Using Delays for Digital Implementation of Derivative-Dependent Control of Stochastic ⋆ Multi-Agents

Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

Using delays for digital implementation of derivative-dependent control of stochastic ⋆ multi-agents

Jin Zhang ∗ Emilia Fridman ∗

∗ School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (e-mails: [email protected], [email protected]).

Abstract: In this paper, we study the digital implementation of derivative-dependent control for consensus of the nth-order stochastic multi-agent systems. The consensus controllers that depend on the output and its derivatives up to the order n 1 are approximated as delayed sampled-data controllers. For the consensus analysis, we propos− e novel Lyapunov-Krasovskii functionals to derive linear matrix inequalities (LMIs) that allow to ﬁnd admissible sampling period. The eﬃciency of the presented approach is illustrated by numerical examples.

Keywords: Sampled-data control, stochastic multi-agent systems, consensus, LMIs.

1. INTRODUCTION consensus controllers that depend on the output and its derivatives up to the order n 1 as delayed sampled- − During the last decade, consensus of multi-agent systems data controllers. Note that results of Zhang and Frid- has received much attention due to its wide applications man (2019) cannot be directly applied to the multi-agent (Olfati-Saber et al., 2007). Consensus requires all agents to case because of an additional term in the closed-loop achieve a desired objective via neighbors’ information. For system. To compensate the additional term, we propose example, the second-order consensus problem was studied novel Lyapunov-Krasovskii functionals that lead to LMIs. by the position and velocity information (Yu et al., 2010; Finally, we present numerical examples to illustrate the Gao and Wang, 2011). If the velocity (i.e. the derivative efficiency of the presented approach. of the position) is not available, it can be approximated T n Notations: Throughout this paper, 1n = [1,..., 1] R , by finite differences leading to a delayed feedback (see e.g. T n ∈ 0n = [0,..., 0] R , In is the identity n n matrix, Niculescu and Michiels (2004); Kharitonov et al. (2005); stands for the Kronecker∈ product, the superscript× T stands⊗ Fridman and Shaikhet (2016); Ram´ırez et al. (2016); for matrix transposition. Rn denotes the n dimensional Fridman and Shaikhet (2017) and reference therein). This n m Euclidean space with Euclidean norm , R × denotes idea has been employed for the second-order deterministic the set of all n m real matrices. Denote| · by | diag ... and multi-agent systems in Yu et al. (2013); Ma et al. (2014); col ... block-diagonal× matrix and block-column{ vector,} Cui et al. (2019). respectively.{ } X > 0 implies that X is a positive definite In many areas of applications, e.g. aircraft engineering, symmetric matrix, X 2 denotes the expression XT SX | |S process control, population dynamics, multiplicative noises with matrix S and vector X of appropriate dimensions, that occur due to the parameter uncertainties and nonlin- EX denotes the mathematical expectation of stochastic earities cannot be avoided (Mao, 2007; Shaikhet, 2013). variable X. Consensus of stochastic multi-agent systems was studied in Li and Zhang (2010); Ding et al. (2015); Ma et al. 2. PROBLEM FORMULATION AND (2017). In our recent paper (Zhang and Fridman, 2019), PRELIMINARIES the delayed implementation of derivative-dependent control for stochastic systems was considered. However, the 2.1 Graph theory idea of using the delayed position information has not been studied yet for the nth-order deterministic multi-agents The communication topology among N agents is repre- (n 3) or stochastic multi-agents (n 2). sented by a directed weighted graph = ( , , ), where ≥ ≥ = 1, 2,...,N is the node set, G V E isA the edge V { } N N E⊆V×V In this paper, we study digital implementation of derivative- set, = [aij ] R × is the weighted adjacency matrix A ∈ dependent control by using delays for the nth-order with aij 0, i, j . It is assumed that aii = 0, i . ≥ ∀ ∈V ∀ ∈V stochastic multi-agent systems. Following the improved Notice that (i, j) when aij > 0. An edge (i, j) approximation method by using consecutive sampled out- implies that node ∈i can E receive information from node∈j E. N N puts (Selivanov and Fridman, 2018), we approximate the Correspondingly, the Laplacian matrix L = [Lij ] R × N ∈ ⋆ of graph is defined by Lii = aij and Lij = aij This work was supported by Israel Science Foundation (grant no. G j=1 − when i = j. Graph is said toP have a spanning tree if 673/19), by C. and H. Manderman Chair at Tel Aviv University, and 6 G by the Planning and Budgeting Committee (PBC) Fellowship from there exists node i such that node i is reachable from the Council for Higher Education, Israel. any other nodes. ∈V

