singapore. (e-mail: yehefu@cqu.edu.cn; ydsong@cqu.edu.cn)
Changyun Wen is with the School of Electrical and Electronic Engineering, Nanyang Technological University,  639798, Singapore. (e-mail:  ecywen@ntu.edu.sg). 

    
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In this paper, we address the problem of prescribed-time leader-following consensus of heterogeneous multi-agent systems (MASs) in the presence of unknown sensor sensitivity. Under a connected undirected topology, we propose a time-varying dual observer/controller design framework that makes use of regular local and inaccurate feedback to achieve consensus tracking within a prescribed time. In particular, the developed analysis framework is applicable to MASs equipped with sensors of different sensitivities. 
One of the design innovations involves constructing a distributed matrix pencil formulation based on worst-case  sensors, yielding control parameters with sufficient robustness yet relatively low conservatism.  Another novelty is the construction of the control gains, which  consists of the product of a proportional coefficient obtained from the matrix pencil formulation and a classic time-varying function that grows to infinity or a novel bounded time-varying function.
Furthermore, it is possible to extend the prescribed-time distributed protocol to infinite time domain by introducing the bounded time-varying gain technique without sacrificing the ultimate control accuracy, and the corresponding technical proof is comprehensive.  The effectiveness of the method is demonstrated through a group of 5 single-link robot manipulators.


Prescribed-time consensus,  sensor sensitivity, output-feedback, distributed matrix pencil formulations




§ INTRODUCTION
  


IN recent years, there has been a notable surge of interest in the field of prescribed-time control, which allows arbitrarily prescribe the finite convergence time of a closed-loop system regardless of initial conditions and any other design parameter. It is wroth noting that the idea of prescribed-time control can be traced back to the proportional navigation in tactical missile guidance <cit.>, and early explorations of prescribed-time stability primarily revolved around the optimal control of linear systems <cit.>. 
Prescribed-time control for nonlinear systems was thoroughly developed in <cit.>, and since then, several appealing techniques have been introduced for achieving prescribed-time stabilization for complex single-plant nonlinear systems. These methods include dynamic high-gain feedback<cit.>, bounded linear time-varying feedback<cit.>, time-varying and non-Lipschitz synergistic feedback<cit.>, among others. An objective fact is that the current research on prescribed-time control for MASs  is primarily focused on linear systems. For instance, Wang et al. in <cit.> and <cit.> developed distributed prescribed-time containment and consensus protocols for single and n-th order integrators, respectively. Kan et al. in <cit.> and Yucelen et al. <cit.> established a prescribed-time consensus framework for linear MASs based upon a time-axis transformation. 
In addition, distributed finite-time and predefined-time protocols were constructed for MASs with directed communication in  <cit.> and <cit.>, respectively. 
Recently, based upon the solution of a parametric
Lyapunov equation, fully distributed state-feedback protocols were developed to solve the problems of asymptotic and prescribed-time  consensus of linear MASs in <cit.> and <cit.>, respectively.
In contrast to the aforementioned consensus results where only cooperative communication exist among agents, Ning et al. in <cit.> introduced a prescribed-time bipartite consensus protocol for double integrators where antagonism exists in the communication. Later on, Chen et al. extended the prescribed-time bipartite consensus result to single integrators with event-triggered communications in <cit.>.

 

A significant challenge occurs in distributed prescribed-time consensus tracking when the considered system model becomes nonlinear and contains uncertainties. This is particularly true in cases where only inaccurate output measurements of each agent are available. Based upon accurate measurement signals, some preliminary efforts on prescribed-time output-feedback control for nonlinear MASs have been made in recent studies. In <cit.>, a practical prescribed-time control approach was proposed for second-order leader-following systems. In <cit.>, prescribed-time precise consensus was achieved for second-order disturbed MASs based upon a mismatched disturbance observer. In <cit.>, Chen et al. made remarkable progress in the prescribed-time control of high-order MASs in strict-feedback form by modifying feedback domination design approach. To the best of the authors' knowledge, research on distributed prescribed-time consensus of heterogeneous MASs in the presence of unknown measurement sensitivity remains limited so far. On the one hand,  it is not straightforward to develop a stability analysis framework suitable for inaccurate measurements, especially when each agent is equipped with sensors of different sensitivities. On the other hand, the aforementioned distributed methods are invalid beyond the prescribed time interval since the control gain escapes to infinity as t approaches the terminal time. It is therefore of theoretical significance and practical importance to develop a modified scheme involving bounded gains that drives the closed-loop system not to the destination, but to its neighborhood within a prescribed time, and ultimately to the origin. 



In the present work, the high-gain domination approach is revisited for prescribed-time distributed control of heterogeneous leader-following MASs. Unlike classical literature that relies on high-gain feedback and can only accommodate uncertainties with known growth rates, the system  under consideration allows for uncertainties with unknown growth rates and coupled unmeasurable states. In addition, inspired by the recent works of Krishnamurthy and  Khorrami<cit.>, we robustify the standard matrix pencil-based design procedure in the presence of multiplicative measurement noise and propose a class of novel distributed matrix pencil formulations for the prescribed-time control design. The basic design idea is to convert the problem of prescribed-time leader-following consensus into the asymptotic stabilization of related errors with the help of state scaling and time-axis transformation, and then cast the Lyapunov inequality into an equivalent matrix pencil form to solve for suitable design parameters. 
Particularly, the most conservative sensitivity parameters of sensors are used as a benchmark for designing controller parameters to ensure that the resulting algorithm is effective for all agents equipped with different sensors. The stability analysis is completed in an elegant way that once the boundedness of the Lyapunov function is obtained, the boundedness of all internal signals including inputs and the prescribed-time convergence of  tracking errors can be obtained.  The contributions of this article are summarized as follows:

	
  1) Prescribed-time leader-following consensus is achieved for heterogeneous  MASs by employing the time-varying feedback and high-gain design, and the corresponding practical prescribed-time solution is then formulated via a modified bounded time-varying feedback; 
	
  2) The mathematical expression of agent-based sensor uncertainty is constructed, and the philosophy of using high-gain design to solve the uncertainty of different sensors is given based on the worst sensor performance; 
	
  3) A distributed matrix pencil formulation that is robust to multiplicative noise is proposed to reduce the conservatism of control parameters.  




 

















§ PROBLEM FORMULATION AND PRELIMINARIES
 
We first introduce some notations and definitions. Given a vector q=[q_1,…,q_n]^⊤, q^⊤ and |q| denote  the transpose  and the Euclidean norm of the vector q, respectively,  the notation |q|_e is used to denote the vector of element-wise magnitudes [|q_1|,…,|q_n|]^⊤, and the relation |q|_e ≤_e |x|_e indicates the set of element-wise inequalities |q_i|_e≤ |x_i|_e, where x is a vector (or a matrix) of the same dimension as q.   Given a matrix 𝐐 (where bold notations denote matrices), 
(𝐐) denotes the determinant of matrix 𝐐, |𝐐| denotes the  Frobenius norm of matrix 𝐐,  |𝐐|_e denotes the matrix of the same dimensions with each element being the magnitude of the corresponding element of 𝐐,  λ_max(𝐐) and λ_min(𝐐) denote its maximum and minimum eigenvalues, respectively, and   diag{𝐐} denotes the diagonal matrix whose diagonal elements are equal to the diagonal elements of 𝐐. The diagonal matrix with diagonal elements q_1,⋯,q_n is denoted by diag{q_1,⋯,q_n}. If q_1,⋯,q_n are matrices, diag{q_1,⋯,q_n} denotes the block diagonal matrix with blocks on the principal diagonal being q_1,⋯,q_n. 


