https://doi.org/10.1080/00207179.2019.1710257
Distributed output feedback control of decomposable LPV systems with delay and
switching topology: application to consensus problem in multi-agent systems
Muhammad Zakwan and Saeed Ahmed
Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey
ABSTRACT
This paper presents distributed output feedback control of a class of distributed linear parameter varying systems with switching topology and parameter varying time delay. To formulate the synthesis condi-tions for the distributed controller in terms of LMIs, the delay dependent bounded-real lemma based on parameter-dependent Lyapunov–Krasovskii functionals is used. The efficacy of the result is illustrated by applying it to two real-world examples pertaining to the consensus problem of multi-agent systems.
ARTICLE HISTORY Received 16 May 2019 Accepted 25 December 2019 KEYWORDS
Output feedback; distributed systems; linear parameter varying; delay; switching; consensus
1. Introduction
Distributed systems have become a field of attraction for many researchers in the recent years. On the other hand, due to the difficulty involved in the implementation of cen-tralised controllers for large-scale systems, the choice of dis-tributed controller architectures is generally preferred; see Wang and Davison (1973). Decomposable systems are a subclass of distributed systems with identical subsystems interacting in a well-defined pattern modelled by a matrix called the pat-tern matrix; see Massioni and Verhaegen (2009), Ghadami and Shafai (2013) and Ghadami (2012) for motivation of decomposable systems. Decomposable systems find their appli-cations in many engineering disciplines, e.g. formation prob-lem of UAVs (Betser, Vela, Pryor, & Tannenbaum, 2005), vehicle platooning (Jovanovic & Bamieh, 2005), satellite for-mations (Carpenter,2000; Mesbahi & Hadaegh, 2001), paper machines (Stewart, Gorinevsky, & Dumont,2003), models com-ing from the discretisation of partial differential equations (Brockett & Willems, 1974) and large segmented telescopes (Jiang, Voulgaris, Holloway, & Thompson,2006). Tools from algebraic graph theory such as Laplacian matrices or graph-adjacency matrices can be used to represent the interactions among the subsystems of a decomposable system; see Borrelli and Keviczky (2008) and Diestel (1996) for more details on pattern matrix representations for decomposable systems in ubiquitous control theoretic applications.
Motivated by the fact that time delay can affect dis-tributed system performance in many practical scenarios due to non-ideal signal transmission, distributed networks with communication delays have been widely studied in the lit-erature. Many contributions have been made to tackle the issue of stability and performance degradation caused by time delay during the exchange of information (Atay, 2013;
CONTACT Saeed Ahmed saeed.ahmed@bilkent.edu.tr, seyit.ahmed565@gmail.com
A preliminary version of this work was presented in the 2019 European Control Conference, Naples, Italy; see the end of Section1for a comparison between this work and the conference version.
Ghaedsharaf, Siami, Somarakis, & Motee, 2016; Olfati-Saber & Murray,2004; Papachristodoulou, Jadbabaie, & Munz,2010; Qiao & Sipahi,2016; Seuret, Dimarogonas, & Johansson,2008; Sun & Wang, 2009), and see also Ghadami (2012) for dis-tributed H∞output feedback control of decomposable LTI sys-tems with a constant delay. On the other hand, linear parameter varying (LPV) systems can be used to model various real-life phenomena. The significance of LPV systems has been highlighted in various contributions; see, e.g. Mohammadpour and Scherer (2012) and Toth (2010).
In this paper, we aim to present distributed output feedback control of decomposable LPV systems with a parameter varying time delay and switching topology. One of the key contribution of our work is that we consider a switching topology, i.e. we allow arbitrary switching among a set of pattern matrices. This problem is motivated by the fact that in many practical applica-tions, the interconnection links may change over time due to various reasons. For instance, an existing link among mobile agents communicating together may be lost due to the presence of an obstacle or a new connection can be formed between them when they come in an effective range of detection with respect to each other (Ghadami & Shafai,2010). To the best of authors’ knowledge, distributed output feedback control of decompos-able LPV systems under the influence of parameter varying delay and switching topology was an open problem, which we have solved for the first time. In order to solve this chal-lenging problem, we employ delay-dependent bounded-real lemma based on parameter-dependent Lyapunov–Krasovskii functionals.
Distributed control of decomposable LPV systems with time varying topology has been studied in Eichler, Hoffmann, and Werner (2013, 2014). However, no delay is present in Eichler et al. (2013, 2014) whereas we consider a parameter
varying delay. Second, full block S-procedure is used in Eich-ler et al. (2013,2014) for the synthesis of distributed controller, whereas we adopt a Lyapunov–Krasovskii functional approach. Distributed control of decomposable LTI systems under a con-stant delay and switching topology is also proposed in Ghadami and Shafai (2010) but there is no direct way to extend the results of Ghadami and Shafai (2010) to tackle parameter variation in dynamics and delay. We point out that this paper is an extension of our conference paper (Zakwan & Ahmed,2019). While the scope of our previous work in Zakwan and Ahmed (2019) was limited to the study of delayed LPV decomposable systems with a fixed pattern matrix, we provide a significant extension of it by allowing arbitrary switching among a set of pattern matrices and two new real-world examples that apply this switching topology, which were not included in Zakwan and Ahmed (2019).
The paper is organised as follows. Section2provides prob-lem formulation and some preliminary results. We present the main result on distributed output feedback control of decom-posable LPV systems with delay and switching topology in Section3. In Section4, we apply our main result to two real-world examples related to the consensus problem of multi-agent systems. Finally, we provide some concluding remarks and future research directions in Section5.
We use standard notation, in which the dimensions of our Euclidean spaces are arbitrary unless otherwise noted, and which will be simplified whenever no confusion would arise. | · | denotes the usual Euclidean norm and the induced matrix norm. In denotes the identity matrix of dimension n and On denotes the null matrix of dimension n. R denotes the set of real numbers, Rn×m denotes real matrices of dimension
n× m and M−T denotes the inverse of the matrix MT. ⊗ denotes the Kronecker product. Given any constantτ > 0, we letC([−τ, 0],Rn) denote the set of all continuousRn-valued functions that are defined on [−τ, 0]. We call it the set of all
ini-tial functions. Also, for any continuous function x : [−τ, ∞) →
Rnand all t≥ 0, we define x
t by xt(θ) = x(t + θ) for all θ ∈ [−τ, 0], i.e. xt ∈ φ is the translation operator.Snrepresents the set of real symmetric matrices of dimension n× n, Sn++ rep-resents the set of real symmetric positive definite matrices of dimension n× n andSn+ represents the set of real symmetric semi-positive definite matrices of dimension n× n.