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2.2 Derivative-dependent control of stochastic multi-agents where xj,0(t)= xj,0(0) for t< 0 and n 1 Consider the nth-order stochastic dynamic for each agent l − m 1 Kl = ( 1) K¯m. (6) i governed by − l hm mX=l ∈V n 1 n − j For practical application of the delayed feedback (5), we y( ) t a c w t y( ) t bu t , i ( )= j + j ˙ ( ) i ( )+ i( ) (1) suggest its sampled-data implementation. We suppose that Xj=0 xj,0(t) is only available at the time instants tk = kh, (0) Rk (j) N where yi(t)= yi (t) is the output, yi (t) is the jth k 0, where h > 0 is the sampling period. Then we ∈ m ∈ derivative of yi(t), ui(t) R is the control input, w(t) is have the following sampled-data controller the scalar standard Wiener∈ process (Mao, 2007; Shaikhet, N n 1 Rk k Rk m − 2013), and ai, ci × , b × are constant matrices. ui(t)= Lij K¯lx¯j,l(tk) ∈ ∈ − Denoting Xj=1 Xl=0 (0) (n 1) N n 1 (7) xi(t) = col yi (t),...,yi − (t) − { } nk = Lij Klxj,0(tk l), = col xi,0(t),...,xi,n 1(t) R , − − { − } ∈ Xj=1 Xl=0 0 Ik 0 0 ··· t [tk,tk+1), k N0, 0 0 Ik 0 ∈ ∈  ···  nk nk A = ... R × , wherex ¯j,l(t) and Kl are given by (4) and (6), respectively. ············ ∈  0 0 0 Ik  For the sampled-data controller (7), we introduce the  ···   a0 a1 a2 an 1 errors due to sampling  nk··· m −  t B = col 0,b R × , { } ∈ nk nk x¯j,0(tk) = xj,0(t) x˙ j,0(s)ds, C = col 0, C¯ R × − Ztk { } ∈ t (8) with C¯ = [c0,...,cn 1], we present (1) as − x¯j,i(tk) =¯xj,i(t) x¯˙ j,i(s)ds, i =1,...,n 1. dxi(t)= Axi(t)+ Bui(t) dt + Cxi(t)dw(t), (2) − Ztk − 0 where the initial condition is given by xi(0) = xi . Then we follow the idea of Selivanov and Fridman (2018) to present the approximation errors xj,i(t) x¯j,i(t) (i = For multi-agent system (2), it is common to look for a 1,...,n 1) as − consensus controller of the form (see e.g. Sun and Wang − t (2009)) x¯j,l(t)= xj,l(t) ϕl(t s)x ˙ j,l(s)ds, (9) N N n 1 − Zt lh − − − ui(t)= K¯ Lij xj (t)= Lij K¯lxj,l(t) (3) h v − − where ϕ1(v)= −h , v [0,h] and for i =1,...,n 2 Xj=1 Xj=1 Xl=0 v ∈ − m nk 1 h v with K¯ = [K¯0,..., K¯n 1] R × such that for any ϕi(λ)dλ + − , v [0,h] − h Z h ∈ initial condition, the consensus∈ of (2) can be exponentially 0v  1 mean-square achieved, i.e. ϕi+1(v)=  ϕi(λ)dλ, v (h,ih). αt h Zv h ∈ E xi(t) xj (t) ce− E xi(0) xj (0) , i, j ,  −ih | − |≤ | − | ∀ ∈V 1 where c > 0 is a constant and α > 0 is called the decay  ϕi(λ)dλ, v [ih,ih + h]. h Zv h ∈ rate.  − For further proceeding, we present several properties of Differently from the state-feedback case with the full the functions ϕi(s) (i =1,...,n 1). knowledge of the system state, we consider the output- − Proposition 1. The functions ϕi(s) (i = 1,...,n 1) feedback control where the derivatives xj,l(t) (l = − 1,...,n 1) in (3) are not available. As in Selivanov and have the following properties (Zhang and Fridman, 2019; Fridman− (2018), we employ their finite-difference approx- Selivanov and Fridman, 2018): imations: (1) 0 ϕi(v) 1, v [0,ih], ≤ ≤ ∈ xj,0(t)=¯xj,0(t), (2) ϕi(0) = 1, ϕi(ih) = 0, x¯j,l 1(t) x¯j,l 1(t h) ih ih xj,l(t) x¯j,l(t)= − − − − (3) 0 ϕi(v)dv = 2 , ≈ h Rd ϕ v 1 , v ,ih l (4) (4) dv i( ) [ h 0], [0 ]. 1 l p ∈ − ∈ = ( 1) xj, (t ph), hl p − 0 − Based on these properties, it follows from (9) that Xp=0 l =1, ..., n 1 t − x¯˙ j,i(t)= ψi(t s)x ˙ j,i(s)ds, i =1,...,n 1 with a constant delay h> 0 and the binomial coefficients Zt ih − − l l! − p = p!(l p)! . By replacing xj,l(t) in (3) with their (10) − d approximations, we have the following delay-dependent where ψi(t s)= ϕi(t s). − ds − controller Denote N n 1 − ¯ x(t) = col x1(t),...,xN (t) , χ(t) = col χ2(t),...,χN (t) , ui(t) = Lij Klx¯j,l(t) { } { } − x¯(t) = col x¯1(t),..., x¯N (t) , χ¯(t) = col χ¯2(t),..., χ¯N (t) Xj=1 Xl=0 { } { } N n 1 (5) with − = Lij Klxj,0(t lh), χi(t)= x1(t) xi(t) = col χi,0(t),...,χi,n 1(t) , − − − { − } Xj=1 Xl=0 χ¯i(t)=¯x1(t) x¯i(t) = col χ¯i,0(t),..., χ¯i,n 1(t) . − { − }