Consider a multi-agent system comprising a leader and N agents. The dynamics of each agent are described by:

    ẋ_k   =𝐀x_k+Bu_k+F_k(t,x_k), 
    
    y_k   =θ_k(t)Cx_k, k∈{0,1,⋯,N},
 

    𝐀 =[[     0 𝐈_n-1;     0     0 ]] , B=[[ 0_(n-1)× 1;          1 ]],  C=[[          1; 0_(n-1)× 1 ]]^⊤,

where x_k∈ℝ^n, y_k∈ℝ, and u_k∈ℝ are the agent k's state, output and input, respectively, and u_0=0. The function θ_k(t), called sensor sensitivity, is the ratio of the k-th sensor's output to its input, and the nonlinear terms F_k =[f_k,1 ,⋯,f_k,n]^⊤∈ℝ^n model the system uncertainties where f_k,i is piecewise continuous in t and smooth w.r.t. x. For system (<ref>), we need the following assumptions.

	For i=1,⋯,n, there exist constants ϱ_k,i such that
	
    |f_k,i(t,x_k)-f_0,i(t,x_0)|≤∑_j=1^iϱ_k,j |x_k,j-x_0,j|.
	


	For k=0,1,⋯,N, the sensor sensitivity θ_k(t) is an unknown continuous function satisfying θ_k(t)∈ [1-Δ_θ_k, 1 +Δ_θ_k], where 0 < Δ_θ_k< 1 denotes the sensitivity error since it allows the sensor sensitivity has a  fluctuation of ±Δ_θ_k.





It is evident that the multi-agent system (<ref>) can be heterogeneous since both the nonlinearity of each agent and the sensor used to measure the output of each agent can be different. 
		

















We use the graph to describe the information exchange between agents and the leader. The agents indexed by  1, ⋯,N are followers, while the agent indexed by 0 is the leader. The communication topology among N followers is represented by a weighted undirected graph 𝒢={𝒱,ℰ,𝒜}, where 𝒱={1,⋯,N} denotes the agent in the group, the set of edges ℰ={(i,j)∈𝒱×𝒱} represents neighboring relations among  agents, and 𝒜=(a_i,j)∈ℝ^N× N is the weighted adjacency matrix of graph 𝒢 as a_i,j=1 if (i,j)∈ℰ and a_i,j=0 otherwise. The Laplacian matrix of 𝒢 is given by ℒ_𝒢=𝒟-𝒜 where 𝒟 is the degree matrix with ith diagonal elements d_i=∑_j=1,j≠ i^Na_i,j and 0 elsewhere. Let 𝒢̅ be a graph generated by graph 𝒢 and the leader. Assuming that the graph 𝒢 is connected, and there is at least one directed edge from a follower to the leader, with no path leading from followers to  the leader. Furthermore, it is either known to all followers what the input of the leader is <cit.>, or the input of the leader is zero<cit.>. The connection weight matrix is given by ℬ=diag{𝖻_1,⋯,𝖻_N} as 𝖻_i=1 if agent i is a neighbor of the leader and 𝖻_i=0 otherwise. The Laplacian matrix   ℒ_𝒢̅= ℒ_𝒢+ℬ associated with 𝒢̅ is positive definite<cit.>.








	The prescribed-time globally uniformly  consensus  problem of the leader-following MAS (<ref>) is said to be solved if, for  k∈{1,⋯,N},  there exist a class 𝒦ℒ function β and a time-varying function μ:[0,T)→[0,∞) such that μ(t) tends to infinity as t goes to T, and ∀ x_k,x_0∈ℝ^n
		
    |x_k(t)-x_0(t)|≤β(|x_k(0)-x_0(0)|,μ(t)), ∀ t∈ [0,T),
  
		where  T>0 is an arbitrary time-independent constant.

































§ PRELIMINARY RESULT: STATE-FEEDBACK WITHOUT UNKNOWN MEASUREMENT SENSITIVITY


The matrix pencil based design framework was originally proposed in <cit.>, and the robust and decentralized formulations of such procedure in the presence of additive noise and interactions were studied in <cit.> and <cit.>, respectively. In this paper, we propose for the first time a class of matrix pencil formulation robust to multiplicative noise for distributed prescribed-time consensus protocol design. 
For ease of exposition, the matrix pencil based approach is first tested under state-feedback design. 

To begin with, we introduce the time-axis transformation, inspired by <cit.>, to facilitate the convergence analysis of the prescribed-time consensus. Define, ∀ t∈[0,T), 

    t=T(1-e^-τ)  ⇔ τ=ln(T/T-t).
   
Then, it can be verified that when t=0, τ=0, and when  t=T, τ=∞, and

    dτ/dt=1/Te^τ≜α(τ) .

Inserting the expression of τ into (<ref>), it is seen that the equivalent form of α(τ) on the t-axis is[Given a general signal q of appropriate dimension, q(τ)=q(t) holds since q(τ) and q(t) refer to the value	of the same signal  at the τ-axis and t-axis, respectively.] 

    α(t)=1/T-t  .
 
With (<ref>), the dynamics of  ẋ=f(x,u,t) is converted into 

    dx/dτ=dt/dτf(x,u,τ)=1/α(τ)f(x,u,τ).

Consequently, the prescribed-time control of ẋ=f(x,u,t) can be equivalently converted into the  asymptotic control of (<ref>), as formally stated in the following Proposition with its proof provided in the Appendix A.







	If the coordinate transformation (<ref>) is applied,
	
    η_k=diag{r^n,r^n-1,⋯,r }(x_k-x_0) :=Λ_r(x_k-x_0),

	and the transformed dynamic system (<ref>) is asymptotically stabilized,[For notational convenience, we drop the arguments of functions whenever no confusion will occur.]
	
    dη_k/dτ =b(𝐀η_k+B u_k +Φ_k)+𝐃_cη_k,
 
	where r=be^τ/T with   b ≥ T, 𝐃_c=diag{n,n-1,⋯,1}, Φ_k=[ϕ_k,1,⋯,ϕ_k,n]^⊤ with ϕ_k,i=r^n-i(f_k,i-f_0,i), namely, there exists a class 𝒦ℒ function β such that, for η=[η_1^⊤,⋯,η_N^⊤]^⊤∈ℝ^nN,
	
    |η (τ)|≤β(|η (0)|,τ), ∀τ∈[0,∞),
 
	then the prescribed-time consensus of the original leader-following MAS (<ref>) is achieved in the sense of Definition 1.





Now, we are in the position to state our preliminary result.




































































































	Suppose that the graph 𝒢 is connected and Assumption <ref> is satisfied with unknown growth rates ϱ_k,j. A proportional coefficient b can be found such that system (<ref>) is asymptotically stabilized by a distributed protocol:
	
    u_k =-KΛ_r( ∑_j=1^Na_k,j( x_k- x_j)+𝖻_k( x_k- x_0) ) ,

	where  K is chosen such that 𝐀_c=𝐈_N⊗𝐀-ℒ_𝒢̅⊗(BK) is Hurwitz.  	