2. Problem formulation and preliminaries
We first define the notion of switching topology and then provide the general framework of decomposable LPV systems under delay and switching topology. Some preliminary results that will be substantial in proving our main result are also provided in this section.
We consider N identical subsystems each of order n. While the scope of our previous work in Zakwan and Ahmed (2019) was limited to the study of decomposable LPV systems with delay under fixed pattern matrix; here we allow the case where the pattern matrix can switch among a set of pattern matri-cesL = {Pσ :σ = 1, 2, . . . , m} under an arbitrary switching signal σ(t) ∈ = {1, 2, . . . , m}. We use the superscript σ to specify time varying interconnection.
We describe decomposable LPV systems under delay and switching topology as follows. Consider an Nnth-order LPV
system with parameter dependent time varying delay ˙x(t) = Aσ1(ρ)x(t) + Aσ2(ρ)x(t − τ(ρ)) + Bσ 1(ρ)w(t) + Bσ2(ρ)u(t) z(t) = Cσ11(ρ)x(t) + Cσ12(ρ)x(t − τ(ρ)) + Dσ11(ρ)w(t) + Dσ12(ρ)u(t) y(t) = Cσ21(ρ)x(t) + Cσ22(ρ)x(t − τ(ρ)) + Dσ21(ρ)w(t) (1) where x∈RNn, w∈RNnw, u∈RNnu, z∈RNnz and y∈RNny
are the state, exogenous input, control input, regulated output and measured output, respectively. We assume that the exoge-nous input w is of finite energy in the space L2[0∞) and an
initial condition inC([−τ, 0],RNn). The time-varying param-eterρ belongs to a setFPv defined asFPv := {ρ ∈C(R+,Rs) :
ρ(t) ∈P,| ˙ρi(t)| ≤ vi} where s is the size of parameter varying vector andP is a compact subset ofRs. We assume that the parameter varying delayτ is differentiable and it belongs to a set T defined asT = {τ ∈C(Rs,R) : 0 ≤ τ(ρ) < ¯τ < ∞, ∀t ∈
R+}. We define the worst case performance cost for system (1)
from w to z with (u≡ 0) as J= sup ρ∈Fv P sup |w|=0 |z| |w|. (2)
Definition 2.1: A matrix-valued function Mσ(ρ) :FPv →
RNp×Nqis called decomposable if there exist matrix-valued
func-tions Ma(ρ), Mb(ρ) :FPv →Rp×qsuch that
Mσ(ρ) = IN⊗ Ma(ρ) +Pσ⊗ Mb(ρ), ∀ρ ∈FPv (3) and Pσ ∈RN×N is diagonalisable for every σ(t) ∈ = {1, 2, . . . , m}, i.e.Pσ = Uσ σ(Uσ)−1where σ is a diagonal matrix. The system (1) is called decomposable if all of its sys-tem matrices are decomposable, i.e. they can be written as (3). The subscript a represents the decentralized part and subscript
b represents the interconnected part. Moreover, ifPσ ’s can be simultaneously diagonalised with a common U= Uσ, we call the system (1) simultaneously decomposable system.
Assumption 2.1: All of the system matrices of (1) are
decomposable.
To prove our main result in the following section, we provide two preliminary results; see Massioni and Verhaegen (2009) and Ghadami and Shafai (2013) for LTI corollaries of these results.
Lemma 2.1: Consider a decomposable matrix-valued func-tion Mσ(ρ) :FPv →RNp×Nq of the form (3) as given in
Definition 2.1, then the matrix-valued functions ¯Mσ(ρ) = (Uσ⊗ Ip)−1Mσ(ρ)(Uσ ⊗ Iq) are block diagonal and have the
following structure:
¯Mσ(ρ) = I
N⊗ Ma(ρ) + σ ⊗ Mb(ρ), (4)
where each of the block has the form ¯Miσ(ρ) = Ma(ρ) + λσ
iMb(ρ) where λσi is the ith eigenvalue of the matrix Pσ.
structure of the form given in (4), we have Mσ(ρ) = (Uσ⊗ Ip) ¯Mσ(ρ)(Uσ ⊗ Iq)−1= IN⊗ Ma(ρ) +Pσ⊗ Mb(ρ). Proof: From Definition 2.1, we can write
¯Mσ(ρ) =Uσ ⊗ I p −1 ×IN⊗ Ma(ρ) +Pσ ⊗ Mb(ρ) Uσ ⊗ Iq (5) then from the properties of the Kronecker product (Brewer,
1978), we have ¯Mσ(ρ) = ((Uσ)−1I NUσ ⊗ IpMa(ρ)Iq) + ((Uσ)−1PσUσ ⊗ IpMb(ρ)Iq) ⇔ ¯Mσ(ρ) = I N⊗ Ma(ρ) + σ ⊗ Mb(ρ). (6) Since INand σare diagonal, then we have that ¯Mσ(ρ) is block diagonal. The converse can be proved analogously.
Theorem 2.1: The Nnth-order LPV system (1) satisfying Assumption 2.1 for eachPσ is equivalent to N independent sub-systems each of order n where the ith modal subsystem is given by
˙ˆxσ
i(t) = ¯Aσ1,i(ρ)ˆxσi(t) + ¯Aσ2,i(ρ)ˆxσi(t − τ(ρ)) + ¯Bσ1,i(ρ) ˆwσi(t) + ¯Bσ2,i(ρ)ˆuσi (t) ˆzσ
i (t) = ¯Cσ11,i(ρ)ˆxσi(t) + ¯Cσ12,i(ρ)ˆxσi(t − τ(ρ)) + ¯Dσ11,i(ρ) ˆwσi(t) + ¯Dσ12,i(ρ)ˆuσi (t) ˆyσi(t) = ¯Cσ21,i(ρ)ˆxσi(t) + ¯Cσ22,i(ρ)ˆxσi(t − τ(ρ))
+ ¯Dσ21,i(ρ) ˆwσi(t) for i = 1, . . . , N and σ = 1, . . . , m, (7)
where ˆxσi ∈Rn, ˆwσi ∈Rnw, ˆuσ
i ∈Rnu, ˆzσi ∈Rnz andˆyσi ∈Rny
are the state, exogenous input, control input, regulated output and measured output of the ith modal subsystem, respectively. Moreover, the matrices ¯Aσ1,i, ¯Aσ2,i,. . . , ¯Dσ21,iare defined as
¯Aσ 1,i(ρ) = Aa1(ρ) + λσiAb1(ρ) ¯Aσ 2,i(ρ) = Aa2(ρ) + λσiAb2(ρ) .. . ¯Dσ 21,i(ρ) = Da21(ρ) + λσiDb21(ρ) for i= 1, 2, . . . , N and σ = 1, 2, . . . , m,
whereλσi is the ith eigenvalue of the matrixPσ.