3665 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

n From Sun and Wang (2009), it follows that χ(t) = (E1 2 2 ⊗ T 2 − (ih) 2 Ink)x(t), x(t) = (E Ink)χ(t) + (1N Ink)x (t) and Φ11 = PD + D P +2αP + + Hi+1 2 ⊗ ⊗ 1 |C|P 4 | |Ri x¯(t) = (E2 Ink)¯χ(t) + (1N Ink)¯x1(t), where E1 = Xi=1 ⊗ ⊗ T n 2 [1N 1, IN 1] and E2 = [0N 1, IN 1] . Then the sys- − (n 1)h − − − − − − 2 2αih 2 2 tem (2), (3) takes the form + h e Hi+1 i + − Hn 1 F1+F2 , | |W 2 | − C| Xi=0 dχ(t)= Dχ(t)dt + g(t)dw(t), (11) Φ12 = P [( BK¯1),..., ( BK¯n 1)], L⊗ L⊗ − where Φ14 = P ( BK¯ ), ¯ TL⊗T D = IN 1 A BK, = E1LE2, Φ15 = D Hn 1(Rn 1 + Q), − ⊗ −L⊗ L (12) − − g(t) = (IN 1 C)χ(t). e 2αhR ,...,e 2α(n 1)hR , − ⊗ Φ22 = diag − 1 − − n 1 − { 2α(n 1)h T − } Using (8) and (9), the system (2), (7) takes the form Φ23 = [0k (n 2)k,e− − Rn 1] , × − ¯ ¯− T T dχ(t)= f(t)dt + g(t)dw(t), (13) Φ25 = [( BK1),..., ( BKn 1)] Hn 1(Rn 1 + Q), L⊗2α(n 1)h L⊗ − − − where Φ33 = e− − (Rn 1 + F1), − 2 − n 1 π 2αh − Φ44 = e− IN 1 diag W0,...,Wn 1 , f(t)= Dχ(t)+ ( BK¯i)κi(t) + ( BK¯ )δ(t), − 4 − ⊗ { − } L⊗ L⊗ ¯ T T T Xi=1 Φ45 = ( K B )Hn 1(Rn 1 + Q), t t L⊗ 4 − − R Q , κi(t)= ϕi(t s)Hiχ˙(s)ds, δ(t)= χ¯˙(s)ds Φ55 = 2 ( n 1 + ) − −(nh h) − Zt ih Ztk − 2 − (14) (n 1) 2α(n 1)h Ω11 = n 1 − e− − Q, with D, g(t) given by (12) and W − − 4 Ω12 = n 1, Hi = IN 1 εi, εi = [0k ik, Ik, 0k (n i 1)k]. W − n 1 − × × − − 2α(n 1)h ⊗ Ω22 = n 1 − e− − F2 As in the determinstic case (see e.g. French et al. (2009)), W − − 2 with = IN 1 C and i = IN 1 Wi. Then consensus we will show in the next section that for small enough C − ⊗ W − ⊗ stochastic perturbations (i.e. small enough C ), if the of multi-agent system (2) under sampled-data controller system (11) is exponentially mean-square stable| | with a (7) with controller gains (6) is exponentially mean-square α> decay rateα> ¯ 0, then for any α (0, α¯) the system (13) achieved with a decay rate 0. ∈ is exponentially mean-square stable with a decay rate α (ii) Given any α (0, α¯), the LMI Φ < 0 is always and small enough h> 0. feasible for small∈ enough stochastic perturbations and Remark 1. Comparatively to the single-agent system in h > 0 (meaning that consensus of multi-agent system (2) Zhang and Fridman (2019), the