Proof: Let 𝐃 =𝐈_N⊗𝐃_c∈ℝ^nN× nN and Φ=[Φ_1^⊤,⋯,Φ_N^⊤]^⊤∈ℝ^nN. Since the equivalent form  of protocol (<ref>) is u_k=  -(Γ_k⊗ K)η where Γ_k is the kth row of ℒ_𝒢̅,  then system (<ref>) can be written in a compact form as follows

    dη/dτ=b (𝐀_cη+Φ) +   𝐃η.
 
Choose a Lyapunov-like  function candidate as

    V_c(τ) =η^⊤𝐏_cη,

where 𝐏_c∈ℝ^nN× nN is symmetric positive-definite (SPD), then the derivative of V_c along the trajectory of (<ref>) is

    dV_c/dτ   =b η^⊤(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)η+2bη^⊤𝐏_cΦ
          +  η^⊤(𝐏_c𝐃+𝐃𝐏_c) η.

Now the design purpose is to find parameters b, κ_0, and κ_1 such that (<ref>) satisfies the following form:

    dV_c/dτ≤-(κ_0-κ_1/α(τ))V_c.

To this point, the design procedure can be divided into  two steps.

Step 1: b is selected to ensure that the following inequality is satisfied with κ_0 being an arbitrary positive constant:

    b η^⊤(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)η   
    
    	+  η^⊤(𝐏_c𝐃+𝐃𝐏_c) η +κ_0η^⊤𝐏_cη   ≤ 0.
 
Notice that 𝐀_c is Hurwitz and 𝐃 is SPD, hence  a SPD matrix 𝐏_c can be found such that

    𝐏_c𝐀_c+ 𝐀_c^⊤𝐏_c   ≤ - 𝐈_nN, 
    𝐏_c𝐃+𝐃𝐏_c     ≥𝐈_nN.
 
Therefore, one can conclude that 𝐏_c𝐃+𝐃𝐏_c+κ_0𝐏_c is SPD and 𝐏_c𝐀_c+𝐀_c^⊤𝐏_c is symmetric negative-definite (SND). Then, pursuant to Lemma <ref>, we can pre-give a κ_0 and choose

    b=max{σ(𝐐_1,-𝐐_2),T},

where 𝐐_1=𝐏_c𝐃+𝐃𝐏_c+κ_0𝐏_c and 𝐐_2=𝐏_c𝐀_c+𝐀_c^⊤𝐏_c, such that (<ref>) always holds. Since the calculation of b only requires local information 𝐀, B and K, and does not require information about neighbor agents and the leader, the resulting controller is parameterly decentralized. 



Step 2: Here we aim to verify that the following inequality  holds for some positive constant κ_1,

    2bη^⊤𝐏_cΦ≤κ_1/α(τ) V.
 
Under Assumption <ref> and the fact r≥ 1, it can be obtained that

    |Φ|_e  ≤_e  1/rdiag{𝐀_ϱ_1,⋯,𝐀_ϱ_N}|η|_e:=1/r𝐀_ϱ|η|_e,

where

    [ 𝐀_ϱ_k =
    [[ ϱ_k,1     0     ⋯     0; ϱ_k,1 ϱ_k,2     ⋯     0;     ⋮     ⋮     ⋱     ⋮; ϱ_k,1 ϱ_k,2     ⋯ ϱ_k,n ]]∈ℝ^n× n. ]
 
Therefore, by recalling r=bα(τ), we get 

    2bη^⊤𝐏_cΦ≤1 /α(τ)|η|_e^⊤(𝐀_ϱ^⊤|𝐏_c|_e+|𝐏_c|_e𝐀_ϱ)|η|_e .
 
Let 𝐏̅_c=diag{𝐏_c}   be a positive definite diagonal matrix. From Lemma <ref>,  there exist two positive constants κ_1 and κ_2, such that,

    -κ_2 𝐏̅_c 
    	+  	   (𝐀_ϱ^⊤|𝐏_c|_e+|𝐏_c|_e𝐀_ϱ)   ≤ 0,
    
    	-κ_1  𝐏_c +
    	κ_2𝐏̅_c    ≤ 0 .

Specifically, the suitable κ_1 and κ_2 are given by

    κ_1    =max{σ(κ_2𝐏̅_c,𝐏_c)},
    κ_2    =max{σ( 𝐀_ϱ^⊤|𝐏_c|_e+|𝐏_c|_e𝐀_ϱ, 𝐏̅_c)}.
 
Now we can upper-bound the right-hand side of (<ref>), in view of (<ref>) and the fact η^⊤𝐏̅_c η=| η|_e^⊤𝐏̅_c| η|_e,  as 
 
    1 /α(τ)|η|_e^⊤(𝐀_ϱ^⊤|𝐏_c|_e+|𝐏_c|_e𝐀_ϱ)|η|_e   ≤κ_2/α(τ) |η|_e^⊤𝐏̅_c|η|_e,
    κ_2/α(τ) |η|_e^⊤𝐏̅_c|η|_e=κ_2/α(τ)η ^⊤𝐏̅_c η   ≤κ_1/α(τ)η ^⊤𝐏_c η .
  
Recalling (<ref>), (<ref>), and (<ref>),  we know that (<ref>) holds. Combining (<ref>), (<ref>), and (<ref>), it is verified that (<ref>) holds.  It follows from Lemma <ref> that

    V_c(τ)≤ e^-κτMV_c(0),

where κ=κ_0-κ_1/α(τ_1)>0 and M=e^Tκ_1τ_1 is a finite number with 0<τ_1<∞. Insert τ defined in (<ref>) into (<ref>) yields

    V_c(t)≤(T-t/T)^κMV_c(0) , t∈[0,T).
 
Therefore, it holds that lim_t→TV_c(t)=0 and V_c(t)∈ℒ_∞[0,T). Additionally, it follows from (<ref>) that[This estimate does not lead to conservative selection of control parameters since it  is only involved in the stability analysis, that is, only the  transient performance  of system state and   the input are conservatively estimated.]

    |η(t)| ≤√(V_c(t)/λ_min(𝐏_c) ) .
  
With Λ_r=diag{r^n,r^n-1,⋯,r} and the fact r≥1, it holds that

    |x_k(t)-x_0(t)|≤|η(t)|/r  =(T-t)^κ+2/2/bT^κ/2√(MV_c(0)/λ_min(𝐏_c) ) .

Hence, we obtain x_k-x_0 ∈ℒ_∞[0,T) and lim_t→ T|x_k-x_0|=0.  By recalling (<ref>), we know that the designed control inputs satisfy u_k ∈ℒ_∞[0,T).   
The proof is completed. ▪






	Note that calculating κ_2 requires a prior knowledge of the growth rate ϱ_k,i that is contained in 𝐀_ϱ, and calculating κ_1 requires a prior knowledge of κ_2. Since we use κ_1 and κ_2 only for stability analysis rather than controller design, a prior knowledge of ϱ_k,i is not required, which makes the proposed algorithm more general compared to the algorithms presented in <cit.> and <cit.>. 

