Proof: Using Lemma 2.1, we can rewrite (1) as
(Uσ ⊗ I
n)−1˙x(t)
= ¯Aσ1(ρ)(Uσ⊗ In)−1x(t) + ¯Aσ2(ρ)(Uσ ⊗ In)−1x(t − τ(ρ)) + ¯Bσ1(ρ)(Uσ⊗ Inw)−1w(t) + ¯Bσ2(ρ)(Uσ ⊗ Inu)−1u(t) (Uσ ⊗ Inz)−1z(t) = ¯Cσ 11(ρ)(Uσ⊗ In)−1x(t) + ¯Cσ12(ρ)(Uσ ⊗ In)−1x(t − τ(ρ)) + ¯Dσ11(ρ)(Uσ ⊗ Inw)−1w(t) + ¯Dσ12(ρ)(Uσ ⊗ Inu)−1u(t) (Uσ⊗ Iny)−1y(t) = ¯Cσ 21(ρ)(Uσ ⊗ In)−1x(t) + ¯Cσ22(ρ)(Uσ ⊗ In)−1x(t) + ¯Dσ21(ρ)(Uσ ⊗ Inw)−1w(t). (8)
Then, with the following change of variables
x(t) = (Uσ ⊗ I n)ˆxσ(t) x(t − τ(ρ)) = (Uσ ⊗ In)ˆxσ((t − τ(ρ))) w(t) = (Uσ ⊗ I nw) ˆwσ(t) u(t) = (Uσ ⊗ Inu)ˆuσ(t) z(t) = (Uσ ⊗ Inz)ˆzσ(t) y(t) = (Uσ ⊗ Iny)ˆyσ(t) (9)
it follows from (8) that forσ = 1, . . . , m, ˙ˆxσ(t) = ¯Aσ 1(ρ)ˆxσ(t) + ¯Aσ2(ρ)ˆxσ(t − τ(ρ)) + ¯Bσ1(ρ) ˆwσ(t) + ¯Bσ2(ρ)ˆuσ(t) ˆzσ(t) = ¯Cσ11(ρ)ˆxσ(t) + ¯Cσ12(ρ)ˆxσ(t − τ(ρ)) + ¯Dσ11(ρ) ˆwσ(t) + ¯Dσ12(ρ)ˆuσ(t) ˆyσ(t) = ¯Cσ21(ρ)ˆxσ(t) + ¯C22σ (ρ)ˆxσ(t − τ(ρ)) + ¯Dσ21(ρ) ˆwσ(t), (10) where ¯Aσ1, ¯Aσ2,. . . , ¯Dσ21are block diagonal matrices. Therefore, the system (10) is equivalent to N independent nth-order modal subsystems given in (7). This concludes the proof.
Remark 2.1: The idea of all the interconnected systems having the same delay has been widely used in the literature, see exam-ples in Zhou and Lin (2014) for the motivation and justification of this assumption.
Assumption 2.2: The pattern matrixPσ belongs to a finite set of symmetric matricesL = {P1,P2,. . . ,Pm} that can com-mute with each other.
Under Assumption 2.2, we state an auxiliary result in the fol-lowing theorem that will be helpful in proving our main result in the next section.
Theorem 2.2 (Horn Johnson, 2012, Theorem 1.3.12): Con-sider a set of commuting symmetric matrices L = {Pσ :σ = 1, 2,. . . , m} in whichPiPj=PjPifor∀i, j = 1, 2, . . . , m. Then
there is a unitary matrix U that simultaneously diagonalises all the matrices in the set.
Finally, we recall sufficient condition for asymptotic stability of a switched LPV system.
Theorem 2.3 (He, Dimirovski, & Zhao, 2010): Consider a switched LPV system
˙α =Aσ(ρ)α, σ(t) ∈ = {1, 2, . . . , m}, ρ ∈Fv P (11)
such that the matricesAσ(ρ) are Hurwitz and the set {Aσ(ρ) : σ ∈ } is compact inRn×n. If all the subsystems of this switched
LPV system share a parameter-dependent quadratic common Lyapunov function, then the switched LPV is uniformly asymp-totically stable.
In what follows, we drop the dependence onρ in the notation for the sake of brevity.
3. Main results
In this section, we design a distributed output feedback con-troller for decomposable LPV systems with a parameter varying delay and switching topology guaranteeing asymptotic stabil-ity as well as minimising the worst case performance cost given in (2). Our goal is to design a controller that has the same inter-connection structure as the plant (following the idea of Massioni & Verhaegen, 2009). Although in some works (e.g. Brockett & Willems,1974or Hovd, Braatz, & Skogestad,1997), decom-position is used to simplify the computations, but they all yield a full centralised controller as a final result; the decomposi-tion approach used in this paper makes sure that the controller has the same interconnection structure as the plant, allowing a distributed implementation of the controller. We propose the controller in the following decomposable form.
˙xk(t) =M1xk(t) +Mσ2xk(t − τ) +By(t)
u(t) =C11xk(t) +C12σxk(t − τ) +Dy(t),
(12) where xkis valued inRNn, and
M1 = IN⊗Ma Mσ2 =Pσ ⊗Mb B= IN⊗Ba C11= IN⊗Ca Cσ 12=Pσ ⊗Cb D= IN⊗Da forσ = 1, 2, . . . , m, (13)
where the controller matricesMa,Mb,Ba,Ca,CbandDaare to be determined. Since the controller has decomposable structure, we can apply similar arguments as in the proof of Theorem 2.1 to decompose the controller in (12) into N controllers. The controller structure corresponding to the ith subsystem in (7) is
˙ˆxσ
k,i(t) = ¯M1,iˆxk,iσ(t) + ¯M2,iσ ˆxσk,i(t − τ) + ¯Biˆyσ(t) ˆuσ
i(t) = ¯C11,iˆxσk,i(t) + ¯C12,iσ ˆxσk,i(t − τ) + ¯Diˆyσ(t)
(14) for i= 1, 2, . . . , N and σ = 1, 2, . . . , m. Note that the con-trollers have the same interconnection structures as the plant at each instant. Therefore, whenever a change in network topology happens between the subsystems of the plants, the controllers instantly follow this change.