multi-agent system (13) under sampled-data controller (7) with controller gains (6) contains additional term δ(t) due to the sampling. This is exponentially mean-square achieved with a decay rate term will be further compensated by additional terms in α> 0). Lyapunov functionals (see e.g. (24), (26), (28), (30)). Proof: (i) Let L be the generator of the system (13) (Mao, 2007; Fridman and Shaikhet, 2019). For the term 3. CONSENSUS ANALYSIS 2 VP = χ(t) , P> 0, (16) | |P In this section, we will analyze the exponential stability of we have L T 2 2 system (13). First, we follow arguments of Sun and Wang VP +2αVP = χ (t)Pf(t)+2α χ(t) P + g(t) P . (17) (2009) to present following result: | | | | To compensate the terms κi(t) (i = 1,...,n 2), we Proposition 2. Assume that directed graph has a span- consider − ning tree. Consensus of multi-agent systemG (2) under t ih 2α(t s) 2 sampled-data controller (7) with controller gains (6) can be VRi = e− − ϕi(v) Hi+1χ(s) Ri dvds, Zt ih Zt s | | (18) exponentially mean-square achieved if and only if system − − Ri > 0, i =1,...,n 2. (13) is exponentially mean-square stable. − Using Jensen’s inequality (Gu et al., 2003; Solomon and It is clear from Proposition 2 that consensus of multi- Fridman, 2013) and Proposition 1 with Hiχ˙(t)= Hi+1χ(t) agent system (2) under sampled-data controller (7) with (i =1,...,n 2), we have − controller gains (6) is converted into stability of (13). ih L 2 VRi +2αVRi = Hi+1χ(t) Ri We now present the following LMI conditions: t 2 | | ¯ ¯ ¯ 2α(t s) 2 Theorem 1. Given K = [K0,..., Kn 1], let system (11) e− − ϕi(t s) Hi χ(s) ds − +1 Ri (19) with C = 0 be exponentially mean-square stable with a − Zt ih − | | − α> ih 2 2 2αih 2 decay rate ¯ 0. Hi χ(t) e− κi(t) , ≤ 2 | +1 |Ri − ih | |Ri (i) Given tuning parameters h > 0 and α (0, α¯), let i =1,...,n 2. ∈ − there exist matrices P > 0, W0 > 0, Ri > 0, Wi > 0 For the term κn 1(t), we consider (i =1,...,n 1), Q> 0, F > 0 and F > 0 of appropriate − − 1 2 t (n 1)h dimensions that satisfy − 2α(t s) VRn−1 = e− − ϕn 1(v) Φ < 0, Ω < 0, (15) Zt (n 1)h Zt s − (20) − − −2 Hn 1f(s) Rn−1 dvds, Rn 1 > 0. where Φ and Ω are, respectively, the symmetric matrices ×| − | − composed from Then we have