 
	Note that the scaling order of the coordinate transformations (<ref>) in our scheme has been subtly changed compared to that of the state-of-the-art works <cit.>. One of the benefits of this is that it technically allows the design of a static rather than dynamic feedback control law thus directly avoiding the computational problem of integrating over an infinite number. Another advantage is that the prescribed-time convergence property of the tracking error (i.e., x_k,i-x_0,i) can be obtained directly from the asymptotic convergence property of η_k. Specifically,  the coordinate transformation might be defined as x_k,i-x_0,i=r^iη_k,i if we adopt the control scheme proposed in <cit.>. It can be seen that further proof that η_k,i decays faster than the growth rate of r^i is needed to obtain the conclusion that the tracking error converges to zero.






















































































































§ MAIN RESULTS: OUTPUT-FEEDBACK WITH UNKNOWN MEASUREMENT SENSITIVITY




 §.§ Prescribed-time exact consensus tracking





In this section, we show that  using only inaccurate output measurements y_k=θ_k(t)x_k,1, the prescribed-time consensus can be achieved. Using the time-varying parameter r and the diagonal matrix Λ_r defined in (<ref>),   the prescribed-time distributed observer can be constructed as, 

    ẋ̂̇_k   =𝐀x̂_k+Bu_k+r^n+1Λ_r^-1G(y_k-x̂_k,1) ,

where G=[g_1,⋯,g_n]. Correspondingly, the prescribed-time consensus tracking problem can be transformed into an asymptotic stabilization problem of the neighborhood error's dynamics, as formally stated in the following proposition. 

	If the coordinate transformation (<ref>) is applied,
	
    η̂_k   =Λ_r(x̂_k-x̂_0), 
    ε_k    =Λ_r(x_k-x_0)-η̂_k,

	and the transformed dynamic system (<ref>) is asymptotically stabilized,  
	
    dη̂_k/dτ   =b(𝐀η̂_k+B u_k)+𝐃_cη̂_k
           +bG^⊤θ_k ε_k,1 +bG^⊤(θ_k-1)η̂_k,1,
    dε_k/dτ   =b( (𝐀-G^⊤C)ε_k+Φ_k)+ 𝐃_cε_k,
 
	namely, there exists a class 𝒦ℒ function β such that, for η̂=[η̂_1^⊤,⋯,η̂_N^⊤]^⊤∈ℝ^nN and ε=[ε_1^⊤,⋯,ε_N^⊤]^⊤∈ℝ^nN, 
	
    |η̂(τ)+ε(τ) |≤β(|η̂(0)+ε(0)|,τ), ∀τ∈[0,∞),
  
	where 𝐀, B, C, 𝐃_c, Φ_k, and τ follow the definitions in the previous Section, and G is chosen such that 𝐀_0=𝐈_N⊗(𝐀-G^⊤C) is Hurwitz,	then the prescribed-time consensus of the original leader-following MAS (<ref>) is achieved in the sense of Definition 1.


Proof: This proposition can be proven by a direct extension of the proof of Proposition <ref> and the details are omitted for brevity.








With Proposition <ref>, we can establish the following theorem.

	Suppose that the graph 𝒢 is connected and Assumptions <ref> and  <ref> are satisfied with unknown growth rates ϱ_k,j. If the maximum value of the allowable sensitivity error satisfies 
	
    0<max{Δ_θ_1,⋯,Δ_θ_N}≤1/ |𝐏_c𝐀_g|,

	where 𝐀_g is defined in (<ref>), then a  proportional coefficient b can be found such that system (<ref>) is asymptotically stabilized by the distributed observer (<ref>) and the following output-feedback distributed protocol
	
    u_k =-KΛ_r( ∑_j=1^Na_k,j(x̂_k-x̂_j)+𝖻_k( x̂_k- x̂_0) ,

	where K is chosen such that 𝐀_c= 𝐈_N⊗𝐀-ℒ_𝒢̅⊗ (BK) is Hurwitz. 	


Proof:  Since the protocol (<ref>) can be rewritten as u_k=-(Γ_k⊗ K)η̂ where η̂=[η̂_1^⊤,⋯,η̂_N^⊤]^⊤∈ℝ^nN and Γ_k is the kth row of ℒ_𝒢̅, then the following compact form is obtained, in view of (<ref>), 

    dη̂/dτ   =b(𝐀_cη̂+G^⊤⊗(𝐈_θΨ_ε+(𝐈_θ-𝐈_N)Ψ_η̂)) + 𝐃η̂, 
    dε/dτ   =b(𝐀_0ε+Φ) + 𝐃ε ,
 
where  𝐃 and Φ are defined in (<ref>), 𝐈_θ=diag{θ_1,⋯,θ_N}, Ψ_ε =[ε_1,1,⋯,ε_N,1]^⊤ , and Ψ_η̂ =[η̂_1,1,⋯,η̂_N,1]^⊤ .

Choose a Lyapunov-like function defined on [0,∞) as follows,

    V(τ)=η̂^⊤𝐏_cη̂+cε^⊤𝐏_0ε,

where 𝐏_c and 𝐏_0 are SPD, and c is a positive constant to be designed later in (<ref>). Calculating the derivative of V along the trajectories of (<ref>) yields 

    dV/dτ=     b  η̂^⊤(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)η̂+η̂^⊤(𝐏_c𝐃+𝐃𝐏_c)η̂
       +b  cε^⊤(𝐏_0𝐀_0+𝐀_0^⊤𝐏_0)ε+cε^⊤(𝐏_0𝐃+𝐃𝐏_0)ε
       +2bcε^⊤𝐏_0Φ+ 2b η̂^⊤𝐏_c(G^⊤⊗(𝐈_θΨ_ε)) 
       + 2bη̂^⊤𝐏_c(G^⊤⊗((𝐈_θ-𝐈_N)Ψ_η̂)).
 	 
To deal with the last two uncertain terms in  (<ref>), we define

    𝐀_g=diag{𝐀_g,1,⋯,𝐀_g,N},
 
where each partitioned matrix 𝐀_g,k has the same structure, i.e.,  𝐀_g,k=[G^⊤,0,⋯,0]∈ℝ^n× n, k=1,⋯,N. Then G^⊤⊗(𝐈_θΨ_ε) and G^⊤⊗((𝐈_θ-𝐈_N)Ψ_η̂) can be equivalently written as

    G^⊤⊗(𝐈_θΨ_ε)    =𝐀_g(𝐈_n⊗𝐈_θ)ε,
    
    G^⊤⊗((𝐈_θ-𝐈_N)Ψ_η̂)    =𝐀_g(𝐈_n⊗(𝐈_θ-𝐈_N))η̂.

By Assumption <ref>,  it holds that θ_k≤ 1+Δ_θ_k and |θ_k-1|≤Δ_θ_k. Therefore,

    |𝐈_θ|_e    ≤_e |𝐈_N+ 𝐈_Δ_θ |_e,  
    
    	|𝐈_θ-𝐈_N|_e     ≤_e |𝐈_Δ_θ|_e,

where 𝐈_Δ_θ=diag{Δ_θ_1,⋯,Δ_θ_N}∈ℝ^N× N. With (<ref>), one can verify

    0<|𝐈_N⊗𝐈_Δ_θ|≤max{Δ_θ_1,⋯,Δ_θ_N}≤1/|𝐏_c𝐀_g|.