Assumption 3.1: Bb2= 0, Cb21= 0, C22b = 0, Db12= 0 and Db21= 0.
Remark 3.1: The conditions Bb2 = 0, C21b = 0 and Cb22= 0 in Assumption 3.1 avoid multiplication ofλis for everyσ. Under
conditions Bb2 = 0, Db12= 0 and Db21= 0, the set of LMIs con-taining eigenvalues ofPσ (i.e.λσi for i= 1, 2, . . . , N and σ = 1, 2,. . . , m) can be expressed as a convex combination of just two LMIs which contain the extreme (maximum and mini-mum) values of eigenvalues of Pσ (i.e. λ = min∀i,σλσi, ¯λ = max∀i,σλσi ). Since LMIs are a convex optimisation problem, the feasibility of just these two LMIs will automatically grant the feasibility of all other LMIs. Assumption 3.1 is a widely used assumption in the literature for full-order output feedback control of decomposable systems (Ghadami & Shafai, 2013; Massioni & Verhaegen,2009).
Theorem 3.1: Let the system (1) satisfy Assumptions 2.1, 2.2
and 3.1. A sufficient condition for the existence of a distributed
dynamic output feedback controller of the form (12) guarantee-ing uniform asymptotic stability of the closed-loop system as well as yielding satisfactory performance level J < γ is that the pair of coupled LMIs (15) has a feasible solution for some continu-ously differentiable matrix-valued functions X, Y :Rs→Sn++,
and P˜ :Rs→S2n++, constant matrices Q˜∈S2n++, and ˜R∈
S2n
++, parameter-dependent matrices ˆAa, ˆAb, ˆB, ˆCa, ˆCbandR,
and a scalarγ > 0. (ρ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −2 ˜V P˜− ˜V +A1, − ˜V +A2, ∗ ˜ 22+A1,+AT R˜+A1,T+A2, ∗ ∗ ˜22+A2,+A2,T ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ B 0 V˜+ ¯τ ˜R B C11,T V˜− ˜P B C12,T V˜ −γ Inw DT 0 ∗ −γ Inz 0 ∗ ∗ (−1 − 2 ¯τ) ˜R ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0 (15)
for = 1, 2 and for all ρ ∈FPv , where ˜ V = Y In In X A1,= ¯A1,Y+ B a 2ˆCa ¯A1,+ Ba2R¯C21, ˆAa X ¯A 1,+ ˆB ¯C21, A2,= ¯A2,Y+ B a 2ˆCb ¯A2,+ ¯Ba2R¯C22, λiˆAb X ¯A2,+ ˆB ¯C22, B=¯B1,+ B a 2R¯D21, X ¯B1,+ ˆB ¯D21, C11,= [ ¯C11,Y+ Da12ˆCa ¯C11,+ Da12R¯C21,] C12,= [ ¯C12,Y+ λiDa12ˆCb ¯C12,+ Da12R¯C22,] D= [ ¯D11,+ Da12R¯D21,] ˜ 22= ˜Q− ˜R+ ⎡ ⎣ s j=1 ± vj∂ ˜P(ρ) ∂ρj ⎤ ⎦
˜22= − ⎡ ⎣1 − s j=1 ± vj∂τ ∂ρj ⎤ ⎦ ˜Q− ˜R. (16)
In the LMIs (15), the matrices with subscript like G have the structure of ¯G1= Ga+ λGband ¯G2= Ga+ ¯λGbwithλ and ¯λ
the minimum and maximum eigenvalues ofPσ(λ = min∀i,σλσi, ¯λ = max∀i,σλσi for i= 1, 2, . . . , N and σ = 1, 2, . . . , m).
More-over, if LMIs have feasible solutions, the matrices appearing in (13) are given by Ma= −N−1(XAa 1Y+ XBa2RCa21Y + NBaCa 21Y+ XBa2CaMT− ˆAa)M−T Mb= −N−1(XAb 2Y+ XBa2RCb21Y + NBaCb 21Y+ XBa2CbMT− ˆAb)M−T Ba= N−1(ˆB − XBa 2R) Ca= ( ˆCa−RCa 21Y)M−T Cb= ( ˆCb−RCb 21Y)M−T Da=R, (17)
where M and N are obtained by factorisation problem In− XY =
NMT.
Proof: The closed-loop expression of the subsystem (7) with the corresponding controller (14) for extreme values of eigen-values ofPσ (λ = min∀i,σλσi, ¯λ = max∀i,σλσi ) is given by
˙ˆxcl
(t) = Acl1,ˆxcl(t) + Acl2,ˆxcl(t − τ) + Bcl1,ˆw(t) ˆz(t) = Ccl11,ˆxcl(t) + C12,cl ˆxcl(t − τ) + Dcl ˆw(t)
for = 1, 2, (18) where ˆxcl = [ˆxT,ˆxTk,]T is the augmented state vector, and the closed-loop matrices for = 1, 2 like G parameterised as G1= Ga+ λGband G2 = Ga+ ¯λGb(λ = min∀i,σλσi and ¯λ = max∀i,σλσi for i= 1, 2, . . . , N and σ = 1, 2, . . . , m) are given by Acl1, = ¯A1,+ ¯B¯ 2,D¯¯C12, ¯B2,C¯11, B¯C12, M¯1, (19) Acl2, = ¯A2,+ ¯B¯ 2,D¯i¯C22, ¯B2,C¯12, B¯C22, M¯2, (20) Bcl1, =¯B1,+ ¯B¯ 2,D¯¯D21, B¯D21, (21) Ccl11, = [ ¯C11,+ ¯D12,D¯¯C12, ¯D12,C¯1,] (22) Ccl12 = [ ¯C11,+ ¯D12,D¯¯C22 ¯D12,C¯2,] (23) Dcl = [ ¯D11,+ ¯D12,D¯¯D21,]. (24)
Consider the Lyapunov–Krasovskii functional
V(ˆxcl(t + θ), ρ) = V1(ˆxcl(t), ρ) + V2(ˆxcl(t + θ), ρ) + V3(ˆxcl(t + θ), ρ) V1(ˆxcl(t), ρ) = ˆxcl T (t)P(ρ)ˆxcl (t) V2(ˆxcl(t + θ), ρ) = t t−τˆx cl T(ξ)Qˆxcl(ξ) dξ V3(ˆxcl(t + θ), ρ) = 0 − ¯τ t t+θ ˙ˆx cl T (η) ¯τR˙ˆxcl (η) dη dθ (25) for allθ ∈ [− ¯τ, 0]. The time derivative of V(ˆxcl(t + θ), ρ) along the trajectories of (18) yields