3666 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

n h ρ t L ( 1) 2 To compensate the term 2( ), we consider VRn−1 +2αVRn−1 − Hn 1f(t) Rn−1 ≤ 2 | − | t 2 (21) 2α(t s) 2 2α(n 1)h 2 VQ = e− − ϕn 1(t s) Hn 1f(s) Qds e− − κn 1(t) ρ1(t) Rn−1 , Z − − | − | −(n 1)h | − − | t (n 1)h − − − where (26) t with Q > 0. Using Jensen’s inequality and Proposition 1 ρ1(t)= ϕn 1(t s)Hn 1g(s)dw(s). yields − − − Zt (n 1)h L 2 2α(n 1)h 2 − − VQ +2αVQ Hn 1f(t) Q e− − ρ2(t) Q. (27) By using Itˆo integral properties (see e.g. Mao (2007); ≤ | − | − | | Fridman and Shaikhet (2019)) and Proposition 1, we have Similarly, for the term ρ3(t), we consider F > t for any matrix 1 0 2α(t s) 2 2α(n 1)h 2 VF2 = e− − ϕn 1(t s) Hn 1g(s) F2 ds Ee− − ρ1(t) F1 Zt (n 1)h − − | − | | t| − − (28) = Ee 2α(n 1)h ϕ2 (t s) H g(s) 2 ds − − n 1 n 1 F1 F > Zt (n 1)h − − | − | with 2 0. Via t − − 2α(n 1)h 2 2α(t s) 2 Ehe− − ρ3(t) F2 E e− − ϕn 1(t s) Hn 1g(s) F1 ds. | t| ≤ Zt (n 1)h − − | − | 2α(n 1)h 2 2 − − = Ee− − h ψn 1(t s) Hn 1g(s) F2 ds For the term Zt (n 1)h − − | − | t (n 1)h t − − − 2α(t s) 2α(t s) 2 VF1 = e− − ϕn 1(v) E e− − ψn 1(t s) Hn 1g(s) F2 ds, − − Zt (n 1)h Zt s − (22) ≤ Zt (n 1)h − | | − − −2 − − Hn 1g(s) F1 dvds, F1 > 0, we have ×| − | we have ELVF2 + E2αVF2 ELV E αV 2 2α(n 1)h 2 (29) F1 + 2 F1 E Hn 1g(t) F2 Ehe− − ρ3(t) F2 . (n 1)h (23) ≤ | − | − | | 2 2α(n 1)h 2 Finally, to cancel the third positive term on the right-hand E − Hn 1g(t) F1 Ee− − ρ1(t) F1 . ≤ 2 | − | − | | side of (25), we consider Denote 0 < W = IN 1 diag W0,...,Wn 1 . Let us t − ⊗ { − } consider V e 2α(t s)ϕ t s H χ s 2 ds, t i = − − i( ) i+1 ( ) i W Zt ih − | |W (30) 2 2α(t s) 2 − VW = h e− − χ¯˙(s) W ds i =1,...,n 2. Ztk | | − 2 t Via Jensen’s inequality and Proposition 1, we have π 2αh 2α(t s) 2 (24) e− e− − δ(s) W ds, L 2 − 4 Ztk | | V i +2αV i Hi+1χ(t) i N W Wt ≤ | |W t [tk,tk+1), k 0 2αih 2 (31) ∈ ∈ e− ψi(t s)Hi+1χ(s)ds i . to compensate δ(t). The exponential Wirtinger’s inequal- − | Zt ih − |W − ity (Selivanov and Fridman, 2016) implies VW 0 since ≥ We now consider the functional δ˙(t) = χ¯˙(t), δ(tk) = 0 and W > 0. Then one can easily n 1 n 2 arrive at − ih − 2 2αih π2 V = VP + VRi + VW + h e V i L 2 2 2αh 2 2 W VW +2αVW = h χ¯˙(t) W e− δ(t) W . Xi=1 Xi=1 (32) | | − 4 | | (nh h)2 (n 1)h From (4) and (10), it follows that + − VQ + VF1 + − VF2 . 4 2 χ¯˙ j,0(t) =χ ˙ j,0(t) t Then in view of (17), (19), (21), (23), (25), (27), (29) and (31), we have χ¯˙ j,i(t) = x¯˙ 1,i(t) x¯˙ j,i(t)= ψi(t s)χ ˙ j,i(s)ds, − Zt ih − T ¯ 2 T − ELV + E2αV Eξ (t)Φξ(t)+ Eh ζ (t)Ωζ(t) i =1,...,n 1. 2≤ − (nh h) 2 (33) which yields +E − Hn 1f(t) Rn−1+Q, − 2 2 2 4 | | χ¯˙(t) W = H1χ(t) 0 + (ρ2(t)+ ρ3(t)) n−1 W W where | | | n 2 | t | | − ξ(t) = col χ(t), ξ¯(t),ρ1(t),δ(t) , ψ t s H χ s ds 2 , { } + i( ) i+1 ( ) i ξ¯(t) = col κ1(t),...,κn 1(t) , | Zt ih − |W { − } Xi=1 − ζ(t) = col ρ (t),ρ (t) , { 2 3 } where ¯ t and Φ is obtained from Φ by taking away the last block- ρ2(t) = ψn 1(t s)Hn 1f(s)ds, column and block-row. By substituting f(t) given by (14) Zt (n 1)h − − − and applying Schur’s complement, it follows from Φ < 0 −t − and Ω < 0 that ELV +E2αV 0 implying the exponential ρ3(t) = ψn 1(t s)Hn 1g(s)dw(s). mean-square stability of the≤ system (13) with a decay Zt (n 1)h − − − − − rate α > 0. Then following arguments of Proposition 2, Thus L 2 2 2 consensus of multi-agent system (2) under sampled-data VW +2αVW = h [ H1χ(t) 0 + (ρ2(t)+ ρ3(t)) n−1 | |W | |W controller (7) with controller gains (6) is thus exponentially n 2 t 2 − 2 π 2αh 2 mean-square achieved with a decay rate α> 0. + ψi(t s)Hi+1χ(s)ds i ] e− δ(t) W . | Zt ih − |W − 4 | | Xi=1 − (ii) If the system (11) with C = 0 is exponentially mean- (25) square stable with a decay rateα ¯ > 0, then for any