Due to (<ref>), we can upper-bound the last term in (<ref>) as 

    2bη̂^⊤𝐏_c   (G^⊤⊗((𝐈_θ-𝐈_N)Ψ_η̂))   
    ≤    2b |η̂| |𝐏_c𝐀_g𝐈_N⊗𝐈_Δ_θ |    |η̂|  
    	≤  2b η̂ ^⊤η̂.

Let 𝐏_ε=𝐏_c𝐀_g(𝐈_n⊗ (𝐈_N+ 𝐈_Δ_θ)). It follows by 
	(<ref>) and (<ref>)  that,

    2bη̂^⊤𝐏_c(G^⊤⊗(𝐈_θΨ_ε))  ≤   2  b|η̂|_e^⊤|𝐏_ε|_e|ε|_e ,
 
By (<ref>) and (<ref>), (<ref>) reduces to  

    dV/dτ≤     b  η̂^⊤(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)η̂+η̂^⊤(𝐏_c𝐃+𝐃𝐏_c)η̂
       +b  cε^⊤(𝐏_0𝐀_0+𝐀_0^⊤𝐏_0)ε+cε^⊤(𝐏_0𝐃+𝐃𝐏_0)ε
       +2bcε^⊤𝐏_0Φ + 2b ( η̂  ^⊤η̂ +  |η̂|_e^⊤|𝐏_ε|_e|ε|_e).
 
Now the deign purpose is to find  parameters b and c such that (<ref>) satisfies the following form:

    dV/dτ≤-(κ_a-κ_b/α(τ))V,

where κ_a and κ_b are two design parameters.
To this point, the design procedure can be divided into  three steps.

Step 1: c is picked to ensure that the following inequality is satisfied with c_1∈(0,1):

    (1-c_1)(η̂^⊤ (𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)η̂.   
    . +cε^⊤ (𝐏_0𝐀_0+𝐀_0^⊤𝐏_0  )ε)  +2   |η̂|_e^⊤|𝐏_ε|_e|ε|_e    ≤ 0.

In order to convert (<ref>) into a matrix pencil structure, we need to find constants δ_A_c and δ_A_0 such that following inequalities hold, 

    δ_A_cη̂^⊤𝐀̅_cη̂≥η̂^⊤ (𝐏_c𝐀_c+𝐀_c^⊤𝐏_c )η̂,
       δ_A_0ε^⊤𝐀̅_0ε≥ε^⊤ (𝐏_0𝐀_0+𝐀_0^⊤𝐏_0 )ε,
 
where 𝐀̅_c =diag{𝐏_c𝐀_c+𝐀_c^⊤𝐏_c} and 
𝐀̅_0 =diag{𝐏_0𝐀_0+𝐀_0^⊤𝐏_0} 
are diagonal matrices. Notice that 𝐀_c and 𝐀_0 are Hurwitz, hence two SPD matrices 𝐏_c and 𝐏_0 can be found such that the following  inequalities hold 

    c_1(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)     ≤ - 2𝐈_nN, 
    𝐏_0𝐀_0+𝐀_0^⊤𝐏_0     ≤ -𝐈_nN.

It follows by, the fact 𝐀̅_c and 𝐀̅_0 are  SND and Lemma <ref>, that (<ref>) holds for

    δ_A_c= min{σ(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c,𝐀̅_c)},
       δ_A_0= min{σ(𝐏_0𝐀_0+𝐀_0^⊤𝐏_0,𝐀̅_0)}.
 
Since 𝐀̅_c and 𝐀̅_0 are diagonal matrices, then we have η̂^⊤𝐀̅_cη̂=|η̂|_e^⊤𝐀̅_c|η̂|_e and ε^⊤𝐀̅_0ε=|ε|_e^⊤𝐀̅_0|ε|_e. With (<ref>), we know that (<ref>) is a consequence of the following inequality:

    (1-c_1)(δ_A_c|η̂|_e^⊤𝐀̅_c|η̂|_e +cδ_A_0 |ε|_e^⊤𝐀̅_0|ε|_e)   
     +2   |η̂|_e^⊤|𝐏_ε|_e|ε|_e    ≤ 0.

Writing (<ref>) as a quadratic form in terms of the expended vector [|ε|_e^⊤,|η̂|_e^⊤]^⊤ yields  

    c[[ (1-c_1)	δ_A_0𝐀̅_0                 0;                 0                 0 ]] ^𝐐_c_1   
    
    	+[[                0        |𝐏_ε|_e^⊤;          |𝐏_ε|_e (1-c_1)δ_A_c𝐀̅_c ]]_𝐐_c_2   ≤ 0 .
	
Note that the block matrix in the upper left corner of 𝐐_c_1 and the block matrix in the lower right corner of 𝐐_c_2 are SND, and both 𝐐_c_1 and 𝐐_c_2 are symmetric, so it is possible to select

    c=max{σ(𝐐_c_2,-𝐐_c_1)} ,

such that (<ref>) holds, which further guarantees that (<ref>) holds.


















 




Step 2:    b is selected to ensure that the following inequality is satisfied with κ_a being an arbitrary  positive constant:

    bc_1(η̂^⊤(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)η̂+cε^⊤(𝐏_0𝐀_0+𝐀_0^⊤𝐏_0)ε)   
    +(η̂^⊤(𝐏_c𝐃+𝐃𝐏_c)η̂+cε^⊤(𝐏_0𝐃+𝐃𝐏_0)ε)    
    +2b η̂  ^⊤η̂ +κ_a(η̂^⊤𝐏_cη̂+cε^⊤𝐏_0ε)   ≤ 0.

Writing (<ref>) as a quadratic form in terms of   [ ε ^⊤, η̂^⊤]^⊤ yields,

    b[[      c_1c(𝐏_0𝐀_0+𝐀_0^⊤𝐏_0 )                           0;                           0 c_1(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c )+2𝐈_nN ]] ^𝐐_b_1   
    +[[ c(𝐏_0𝐃+𝐃𝐏_0) +cκ_a 𝐏_0                      0;                      0   (𝐏_c𝐃+𝐃𝐏_c) +κ_a 𝐏_c ]]_𝐐_b_2   ≤ 0 .
 
Note that 𝐐_b_1 and 𝐐_b_2 are both symmetric matrices and -𝐐_b_1 is SPD. According to Lemma <ref> and noting that b≥ T, it is sufficient to choose

    b=max{σ(𝐐_b_2,-𝐐_b_1),T}
 
such that (<ref>) holds, which further guarantees that (<ref>) holds. 


Step 3: Here we aim to verify that the following inequality  holds for some positive constant κ_b,

    2bcε^⊤𝐏_0Φ≤κ_b/α(τ) V.
 
By recalling (<ref>) and using Assumption <ref> and the fact r≥ 1, we have  

    |Φ|_e≤_e 1/r𝐀_ϱ(|η̂|_e+|ε|_e),

where 𝐀_ϱ= diag{𝐀_ϱ_1,⋯,𝐀_ϱ_N} with 𝐀_ϱ_k (k=1,⋯,N) being defined in (<ref>). Accordingly, the effects of uncertainties on the Lyapunov inequality (<ref>) are bounded by

    2bcε^⊤𝐏_0Φ≤2c /α(τ)|ε|_e^⊤|𝐏_0𝐀_ϱ|_e(|η̂|_e +|ε|_e) .
  