˙V1(ˆxcl(t), ρ) = ˙ˆxcl T (t)P(ρ)ˆxcl (t) + ˆxcl T (t)P(ρ)˙ˆxcl (t) + ˆxcl T (t)∂P(ρ) ∂ρ ˙ρˆxcl(t) ˙V2(ˆxcl(t + θ), ρ) = ˆxcl T (t)Qˆxcl (t) − 1−∂τ ∂ρ ˙ρ ˆxcl T (t − τ)Qˆxcl (t − τ) ˙V3(ˆxcl(t + θ), ρ) = ¯τ2˙ˆxcl T (t)R˙ˆxcl (t) − t t− ¯τ˙ˆx cl T (θ) ¯τR˙ˆxcl (θ) dθ. (26) Sinceτ ≤ ¯τ, we have − t t− ¯τ ˙ˆx cl T(θ) ¯τR˙ˆxcl(θ) ≤ − t t−τ ˙ˆx cl T(θ) ¯τR˙ˆxcl(θ) dθ. Using the Jensen’s inequality, we can bound the integral term of
˙V3(ˆxcl(t + θ), ρ) in (26) as follows: ˙V3(ˆxcl(t + θ), ρ) ≤ ¯τ2˙ˆxcl T(t)R˙ˆxcl(t) − t t−τ˙ˆx cl T(θ) ¯τR˙ˆxcl(θ) dθ ≤ ¯τ2˙ˆxcl T(t)R˙ˆxcl(t) − ¯τ τ t t−τ˙ˆx cl (θ) dθ T R t t−τ˙ˆx cl (θ) dθ = ¯τ2˙ˆxcl T (t)R˙ˆxcl (t) − ¯τ τ[ˆxcl(t) − ˆxcl(t − τ)]TR[ˆxcl(t) − ˆxcl(t − τ)]. (27)
Boundingτ¯τ by 1, it follows from (27) that ˙V3(ˆxcl(t + θ), ρ) = ¯τ2˙ˆxcl T (t)R˙ˆxcl (t) − [ˆxcl(t) − ˆxcl(t − τ)]T ×R[ˆxcl (t) − ˆxcl(t − τ)]. (28) Collecting all the derivative terms ˙V1(ˆxcl(t), ρ), ˙V2(ˆxcl(t +
θ), ρ) and ˙V3(ˆxcl(t + θ), ρ), and letting ˙V(ˆxcl(t + θ), ρ) < 0,
we get the following LMIs: ˙V(ˆxcl (t + θ), ρ) ≤ ζT(t)(ρ, ˙ρ)ζ(t) < 0, (29) where (ρ, ˙ρ) = ⎡ ⎣11 P(ρ)A cl 2,+R P(ρ)Bcl1, ∗ 22 0 ∗ ∗ 0 ⎤ ⎦ + ¯τ2TT(ρ)RT(ρ)
ζ(t) = col[ˆxcl (t), ˆxcl(t − τ), ˆw(t)] T(ρ) = [Acl 1,Acl2,Bcl1,] 11= Acl1, T P(ρ) + P(ρ)Acl1,+∂P(ρ) ∂ρ ˙ρ +Q−R 22= − 1−∂τ ∂ρ ˙ρ Q−R. (30)
To guarantee the performance level J< γ , we further require ˙V(ˆxcl
(t + θ), ρ) − γ2ˆwT(t) ˆw(t) + ˆzT(t)ˆz(t) ≤ 0. (31) Substituting ˆz(t) from (18) into the inequality (31), and then applying the Schur complement yields the following LMIs:
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ M(ρ, v) P(ρ)Acl2,+R P(ρ)Bcl1, ∗ − ⎡ ⎣1 − s j=1 ± vj∂τ ∂ρj ⎤ ⎦Q−R 0 ∗ ∗ −γ Inw ∗ ∗ ∗ ∗ ∗ ∗ Ccl11,T ¯τAcl1,TR Ccl12,T ¯τAcl2,TR Dcl11,T ¯τBcl1,TR −γ Inz 0 ∗ −R ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦< 0, (32) where M(ρ, v) = Acl 1, T P(ρ) + P(ρ)Acl 1, + ⎡ ⎣ s j=1 ± vj∂P(ρ) ∂ρj ⎤ ⎦ +Q−R. Let us define a symmetric positive definite matrixV ∈S2n++ and its inverse as
V = X N NT ∗ , V−1= Y M MT ∗ (33) such that XY+ NMT= Infor X, Y :Rs→Sn++.
Next we will show that following LMIs are equivalent to LMIs in (32) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −2V P(ρ) −V +VA1,cl −V +VAcl2, ∗ 22+VAcl1,+ Acl1, T V R+ Acl1, T V +VAcl2, ∗ ∗ 22+ Acl2, T V +VAcl2, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ VBcl1, 0 V + ¯τR VBcl1, Ccl11,T V − P(ρ) VBcl1, Ccl12,T V −γ Inw Dcl T 0 ∗ −γ Inz 0 ∗ ∗ (−1 − 2 ¯τ)R ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, (34)
where 22=∂P(ρ)∂ρ ˙ρ +Q−R. We can write (34) as
+CTTD+DTC< 0, (35) where = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 P(ρ) 0 0 0 ¯τR ∗ 22 R 0 Ccl11, T −P(ρ) ∗ ∗ 22 0 Ccl12, T 0 ∗ ∗ ∗ −γ Inw Dcl T 0 ∗ ∗ ∗ ∗ −γ Inz 0 ∗ ∗ ∗ ∗ ∗ (−1 − 2 ¯τ)R ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ C= [−I Acl 1, Acl2, Bcl1, 0 I] D= ⎡ ⎣0I 0I 00 00 00 00 0 0 I 0 0 0 ⎤ ⎦ T = [V V V], (36) and the explicit bases of the null spaces ofCandDare given as
Ker(C) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Acl1, Acl2, Bcl1, 0 I I 0 0 0 0 0 I 0 0 0 0 0 I 0 0 0 0 0 I 0 0 0 0 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Ker(D) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 0 0 0 0 0 I 0 0 0 I 0 0 0 I ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . (37)
Employing the projection lemma with respect to in (35) yields (32) which proves that the LMIs in (34) are equivalent to the LMIs in (32). Defining Z := Y In MT 0 (38) and then performing congruence transformationκ = diag(ZT,
ZT, ZT, In, In, ZT) on (34) yields the following LMIs: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −2ZTVZ P˜− ZTVZ +ZTVAcl 1,Z −ZTVZ +ZTVAcl 2,Z ∗ ˜ 22+ Z T(Acl 1,V +VAcl1,T)Z ˜ R+ ZTAcl 1, T Z +ZTVAcl 2,Z ∗ ∗ ˜22+ ZT(Acl2, T V +VAcl2,)Z ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
ZTVBcl 1, 0 ZTVZ+ ¯τ ˜R ZTVBcl1, ZTCcl11,T ZTVZ− ˜P ZTVBcl1, ZTCcl12,T ZTVZ −γ Inw Dcl T 0 ∗ −γ Inz 0 ∗ ∗ (−1 − 2 ¯τ) ˜R ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, (39) where P˜ = ZTP(ρ)Z, ˜R= ZTRZ, ˜ 22= ZT 22Z, ˜22= ZT22Z. Note that ZTVZ= Y I I X =: ˜V, ZTV = I 0 X N . (40)
Using (40), we obtain the following identities:
ZTVAcl1,Z=A1,, ZTVBcl1,=B ZTVAcl2,Z=A2,, ZTCcl11, T =CT 11, ZTCcl12,T=C12,T, Dcl =D, (41)
whereA1,cl,A2,,B,C11,T,C12,T,Dare defined in (16).