3667 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

α (0, α¯) there exists matrix P > 0 of appropriate dimension∈ such that PD + DT P +2αP < 0. Thus Agent Agent Agent Agent 1 4 1 4 PD + DT P +2αP + 2 < 0 (34) |C|P for small enough C . We choose Ri (i =1,...,n), Q, F1, 1 | | 1 F2, as I(N 1)k and Wi (0=1,...,n) as Ik. By using √h − √h Schur’s complement, Φ¯ < 0 given by (33) is equivalent to T 2 PD + D P +2αP + + √h(G + hG ) < 0 (35) Agent Agent Agent Agent |C|P 1 2 where 2 3 2 3 n 1 2 4 2αh G1 = − Hn 1 + e P ( BK¯ ) (a) (b) 2 | − C| π2 | L⊗ | n 2 − 2αih 2α(n 1)h + e P ( BK¯i) +2e − P ( BK¯n 1) , Fig. 1. Directed graphs with a spanning tree. | L⊗ | | L⊗ − | Xi=1 n 2 n 2 Table 1. Maximum values of h for diﬀerent σ − 2 − i 2 2αih 2 and α =0.1 G = Hi + e Hi . 2 4 | +1| | +1| Xi=1 Xi=0 σ 0.02 0.1 0.2 0.5 1 It is clear that (34) implies (35) for small enough h > 0 Re. 2 0.145 0.141 0.135 0.085 0.012 Th. 1 0.079 0.076 0.070 0.037 0.004 since √h(G1+hG2) 0 for h 0, implying the feasibility of Φ¯ < 0 for small enough→ h>→0. Finally, applying Schur’s complement to the last block-column and block-row of Φ 6 3 15 given by (15), we ﬁnd that Φ < 0 for small enough h> 0 x11 x12 x13 x x x 5 21 22 10 23 if Φ¯ < 0 is feasible. Therefore, the LMI Φ < 0 is always 2 x31 x32 x33 feasible for small enough h> 0 and C . 4 x41 x42 x43 | | 5 L 1 From (33), it follows that for small enough h> 0, E V + 3 E2αV 0 always holds provided (34) holds. This implies 0 0 that for≤ small enough h > 0 and C , consensus of multi- 2 2 3 | | i i -5 agent system (2) under sampled-data controller (7) with x x 1 -1 controller gains (6) is exponentially mean-square achieved -10 with a decay rate α> 0. State trajectories 0 -2 -15 -1