Let 𝐏̅_c=diag{𝐏_c} and 𝐏̅_0=diag{𝐏_0} be two positive definite diagonal matrices. Hence, from Lemma <ref>,  there exist two positive constants κ_b and κ̃_b, such that,

    κ̃_b[[ -𝐏̅_0     0;     0 -𝐏̅_c ]]^𝐐_κ_a   
    
    	+c[[ |𝐏_0|_e𝐀_ϱ+𝐀_ϱ^⊤|𝐏_0|_e              |𝐏_0|_e𝐀_ϱ;            𝐀_ϱ^⊤|𝐏_0|_e                       0 ]] _𝐐_κ_b   ≤ 0 ,


    κ_b[[ -c 𝐏_0      0;      0  - 𝐏_c ]]_𝐐_κ_c 
    	+[[ κ̃_b𝐏̅_0        0;        0 κ̃_b𝐏̅_c ]]_𝐐_κ_d≤ 0 .

Specifically, the suitable κ_b and κ̃_b are

    κ_b    =max{σ(𝐐_κ_d,-𝐐_κ_c)}, 
    κ̃_b   =max{σ(𝐐_κ_b,-𝐐_κ_a)}.
  
Similar to (<ref>),  by means of (<ref>) and the fact η^⊤𝐏̅_c η=| η|_e^⊤𝐏̅_c| η|_e and ε^⊤𝐏̅_0ε=|ε|_e^⊤𝐏̅_0|ε|_e, one can conclude that,

    2c /α(τ)|ε|_e^⊤|𝐏_0𝐀_ϱ|_e(|η̂|_e  
    ≤   κ_b/α(τ)( η̂^⊤𝐏_cη̂+cε^⊤𝐏_0ε) .
 
Recalling (<ref>) and (<ref>), we know that (<ref>) holds. Combining (<ref>), (<ref>), and (<ref>), it can be verified that (<ref>) holds.  
	Similar to (<ref>) and (<ref>), we have from Lemma <ref> that  

    V (t)≤(T-t/T)^κ̃M̃V (0) , t∈[0,T),
  
where κ̃=κ_a-κ_b/α(τ_1) and M̃=e^Tκ_bτ_1 is a finite number with 0<τ_1<∞. Additionally, by recalling (<ref>), we get

    |η(t)|+|ε(t)|≤2√(V(t))/√(min{λ_min(𝐏_c),cλ_min(𝐏_0)}) .
 
Similar to the analysis in footnote 3, the above estimate does not lead to conservative controller parameter selections.   With Λ_r = diag{r^n,r^n-1,⋯,r} and the fact r ≥  1, it holds that 

    |x_k-x_0|   ≤1/r (|η|+|ε|)
       ≤2((T-t)^κ̃+2/T^κ̃)^1/2√(M̃V(0))/b√(min{λ_min(𝐏_c),cλ_min(𝐏_0)}),

establishing the same for |x̂_k-x̂_0|.
	Hence, it follows that x_k-x_0 ∈ℒ_∞[0,T), x̂_k-x̂_0 ∈ℒ_∞[0,T), lim_t→ T|x_k-x_0|=0, and lim_t→ T|x̂_k-x̂_0|=0.  By recalling (<ref>), we know that the designed control inputs satisfy u_k ∈ℒ_∞[0,T).    
	The proof is completed. ▪




	Due to each agent being equipped with sensors of different sensitivities θ_k(t), addressing these unknown parameters individually is not straightforward.  The proposed solution is to aggregate all unknown sensitivities into a matrix for unified processing, which is technically feasible but may lead to more conservative design parameters (i.e., smaller allowable sensitivity errors Δ_θ_k). This is because we only incorporate the parameters of the poorest-performing sensor into the design of the entire distributed controller. As the system scale increases (i.e.,  as N becomes larger), the value of 1/|𝐏_c𝐀_g| typically decreases accordingly. According to (<ref>), it can be inferred that in such a scenario, sensors with higher precision are needed for the proposed algorithm to be applicable.


















 §.§ Practical prescribed-time consensus tracking







Note that α(t) in Theorem <ref>   escapes to ∞ as t approaches the terminal time T. Although the existing literature (see, for instance, <cit.>) has preliminarily explored applying the prescribed-time controller to the entire time domain by imposing saturation on the time-varying gain before t approaches the terminal time, its viability has not been thoroughly analyzed. In this subsection, we provide a quantitative relationship between the maximum saturation level of the time-varying gain α(t) and the terminal time T.

In order to facilitate the association of the solution under bounded time-varying feedback with the control scheme introduced earlier, we redefine T=t_f+δ, with t_f and b being two arbitrary positive constants. Then we have the following practical prescribed-time protocol whose proof is provided for completeness.



	Suppose that the graph 𝒢 is connected and Assumption <ref> is satisfied with known growth rates ϱ_k,j. If maximum value of the allowable sensitivity error in Assumption <ref> satisfies 
	
    0<max{Δ_θ_1,⋯,Δ_θ_N}≤1/ |𝐏_c𝐀_g|,
	
	and the time-varying gain of the controller and the observer is replaced by r=bα_0, with
	
    α_0(t)={[ 1/t_f+δ-t  , t∈[0,t_f ),;  1/δ,         t∈[t_f,∞), ].
 
	where 1/δ, denotes the maximum saturation level of the time-varying gain, can be selected as any positive constant, then we can find a proportional coefficient b such that: 1) the  closed-loop dynamic system  composed of (<ref>), (<ref>),  and (<ref>) is asymptotically stable on t∈[0,∞); and 2) the tracking error e_k=x_k-x_0 converge to a compact set Ω as t→ t_f, where
	
    Ω={|e_k|∈ℝ| |e_k|≤2δ√(V(0))(δ/t_f+δ)^γ/2/b√(min{λ_min(𝐏_c),cλ_min(𝐏_0)})},
 
	with  γ>0 being an arbitrarily large  parameter, and V(0) being the initial value of V defined in (<ref>).
	

Proof: The proof is implemented by two stages. Firstly, for t∈[0,t_f), functions α_0(t) and α(t) are equivalent. According to the proof of Theorem <ref>, let V(t)=η̂^⊤𝐏_cη̂+cε^⊤𝐏_0ε, we know that choosing  c and b according to (<ref>) and (<ref>) with κ_a  being

    κ_a>κ_1/inf{α_0}= κ_b(t_f+δ),

where  κ_b is  calculated by (<ref>), can make the derivative of V satisfies 

    dV/dτ≤ -(κ_a-κ_b/α(τ))V≤ -γ V,

where γ=κ_a-κ_b(t_f+δ)>0.
	This indicates that  V(t)∈ℒ_∞[0,t_f) and V(τ)≤ e^-γτV(0), which further implies that, by recalling τ=ln(T/(T-t)) and T=t_f+δ,

    lim_t→ t_fV(t)≤(δ/t_f+δ)^γV(0).
 According to (<ref>), we have

    lim_t→ t_f|e_k|≤2δ√(V(0))(δ/t_f+δ)^γ/2/b√(min{λ_min(𝐏_c),cλ_min(𝐏_0)}) .
 