Applying the following change of variables ˆD=R ˆC11,=RCa21Y+CaMT ˆC12,=RCb21Y+CbMT ˆB= XBa2R+ NBa ˆA1,= XAa1Y+ XBa2RCa21Y+ NBaCa21Y + XBa 2CaMT+ NMaMT ˆA2,= XAb2Y+ XBa2RCb21Y+ NBaCb21Y + XBa 2CbMT+ NMbMT (42)
linearises the bilinear matrix inequalities in (39) with respect to the new variables ( ˆAa, ˆAb, ˆB, ˆCa, ˆCbandR) and yields the LMIs given in (15).
Using the constraints ˆA1,= ˆAa, ˆA2, = λˆAb, ˆB= ˆB, ˆC11,= ˆCa, ˆC12, = λˆCb, and ˆD =R from the controller
structure, we can parametrise the controller parameters ¯M1,,
¯
Mσ
2,, ¯B, ¯C11,, ¯C12,σ , ¯Das ¯M1,=Ma, ¯Mσ2,= λMb, ¯B =
Ba, ¯C
11,=Ca, ¯C12,σ = λCband ¯D=Dawhich yield the dis-tributed dynamic output-feedback controller matrices (M1,
Mσ
2,B,C11,C12σ,D). This completes the proof. 4. Illustration
We illustrate our main result by applying it to the consensus problem of multiagent systems composed of wheeled mobile robots and VTOL helicopters subject to parameter varying delay and switching topology.
4.1 Consensus problem of nonholonomic multi-agent systems composed of wheeled mobile robots
Wheeled mobile robots are usually subject to nonholonomic constraints. Difficulty arises from its underactuated nature,
which is a direct consequence of its non-holonomicsee (Gon-zalez & Werner,2014). We consider a swarm of five uniformly distributed nonholonomic agents orbiting around an obsta-cle circularly. We consider switching communication network topology among these agents as opposed to our previous work in Zakwan and Ahmed (2019) where the communication pattern between the agents was assumed to be fixed. The communica-tion pattern can change from the case where an agent can only communicate with its immediate neighbours to the case where all the agents can communicate with each other. Ifσth topology is that an agent can communicate with itsσ neighbouring agents on each of its side, then there are [(N − 1)/2] communication patterns where [·] indicates floor function. Our aim is to design a distributed LPV controller that stabilises the system for any switching among these patterns and parameter varying commu-nication delay while minimising the following agent’s relative positions zxk,i= 1 2σ σ l=1
(xk,i− xk,i+l) + (xk,i− xk,i−l) for
k= 1, 2 and i = 1, 2, . . . , 4 (43)
forσth topology under the performance criterion given in (2). Since there is no dynamic interaction among the agents, we can model the multi-agent nonholonomic system as the delayed LPV system ˙x(t) = (IN⊗ A1(ρ))x(t) + (Pσ ⊗ A2(ρ))x(t − τ(ρ)) + (IN⊗ B1(ρ))w(t) + (IN⊗ B2(ρ))u(t) z(t) = (IN⊗ C11(ρ))x(t) + (Pσ ⊗ C12(ρ))x(t − τ(ρ)) + (IN⊗ D11(ρ))w(t) + (IN⊗ D12(ρ))u(t) y(t) = (IN⊗ C21(ρ))x(t) + (Pσ ⊗ C22(ρ))x(t − τ(ρ)) + (IN⊗ D21(ρ))w(t), (44)
where x, w, u, z and y all valued inRNnwith N = 5 and n = 2,
ρ = [ρ1,ρ2], and the matrices in (44) are given by
A1(ρ) = 0 ρ1 −ρ1 0 , A2(ρ) = O2, B1= B2= 1 0 0 0.1 , C11= C12= C21= I2, C22= O2, D11= 0, D12= 0.1I2, D21= 0.
We choose ρ1 = ω sin(t) and ρ2 = | cos(ωt)| with v1= v2=
5,τ(ρ) = μρ2 withμ = 0.09, and ω = 5 rad/ s. The
parame-ter space is [ρ1,min ρ1,max]× [ρ2,min ρ2,max] whereρ1,min=
−5, ρ1,max= 5, ρ2,min= 0 and ρ2,max= 1. The
parameter-dependent delayτ varies from 0 to ¯τ = 0.09, and the condition dτ/dt < 1 holds except for a countable number of points. More-over,|dρ1/dt| ≤ 5 and |dρ2/dt| ≤ 5, hence the controller does
not depend on future value of the parameters.
From (43),Pσ’s are symmetric Toeplitz matrices with first row as 1 σ times −1 2σ . . . −1 2σ 0. . . 0 σ times −1 2σ . . . −1 2σ] (45)
and they commute with each other. Hence, Assumptions 2.1, 2.2 and 3.1 are satisfied, and we can apply Theorem 3.1.
Figure 1.Eigenvalues ofLforN = 30.