-3 Remark 2. As in Selivanov and Fridman (2018), we con- -2 -20 sider the scenario that multi-agent system (2) is under -3 -4 -25 continuous-time control (i.e. delay-dependent controller 0 5 0 5 0 5 (5)). Based on Theorem 1, new LMIs are obtained by time (s) time (s) time (s) setting Wi = 0 (i = 0,...,n 1) and Q = F = 0. − 2 Thus, consensus of multi-agent system (2) under delay- Fig. 2. State trajectories under delay-dependent controller dependent controller (5) with controller gains (6) is expo- (5) with h =0.145. nentially mean-square achieved with a decay rate α> 0. second-order deterministic multi-agent systems (i.e. n = 2, 4. NUMERICAL EXAMPLE σ = 0). Instead, our method allows to cope with high-order stochastic multi-agent systems (i.e. n 3, σ = 0). ≥ 6 We consider each agents described by (1) with Case II: n = 3, σ = 0. We choose K¯0 = 1, K¯1 = 2, ¯ 6 aj =0, b =1, cj = σ R, j =0,...,n 1. (36) K2 = 3, and consider the communication topology Fig. ∈ − 1(b). For diﬀerent values of σ and α = 0.1, Table 1 Two communication topologies are given as directed presents the maximum values of h via LMIs in Remark graphs with a spanning tree in Fig. 1. Without loss of 2 and Theorem 1. For the further simulation, we choose generality, all the weights are assumed to be 1. σ = 0.02 and the initial conditions of the four agents T T Case I: n = 2, σ = 0. First, we choose K0 = 1, K1 = 0.5, as x1(0) = [3, 4, 2] , x2(0) = [ 2, 3, 3] , x3(0) = − T − T − − and consider the communication topology Fig. 1(a). Via [4, 2, 2] and x4(0) = [2, 3, 3] . By using the Euler- the frequency-domain approach in Yu et al. (2013), the Maruyama− method (Higham,− 2001) with step size 0.001, maximum value of h is obtained as 1.8137. From (6), Figs. 2 and 3 depict the state trajectories under delay- one obtains K¯0 = 0.5, K¯1 = 0.9069. LMIs in Remark 2 dependent controller (5) with h = 0.145 and sampled- with α = 0 lead to the maximum value of h as 1.1025. data controller (7) with h = 0.079, respectively, showing Second, as in Ma et al. (2014), we choose K¯0 = 2.5, that the consensus is achieved in the presence of stochastic K¯1 = 2. Under the communication topology Fig. 1(b), perturbations. the direct discretization approach in Ma et al. (2014) leads to the maximum value of h as 0.2, whereas LMIs 5. CONCLUSION in Theorem 1 with α = 0 give the maximum value of h as 0.13. It should be pointed out that the approaches in Yu In this paper, the digital implementation of derivative- et al. (2013); Ma et al. (2014) are only applicable to the dependent control by using delays has been investigated

3668 Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020

Kharitonov, V.L., Niculescu, S.I., Moreno, J., and 8 3 20 Michiels, W. (2005). Static output feedback stabiliza- x11 x12 x13 tion: Necessary conditions for multiple delay controllers. x21 x22 15 x23 2 6 x31 x32 x33 IEEE Transactions on Automatic Control, 50(1), 82–86. x41 x42 10 x43 Li, T. and Zhang, J.F. (2010). Consensus conditions of 1 multi-agent systems with time-varying topologies and 4 5 stochastic communication noises. IEEE Transactions

0 0 on Automatic Control, 55(9), 2043–2057. 2 3 2 i i Ma, L., Wang, Z., Han, Q.L., and Liu, Y. (2017). Consen- x x -5 -1 sus control of stochastic multi-agent systems: a survey. Science China Information Sciences, 60(12), 120201. State trajectories 0 -10 -2 Ma, Q., Xu, S., and Lewis, F.L. (2014). Second-order -15 consensus for directed multi-agent systems with sampled -2 -3 data. International Journal of Robust and Nonlinear -20 Control, 24(16), 2560–2573.