Secondly, for t∈[t_f,∞), we have α_0≡1/δ, then r=b/δ becomes a constant rather than a time-varying parameter.  Therefore, the dynamics of η and ε in (<ref>) can be simplified to

    η̇̂̇   =r(𝐀_cη̂+G^⊤⊗(𝐈_θΨ_ε+(𝐈_θ-𝐈_N)Ψ_η̂)) , 
    ε̇   =r(𝐀_0ε+Φ)   ,
 
Similar to the treatment given in (<ref>)-(<ref>), the derivative of V w.r.t. t along (<ref>) can be calculated as 

    V̇≤    b/δη̂^⊤(𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)η̂+bc/δε^⊤(𝐏_0𝐀_0+𝐀_0^⊤𝐏_0)ε
        +  2bc/δε^⊤𝐏_0Φ + 2b/δ( η̂  ^⊤η̂ +  |η̂|_e^⊤|𝐏_ε|_e|ε|_e).

Analogous to the analysis in the proof of Theorem <ref>, the design purpose is to find parameters b and c such that (<ref>) can be reduced to V̇≤ -γ^* V with γ^*>0. From (<ref>)-(<ref>), we know that select  c=max{σ (𝐐_c_2,-𝐐_c_1 ) }
	can ensure that, for c_1∈(0,1), 

    (1-c_1)b/δ( η̂^⊤ (𝐏_c𝐀_c+𝐀_c^⊤𝐏_c)  η̂.    
    .+c ε^⊤ (𝐏_0𝐀_0+𝐀_0^⊤𝐏_0) ε)
    	+   2b/δ  |η̂|_e^⊤|𝐏_ε|_e|ε|_e    ≤ 0.
 
Next, in accordance with (<ref>)-(<ref>), we know that choose κ̃_b=max{σ(𝐐_κ_b,-𝐐_κ_a)}, and κ_b=max{σ(𝐐_κ_d,-𝐐_κ_c)} can ensure that

    2bcε^⊤𝐏_0Φ≤κ_b/α(τ) V.

Since r(t) is always a constant (i.e., ṙ≡ 0) after t≥ t_f, the two SPD matrices  produced by ṙ appearing in 𝐐_b_2 (i.e. 𝐏_0𝐃+𝐃𝐏_c and 𝐏_0𝐃+𝐃𝐏_0 in (<ref>)) can be removed when recalculating b in this part. Namely, picking 

    b=max{σ (𝐐_b_2^*,-𝐐_b_1 ),t_f+δ}
 
with 𝐐_b_2^*=diag{cκ_a𝐏_0,κ_a𝐏_c} can yield

    bc_1/δ(η̂^⊤  (𝐏_c𝐀_c+𝐀_c^⊤𝐏_c ) η̂.    
    .+cε^⊤  (𝐏_0𝐀_0+𝐀_0^⊤𝐏_0 )ε)
    	+2b/δη̂  ^⊤η̂+κ_aV   ≤ 0.
  
Combining (<ref>), (<ref>),  and (<ref>), we know that (<ref>) satisfies

    V̇≤ -κ_a/δ V+κ_bV.

From (<ref>), we know κ_a=κ_b(t_f+δ)>κ_bδ, which indicates that 

    V̇≤ -γ^*V, t∈[t_f,∞),
 
where γ^*=κ_a/δ-κ_b>0. Solving (<ref>) and recalling (<ref>) gives
	
    V(t)≤ e^-γ^* tV(t_f)≤ e^-γ^* t(δ/t_f+δ)^γV(0).
	
	Therefore, the conclusion of V∈ℒ_∞[t_f,∞) and lim_t→∞V=0 can be obtained under the selection of c and b in (<ref>) and (<ref>). It follows from (<ref>) that η̂∈ℒ_∞[0,∞), ε∈ℒ_∞[0,∞), and lim_t→∞|η̂|=lim_t→∞|ε|=0.  
Recalling (<ref>), one can conclude that e_k  ∈ℒ_∞[0,∞) and   lim_t→∞|e_k |=0. By recalling (<ref>), it also holds that  u_k ∈ℒ_∞[0,∞).  
This completes the proof.   
▪


	Since the parameter δ in (<ref>) can be chosen to be sufficiently small, such as δ=0.001, and the power of δ (i.e., γ) can be chosen to be sufficiently large by increasing κ_a, we can consider Ω defined in (<ref>) as a compact set with an adjustable radius. Notice that all parameters including δ,γ,t_f,b,c,λ_min(𝐏_c) and λ_min(𝐏_0) are known, so the radius of the compact set Ω can be derived from the initial condition V(0). In this subsection, the selection of κ_a needs to satisfy the quantitative relationship given by (<ref>) rather than any positive number  allowed by (<ref>).
  


	
	Different from the approach in <cit.> and <cit.> which use dual-domination gains to deal with system nonlinearities and unknown sensor sensitivities, this paper uses only a time-varying gain to comprehensively handle all uncertainties in the closed-loop system. The advantage of doing so is that it accelerates the convergence rate without increasing the controller complexity. In particular, the entire parameter design process is conducted within the matrix pencil based framework, as demonstrated in   <cit.>, which ensures low conservatism of the design parameters. In addition,  the proposed method ensures that all states converge to a compact set within the prescribed time. This extends the practical prescribed-time control scheme presented in <cit.> and <cit.>, which only guarantees the convergence of the outputs to a compact set within the prescribed time.


































§ SIMULATIONS

To verify the effectiveness of our prescribed-time  protocols, we consider a network consisting of a leader and four followers. Each agent  is modeled as a single-link robot manipulator, described by the Lagrangian equation 
  

    J_kq̈_k+B_kq̇_k+m_kgh_ksin(q_k)=u_k, k=0,1⋯,4

where q_k and q̇_k are the general position and velocity, respectively, J_k denotes the rotation inertia, B_k is the damping coefficient, h_k is the length from the axis of joint to the mass center, m_k is the mass of the link, and g is the acceleration of gravity. Furthermore, J_0=J_4=8.5, J_1=J_2=J_3 =10, B_0=B_3=1.4, B_1=B_2=B_4=1.6, h_0=h_1=1, h_2=0.8, h_3=h_4=1.2, m_0= m_2=1.3, m_1=m_3=m_4=1, g=9.8.   Laplacian matrix of graph 𝒢̅ is given as follows

    ℒ_𝒢̅ =[[  2 -1  0  0; -1  2 -1  0;  0 -1  2 -1;  0  0 -1  1 ]].



Our control task is to design a distributed controller such that all the followers' states can track the states of the leader within a pre-given time. Define x_k=[x_k,1,x_k,2]^⊤=[J_kq_k,J_kq̇_k]^⊤ and y_k=J_kq_k, then (<ref>) can be rewritten as the same form as ẋ_k=𝐀x_k+Bu_k+F_k, y_k=θ_k(t)Cx_k with 𝐀=[0 ,1; 0 ,0], B=[0;1], C=[1,0], and F_k=[0;- B_kx_k,2/J_k-m_kgh_ksin(x_k,1/J_k)]. 
It can be verified that Assumption <ref> is satisfied with ϱ_k,1=1.8 and ϱ_k,2=0.19 for k=1,2,3,4. In the simulation, the initial conditions are set to x_0(0)=(0;0), x_1(0)=(1;1), x_2(0)=(2,2), x_3(0)=(3,3), x_4(0)=(4;4),  the input of the leader agent is set to u_0(t)=0, and  the sensor sensitivity θ_k(t) is set to θ_k(t)=1+ϰ_k|sin(10t)| (k=0,1,2,3,4) with ϰ_k∈ (-0.1,0.1).
