Figure 2.Topologies with five agents: (a) graph corresponding toP1 and (b) graph corresponding toP2.
We provide a pictorial representation of eigenvalues of such Toeplitz pattern matrices for N= 30 in Figure1. It is evident from Figure 1that maximum and minimum values of eigen-values ofPσ (i.e.λ = min∀i,σλσi , ¯λ = max∀i,σλσi ) is indepen-dent of number of agents in the system. Hence, the controller designed forλ = 0 and ¯λ = 2 will stabilise the multi-agent sys-tem with an arbitrary number of agents in the presence of external disturbances and time-varying delay subject to an arbi-trary switching signal among the set of pattern matricesL. For this example, we choose pattern matrices asP1andP2as shown
in Figure2, whereP1 = U 1UTandP2= U 2UTare given by P1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 −0.5 0 0 −0.5 −0.5 1 −0.5 0 0 0 −0.5 1 −0.5 0 0 0 −0.5 1 −0.5 −0.5 0 0 −0.5 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦, P2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 −0.25 −0.25 −0.25 −0.25 −0.25 1 −0.25 −0.25 −0.25 −0.25 −0.25 1 −0.25 −0.25 −0.25 −0.25 −0.25 1 −0.25 −0.25 −0.25 −0.25 −0.25 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦.
4.1.1 Application of Theorem 3.1 to (eqn44)
Since the system matrix A1(ρ) has affine dependence on only
ρ1 and A2(ρ) = O2, we can design LPV controller (12) for
the system (44) in polytopic form using two vertices ofρ1, i.e.
[ρ1,min ρ1,max]. The matricesMaandMbin (13) are given by
Ma(ρ
1) = c1(ρ1)Mmaxa + c2(ρ1)Mamin
Mb(ρ
1) = c1(ρ1)Mbmax+ c2(ρ1)Mbmin,
where the matricesMa
max,Mamin,MbmaxandMbmin, obtained
by solving LMIs (15), are given by
Ma max= −3.9444 0.7274 −2.1742 −3.2826 , Ma min= −4.0245 −0.8448 2.5225 −3.2378 Mb max= −0.1748 0.0142 −0.0759 0.1355 , Mb min= −0.2324 −0.0040 0.0571 0.1506 c1(ρ1) = ρ1,max− ρ1 ρ1,max− ρ1,min , c2(ρ1) = ρ1− ρ1,min ρ1,max− ρ1,min, c1(ρ1) + c2(ρ1) = 1,
and the controller matrices Ba, Ca, Cb and Da, obtained by solving LMIs (15), are given by
Ba= −0.0043 −0.0001 0.0008 −0.0074 , Ca= −0.8658 0.0178 4.6066 4.4232 Cb= 0.6888 0.0034 −0.0556 −1.9649 , Da= −3.7088 −0.0046 −0.0046 −25.1133 .
The variation of controller matricesMa(ρ1) andMb(ρ1) as
a function of parameterρ1is demonstrated in Figure3.
4.1.2 Simulation results
Figure4(a) demonstrates that all the respective state trajectories of the agents are achieving consensus in closed-loop configura-tion for an arbitrary initial condiconfigura-tion under step disturbances. Hence, the step disturbances are efficiently rejected by the pro-posed distributed output feedback controller. The switching signalσ(t) is defined as
σ(t) =
0 if ϕ ≤ 0.5,
1 else, (46)
where ϕ is a uniform random variable in U(0, 1) and U is the univariate uniform distribution. The switching signalσ(t) is shown in Figure 4(b) whereas the control energy is shown in Figure 4(c). The convergence of normγ against the itera-tion number is provided in Figure4(d). The simulations cor-roborate overall system stability under parameter-dependent time-varying delay and switching topology.
Figure 3.Variation of controller matricesMa(ρ1) andMb(ρ1) as function of parameterρ1.
4.2 Consensus problem of multi-agent systems composed of VTOL helicopters
A multi-agent system composed of VTOL helicopters subject to parameter varying time delay and switching topology can be described as ˙x(t) = (IN⊗ A1(ρ))x(t) + (Pσ ⊗ A2(ρ))x(t − τ(ρ)) + (IN⊗ B1(ρ))w(t) + (IN⊗ B2(ρ))u(t) z(t) = (IN⊗ C11(ρ))x(t) + (Pσ ⊗ C12(ρ))x(t − τ(ρ)) + (IN⊗ D11(ρ))w(t) + (IN⊗ D12(ρ))u(t) y(t) = (IN⊗ C21(ρ))x(t) + (Pσ ⊗ C22(ρ))x(t − τ(ρ)) + (IN⊗ D21(ρ))w(t), (47)
where the state variables x(t)T= [x
1 x2x3 x4]Tare horizontal
velocity, vertical velocity, pitch rate and pitch angle, respectively. The matrices in (47) are given as
A1(ρ) = ⎡ ⎢ ⎢ ⎣ −0.0366 0.0271 0.0188 −0.4555 0.0482 −1.01 0.0024 −4.0208 0.1002 a32(ρ) −0.7071 a34(ρ) 0 0 1 0 ⎤ ⎥ ⎥ ⎦ , A2(ρ) =O4, B1(ρ) = [I4 04×1], B2(ρ) = ⎡ ⎢ ⎢ ⎣ 0.4422 0.1761 b21(ρ) −7.5922 −5.52 4.49 0 0 ⎤ ⎥ ⎥ ⎦ , C11= 0.1I4 02×4 , C12= C11, D11= 06×5, D12= 04×2 0.01I2 , C21= [0 1 0 0], C22= 01×4, D21= [0 0 0 0 1], a32(ρ) = 0.0670 + 0.4390ρ, a34(ρ) = 0.0479 + 2.3440ρ, b21(ρ) = 0.9263 + 4.0760ρ,
and the parameterρ ∈ [0 1] is defined as ρ = 0.0091(θt− 60), where θt ∈ [60 170] is the airspeed in knots. This example is inspired by Zakwan (2020).
Remark 4.1: Note that the output matrices C21and C22have
incomplete row rank.
