Designing Proportional-Integral Consensus Protocols for Second-Order Multi-Agent Systems Using Delayed and Memorized State Information

Honghai Wang and Qing-Long Han,,

Abstract—This paper is concerned with consensus of a secondorder linear time-invariant multi-agent system in the situation that there exists a communication delay among the agents in the network.A proportional-integral consensus protocol is designed by using delayed and memorized state information.Under the proportional-integral consensus protocol, the consensus problem of the multi-agent system is transformed into the problem of asymptotic stability of the corresponding linear time-invariant time-delay system.Note that the location of the eigenvalues of the corresponding characteristic function of the linear time-invariant time-delay system not only determines the stability of the system,but also plays a critical role in the dynamic performance of the system.In this paper, based on recent results on the distribution of roots of quasi-polynomials, several necessary conditions for Hurwitz stability for a class of quasi-polynomials are first derived.Then allowable regions of consensus protocol parameters are estimated.Some necessary and sufficient conditions for determining effective protocol parameters are provided.The designed protocol can achieve consensus and improve the dynamic performance of the second-order multi-agent system.Moreover, the effects of delays on consensus of systems of harmonic oscillators/double integrators under proportional-integral consensus protocols are investigated.Furthermore, some results on proportional-integral consensus are derived for a class of high-order linear time-invariant multi-agent systems.

I.INTRODUCTION

CONSENSUS is a fundamental issue of cooperative control of multi-agent systems, which has attracted considerable attention because of its widespread variety of applications [1]–[5].For a linear time-invariant (LTI) multi-agent system, one can design a consensus protocol via assigning all the eigenvalues of the characteristic equation of the state error system in the open left-half complex plane [6], [7].Note that such an idea is significant due to the fact that on one hand, it is possible to obtain some necessary and sufficient conditions in relation to the desired results on analysis and synthesis of LTI systems [8], [9]; on the other hand, the dynamic performance of LTI systems directly depends on the distribution of eigenvalues of the corresponding characteristic equation [10].

It is well known that in a multi-agent system, time delay inevitably occurs in the information communication from one agent to another agent.The existence of time delay poses a challenge in designing consensus protocols for an LTI multiagent system via assigning eigenvalues because it leads to infinite eigenvalues in the characteristic equation of the system [11].Several results on consensus protocol design of LTI multi-agent systems via the distribution of eigenvalues by considering delayed information are obtained.Some consensus protocols of first-order LTI multi-agent systems with current and outdated states under an undirected topology or a directed topology are proposed [12]–[14].The synchronization problem of networked harmonic oscillators by applying outdated position data under directed topology is considered[15].Synchronization protocols of coupled harmonic oscillators with current and past sampled position data under directed topology are provided [16].Consensus and quasiconsensus of second-order LTI multi-agent systems by using both current and delay position information under an undirected topology are stressed [17].The delay margin for consensus of networked double integrators with position and velocity states under an undirected topology is studied [18].The consensus problem of networked second-order integrating systems with sampled position information under a directed topology is investigated [19].The consensus analysis for second-order LTI multi-agent systems according to the crossing directions of the characteristic roots is considered[20].The consensus problem for general second-order LTI multi-agent systems with delay position and velocity states under a directed topology is studied [21].Note that most of the existing results mainly seek the delay margin for fixed parameters of consensus protocols, or first determine the parameters of consensus protocols for systems without delay,and then check the effective delay for the chosen consensus protocol parameters [22].

Recently, several criteria for the distribution of roots and Hurwitz stability criteria for quasi-polynomials of neutral type are derived [23], which can be used for analysis and synthesis of LTI time-delay systems of neural type.Some necessary conditions for the determination of the number of roots in the open right-half complex plane for a class of quasi-polynomials are obtained [24], where consensus protocols with delayed state information in the proportional-derivative form for second-order LTI multi-agent systems via assigning eigenvalues are designed.

In this paper, we deal with the problem of consensus for a second-order LTI multi-agent system via assigning eigenvalues.The consensus protocol in the proportional-integral (PI)form of the multi-agent system is designed by applying both delayed position state information and memorized position state information.The main contributions of this paper are listed as follows.