-4 -4 -25 Mao, X. (2007). Stochastic differential equations and 0 5 0 5 0 5 time (s) time (s) time (s) applications. Horwood, Chichester. Niculescu, S.I. and Michiels, W. (2004). Stabilizing a chain of integrators using multiple delays. IEEE Transactions Fig. 3. State trajectories under sampled-data controller (7) on Automatic Control, 49(5), 802–807. with h =0.079. Olfati-Saber, R., Fax, J.A., and Murray, R.M. (2007). for consensus of stochastic multi-agent systems. Simple Consensus and cooperation in networked multi-agent LMIs that allow to find admissible sampling period have systems. Proceedings of the IEEE, 95(1), 215–233. been presented by using appropriate Lyapunov-Krasovskii Ram´ırez, A., Mondié, S., Garrido, R., and Sipahi, R. functionals. The efficiency of the presented approach has (2016). Design of proportional-integral-retarded (PIR) been illustrated by numerical example. The presented controllers for second-order LTI systems. IEEE Trans- method can be further applied to digital implementation actions on Automatic Control, 61(6), 1688–1693. of PID control for the second-order stochastic multi-agent Selivanov, A. and Fridman, E. (2016). Observer-based systems. This may be a topic for future research. input-to-state stabilization of networked control systems with large uncertain delays. Automatica, 74, 63–70. Selivanov, A. and Fridman, E. (2018). An improved time- REFERENCES delay implementation of derivative-dependent feedback. Automatica, 98, 269–276. Cui, B., Xia, Y., Liu, K., and Zhang, J. (2019). Coop- Shaikhet, L. (2013). Lyapunov functionals and stability erative tracking control for general linear multiagent of stochastic functional differential equations. Springer systems with directed intermittent communications: An Science & Business Media. artificial delay approach. International Journal of Ro- Solomon, O. and Fridman, E. (2013). New stability condi- bust and Nonlinear Control, 29, 3063–3077. tions for systems with distributed delays. Automatica, Ding, D., Wang, Z., Shen, B., and Wei, G. (2015). Event- 49(11), 3467–3475. triggered consensus control for discrete-time stochas- Sun, Y.G. and Wang, L. (2009). Consensus of multi- tic multi-agent systems: The input-to-state stability in agent systems in directed networks with nonuniform probability. Automatica, 62, 284–291. time-varying delays. IEEE Transactions on Automatic French, M., Ilchmann, A., and Mueller, M. (2009). Robust Control, 54(7), 1607–1613. stabilization by linear output delay feedback. SIAM Yu, W., Chen, G., and Cao, M. (2010). Some necessary and Journal on Control and Optimization, 48(4), 2533–2561. sufficient conditions for second-order consensus in multi- Fridman, E. and Shaikhet, L. (2016). Delay-induced stabil- agent dynamical systems. Automatica, 46(6), 1089– ity of vector second-order systems via simple Lyapunov 1095. functionals. Automatica, 74, 288–296. Yu, W., Chen, G., Cao, M., and Ren, W. (2013). Delay- Fridman, E. and Shaikhet, L. (2017). Stabilization by induced consensus and quasi-consensus in multi-agent using artificial delays: An LMI approach. Automatica, dynamical systems. IEEE Transactions on Circuits and 81, 429–437. Systems I: Regular Papers, 60(10), 2679–2687. Fridman, E. and Shaikhet, L. (2019). Simple LMIs for Zhang, J. and Fridman, E. (2019). Derivative-dependent stability of stochastic systems with delay term given by control of stochastic systems via delayed feedback imple- Stieltjes integral or with stabilizing delay. Systems & mentation. IEEE Conference on Decision and Control, Control Letters, 124, 83–91. 66–71. Gao, Y. and Wang, L. (2011). Sampled-data based consensus of continuous-time multi-agent systems with time- varying topology. IEEE Transactions on Automatic Control, 56(5), 1226–1231. Gu, K., Chen, J., and Kharitonov, V.L. (2003). Stability of time delay systems. Birkhäuser, Boston. Higham, D.J. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM review, 43(3), 525–546.

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