We chooseρ = sin2(ωt) with v1= 2ω where ω = 0.01 rad/s
and τ(ρ) = 0.25 + 0.20ρ. The parameter-dependent delay τ varies from 0 to ¯τ = 0.45, and the condition dτ/dt < 1 holds. For simulation purposes, we choose N = 4 and interaction topology Pσ randomly switches between normalised com-mutable Laplacians belonging to the set L = {P1,P2,P3}
whereP1,P2,P3are given below
P1= ⎡ ⎢ ⎢ ⎣ 1 −0.5 0 −0.5 −0.5 1 −0.5 0 0 −0.5 1 −0.5 −0.5 0 −0.5 1 ⎤ ⎥ ⎥ ⎦ , P2= ⎡ ⎢ ⎢ ⎣ 1 −0.5 −0.5 0 −0.5 1 0 −0.5 −0.5 0 1.0 −0.5 0 −0.5 −0.5 1 ⎤ ⎥ ⎥ ⎦ , P3= ⎡ ⎢ ⎢ ⎣ 1 0 −0.5 −0.5 0 1 −0.5 −0.5 −0.5 −0.5 1 0 −0.5 −0.5 0 1 ⎤ ⎥ ⎥ ⎦ .
The graphs ofP1,P2,P3are depicted in Figure5. It is clear from the setL that maximum and minimum values of eigenvalues of Pσ (i.e.λ = min∀i,σλσ
i , ¯λ = max∀i,σλσi) areλ = 0 and ¯λ = 2. We aim to design an output feedback controller that stabilises the multi-agent system in the presence of external disturbances and time-varying delay subject to an arbitrary switching signal among the set of pattern matricesL.
4.2.1 Application of Theorem 3.1 to (eqn47)
Since the system matrices A1(ρ) and B2(ρ) in (47) have
linear dependence of ρ, therefore, we cannot employ the polytopic approach straightforwardly as in previous example. The pair of coupled feasibility conditions formulated in (15) are infinite-dimensional semidefinite programs and cannot be solved directly. To make them tractable, we employ determinis-tic gridding approach that will result in an approximate finite-dimensional semidefinite program which can be solved using standard LMI parsers such as YALMIP and solvers such as SeDuMi. We choose 50 gridding points to solve the inequalities in Theorem 3.1. Polynomials of order 1 have been employed to computeρ -dependent unknown matrices in Theorem 3.1. We compute the controller matrices of the form (12) and (13), where the matrices are given as
Ma= ⎡ ⎢ ⎢ ⎣ 0.10356− 0.12051ρ 1.7844− 1.6518ρ 0.77118ρ − 0.37396 10.502ρ − 2.2639 −0.41125ρ − 0.59394 −5.4494ρ − 9.1123 1.1705ρ + 0.36721 15.844ρ + 7.7879
Figure 4.Simulation results: (a) closed-loop state under a step disturbance, (b) switching signal between pattern matricesP1andP2, (c) control effort of the distributed controller and (d) convergence of the performance indexγ against the number of iterations.
Figure 5.Topologies with five agents: (a) graph corresponding toP1, (b) graph corresponding toP2, (c) graph corresponding toP3.
0.55014− 0.44949ρ 0.52978ρ − 0.82897 1.1878ρ + 1.2721 1.2429− 4.4767ρ 3.0683ρ − 3.7061 4.7685ρ + 2.2789 2.6305− 0.55895ρ −8.3141ρ − 2.8824 ⎤ ⎥ ⎥ ⎦ (48) Mb= ⎡ ⎢ ⎢ ⎣ 0.067396− 0.014851ρ 0.066363 − 0.0092064ρ 0.077171ρ − 0.25125 0.047841ρ − 0.32972 0.047081− 0.0019904ρ 0.1053− 0.0012339ρ 0.092147ρ − 0.33285 0.057125ρ − 0.51 0.006335ρ − 0.0065472 0.0055668ρ − 0.050077 0.023699− 0.032919ρ 0.26138− 0.028928ρ 0.00084904ρ + 0.034154 0.00074609ρ − 0.092001 0.038485− 0.039308ρ 0.41192− 0.034541ρ ⎤ ⎥ ⎥ ⎦ (49) Ca= −4.0464 −55.1292 −6.8387 23.0990 −0.2280 0.0813 −3.6125 2.3946 , Cb= −0.4113 −0.2550 0.1755 0.1542 −0.7599 −0.9569 0.0670 0.7577 (50) Ba= ⎡ ⎢ ⎢ ⎣ −0.034353ρ − 0.012871 0.17851ρ + 0.1449 −0.0046041ρ − 0.30301 0.21315ρ + 0.41548 ⎤ ⎥ ⎥ ⎦ , Da= −0.9515 0.5475 . (51) 4.2.2 Simulation results
Figure6(a) demonstrates that all the respective state trajecto-ries of all four VTOL helicopters are achieving consensus in closed-loop configuration for an arbitrary initial condition. The distributed controller ensures that the agents reach a consensus. The switching signalσ(t) is defined as
σ(t) = ⎧ ⎪ ⎨ ⎪ ⎩ 0 ifϕ ≤ 0.33 1 if 0.33≤ ϕ ≤ 0.67 2 else, (52)
whereϕ is a uniform random variable inU(0, 1) andU is the univariate uniform distribution. The switching signal and the
Figure 6.Simulation results: (a) closed-loop state under disturbance, (b) switching signal between pattern matricesP1,P2andP3, (c) control effort of the distributed controller, (d) convergence of the performance indexγ against the number of iterations.
control energy are provided in Figure6(b) and6(c), respectively. We also provide the convergence of normγ against the itera-tion in Figure6(d). The simulations corroborate overall system stability under parameter-dependent time-varying delay and switching topology subject to disturbance w(t) = 0.5(H(t) −
H(t − 5)) where H is the Heavyside step function. Hence, the
disturbances are efficiently rejected by the proposed distributed output feedback controller.
5. Conclusion
We provided synthesis conditions for a distributed output feed-back controller for decomposable LPV systems with switching topology and parameter-dependent time varying delay ensuring uniform asymptotic stability as well as satisfactory performance. The synthesis conditions for distributed controller were derived using Lyapunov–Krasovskii functional and the bounded real lemma. To demonstrate the efficacy of our result, we applied it to consensus problem of multi-agent systems under parameter varying delay and switching topology.
Many extensions of our result can be expected; for instance, extension to other families of distributed systems and
coordination patterns, and extension to the case where the pat-tern matrix and parameter variation are stochastic following the ideas of Briat (2018) and Zakwan (2020).
Acknowledgements
We would like to thank Prof. Hitay Özbay and Dr. Corentin Briat for their insightful comments which helped us in improving the quality of this paper.
Disclosure statement
No potential conflict of interest was reported by the authors. ORCID
Muhammad Zakwan http://orcid.org/0000-0001-6399-3036
Saeed Ahmed http://orcid.org/0000-0001-9545-9313
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