For the purpose of verifying the effectiveness of the protocol stated in Theorem <ref>, we use the distributed prescribed-time output-feedback protocols given in Theorem <ref> for simulation, i.e., u_k=-KΛ_r (∑_j=1^4a_k,j(x̂_k-x̂_j)+b_k(x̂_k-x̂_0) ) where Λ_r=diag{r^2, r}, r=bα(t) with α being defined in (<ref>).
Besides, the distributed observer is given by

    ẋ̂̇_k=𝐀x̂_k+Bu_k+r^n+1Λ_r^-1G(y_k-x̂_k,1),

where  K=[8,9] and  G=[2,2]. Let a_0=0,a_1=0.08,a_2=0.09,a_3=-0.08,a_4=-0.09, and c_1=0.9, we obtain two 8× 8-dimensional SPD matrices 𝐏_c and 𝐏_0 in accordance on the basis of (<ref>). Let κ_a=0.001, δ=0.02  and t_f=1.98, according to (<ref>), (<ref>), (<ref>),  and   the expressions for 𝐏_c, 𝐏_0, 𝐏̅_c, 𝐏̅_0, 𝐃, 𝐀_g, 𝐀_ϱ, 𝐀_c, 𝐀_0, 𝐀̅_c, and 𝐀̅_0, the design parameters can be calculated as δ_A_c=δ_A_0=1,   
	c = max{σ (𝐐_c_2,-𝐐_c_1 ) }=565.79, and b =max{σ (𝐐_b_2^*,-𝐐_b_1 ),t_f+δ} =4.45.
In addition, it can be checked that the sensitivity error under consideration is allowable since  max{Δ_θ_1,⋯,Δ_θ_4}≤ 0.1≤1/|𝐏_c𝐀_g|= 0.1015.   

The simulation results are presented in Fig. <ref>, which shows that the followers' states (with different initial values) tracks the leader's states within the pre-given time, and the control input signals are uniformly bounded over the prescribed time interval.  In addition, it can be seen that all observer errors converge to zero within the prescribed time, and the sensor sensitivity θ_k for each agent is different. These findings confirm our theoretical predictions and affirm the efficacy of the proposed prescribed-time control strategy. 






 




§ CONCLUSIONS

In this paper, we have tackled the leader-following consensus problem by designing distributed state-feedback and output-feedback controllers using local information (might be inaccurate) to ensure that all the agents track the leader's states within a prescribed time. 
Furthermore, we robustify the standard matrix pencil-based design procedure in the presence of multiplicative measurement noise and develop its distributed formulation to select the proportional coefficient of the time-varying control gain, and adopt the time-axis transformation to convert the original system operating over a finite time interval into one that operates over an infinite time interval. 
In this way, the prescribed-time control problem can be recast into the asymptotic control problem, which simplifies the control design and stability analysis.
Additionally, we achieve practical prescribed-time consensus for MAS systems by embedding bounded time-varying gain without sacrificing the ultimate control accuracy. 
In particular, the prescribed-time and the practical prescribed-time leader-following controllers (including the observer) have the same structure. 
The latter can be obtained directly by applying saturation to the unbounded gain of the former, and we quantitatively discuss how to choose this saturation constant.  
The simulation results verify the effectiveness of the proposed method. 
Based on these results, it is of interest to further study prescribed-time formation control for MASs with more general topologies and topology attacks. 










 §.§ Proof of Proposition 1
  
Taking the derivative of (<ref>)   along the trajectory of (<ref>) yields 

    η̇_k =r(𝐀η_k+B u_k +Φ_k)+ṙ/r𝐃_cη_k .
 
Applying the time transformation on the foregoing ODE, we can rewrite (<ref>) as its equivalent form on the time domain [0,∞),

    dη_k/dτ =dt/dτr(𝐀η_k+B u_k +Φ_k)+dr/dτ1/r𝐃_cη_k,
 
Recalling that r=bα(τ)=b (dτ/dt)=be^τ/t_f, we have 
    dt/dτr=b,  dr/dτ1/r=1.
 Accordingly, (<ref>) can be continued as

    dη_k/dτ=b(𝐀η_k+B u_k+Φ_k)+𝐃_c η_k .
 
The proof is completed. ▪








 §.§ Properties of two classes of matrix pencils

We summarize the properties of the two forms of matrix pencils used in this paper into the following lemma. 
	Given real square matrices 𝐐_1 and 𝐐_2, the generalized eigenvalues of matrix pencil 𝐐_1-s𝐐_2 are defined as the values of s that make (𝐐_1-s𝐐_2) = 0. The set of generalized eigenvalues of matrix pencil 𝐐_1-s𝐐_2 are denoted as σ(𝐐_1, 𝐐_2). 
	
		
  1) If 𝐐_1 is symmetric and 𝐐_2 is symmetric positive-definite (SPD), then 𝐐_1-s𝐐_2<0 holds for all s > max{σ(𝐐_1, 𝐐_2)};
		
  2) If  both 𝐐_1 and 𝐐_2 are symmetric negative-definite (SND), 𝐐_1-s𝐐_2< 0 holds for all s < min{σ(𝐐_1, 𝐐_2)}.
	 
 

In implementation,  the  set of the desired generalized eigenvalues  σ(𝐐_1, 𝐐_2) can be obtained through the code s=double(solve(det(𝐐_1-s*𝐐_2))) with Matlab. 




 §.§ A Lyapunov-like function

In this paper, the time-derivative of V(τ) might be indefinite since the sign of -κ_0+κ_1/α(τ) is not required to be negative for all τ,  therefore the function V (τ) in Theorems 1-3 is not a Lyapunov function in the usual sense. For this special type of V, we need the following lemma to facilitate stability analysis.

	Consider a function V(τ):[0,∞)→[0,∞), if there exist two positive constants κ_0 and κ_1 such that
	
    dV/dτ≤- (κ_0 - κ_1/α(τ)) V,

	where α(τ)=e^τ/T is a monotonically increasing function to infinity as defined in (<ref>), T is a positive design parameter. Then, it holds that  V(τ)∈ℒ_∞[0,∞) and  lim_τ→∞V(τ)=0. 

Proof:  
Since κ_0 is a positive constant and κ_1/α(τ) is a monotonically decreasing function to zero, thus there must exist a time τ_1∈[0,∞), such that, for all τ∈[τ_1,∞),

    κ_0-κ_1/α(τ)>0.

Firstly, we analyze the boundedness of V on [0,τ_1). From (<ref>) and the property of α(τ), we know that

    dV/dτ≤κ_1/α(0)V,

where α(0)=1/T, and then by solving (<ref>) it is easily to obtain 

    V(τ)≤ e^Tκ_1 τ_1V(0)∈ℒ_∞[0,τ_1).

Then, for τ∈[τ_1,∞), by means of (<ref>), Eq. (<ref>) satisfies
dV/dτ≤ - κV,
where κ=κ_0-κ_1/α(τ_1) is a positive constant, which yields

    V(τ)≤ e^-κτV(τ_1) ∈ℒ_∞[τ_1,∞).

Combining with (<ref>) and (<ref>), we obtain that 

    V(τ)≤ e^-κτe^Tκ_1 τ_1V(0),

which implies that V(τ)∈ℒ_∞[0,∞) and  lim_τ→∞V(τ)=0. The proof is completed. ▪
 


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