1) Some Hurwitz stability criteria and necessary conditions for Hurwitz stability are derived for a class of quasi-polynomials with complex/real coefficients.

2) Allowable regions of consensus protocol parameters are estimated.Some necessary and sufficient conditions are obtained for determining the effective values of the parameters to achieve consensus of the second-order LTI multi-agent system under a directed topology and an undirected topology,respectively, via assigning eigenvalues in some region to the left of the imaginary axis of the complex plane.The protocol can achieve consensus of the LTI multi-agent system and improve the dynamic performance of the system.

3) The effects of delays on consensus of systems of harmonic oscillators/double integrators under PI control are investigated.For the system of harmonic oscillators, it is proved that consensus cannot be achieved under a PI protocol without introducing delayed information.For the system of double integrators, it is proved that consensus cannot be achieved under a PI protocol with and without introducing delayed information.

4) An consensus issue is briefly addressed for a class of high-order LTI multi-agent systems.

The rest of this paper is organized as follows.The problem is stated in Section II.Section III presents some Hurwitz stability criteria and necessary conditions for Hurwitz stability for a class of quasi-polynomials.The results on consensus protocol design of second-order LTI multi-agent systems are provided in Section IV, where consensus protocol designs of second-order LTI multi-agent systems under a directed topology and an undirected topology are presented in Sections IVA and IV-B, respectively.The effects of delays on consensus of systems of harmonic oscillators/double integrators under PI control are investigated in Section IV-C.The results on consensus protocol design of high-order LTI multi-agent systems are illustrated in Section V.A brief conclusion is finally drawn in Section VI.

Notations:

II.PROBLEM STATEMENT

We consider the following multi-agent system withNidentical general second-order dynamics

In this paper, we consider the situation that there exists a communication delay τ >0 between agentkand agentjin the network.Thus, we make use of delayed state information to design the consensus protocol in the PI form of

Then we have

Therefore, the problem on consensus design of the multiagent system (1) is transformed into the parameter determination of the consensus protocol (2) such that the system (4) is asymptotically stable.

For the convenience of the expression, let

We have

Moreover, letg˜k(t)=gk(t)-g1(t) fork=2,...,Nand

Rewrite the system (4) as follows

It should be pointed out that the system (4) is asymptotically stable if and only if all the eigenvalues of the characteristic function (6) are located in the open left-half plane of the complex plane.Note that the location of the eigenvalues not only determines the stability of the system (4), but also plays a critical role in the dynamic performance of the system (4).In this paper, we intend to place the eigenvalues to the left of the lines=-σ with σ ∈R+.

From (6), one can see that all the eigenvalues of δ (s) can be placed to the left of the lines=-σ if and only if all the roots of the functions δk(s) are placed to the left of the lines=-σ.Thus, we focus on the distribution of roots of the functions δk(s).

Note that the existence of time delay has brought much difficulty to the consensus protocol design.Meanwhile, the value of µkmay be complex, which leads to the fact that some of the functions δk(s) are the functions with complex coefficients.Recently, some Hurwitz stability criteria for quasi-polynomials with complex coefficients are derived [23].In this paper,we provide some slightly different Hurwitz stability criteria for a class of quasi-polynomials.We further derive some necessary conditions for Hurwitz stability of the quasi-polynomials.The necessary conditions for Hurwitz stability play a fundamental role in the estimate of the allowable regions of consensus protocol parameters for the multi-agent system.The derived Hurwitz stability criteria are applied to judging whether the distribution of roots of δk(s) is available for a given set of consensus protocol parameters in the allowable regions.

Remark 1: Note that consensus protocols in the proportional-derivative form are designed in [24].In this paper, we design consensus protocols in the proportional-integral form.Furthermore, one can use the same principle mentioned in Remark 1 in [24] to deal with delay in different situations.

III.HURWITz STABILITY FOR A CLASS OF QUASI-POLYNOMIALS

In this section, we present some preliminary results on Hurwitz stability based on [23] and further derive several results on necessary conditions for Hurwitz stability for a class of quasi-polynomials.

A. Preliminary Results

Consider the quasi-polynomial described by

Several Hurwitz stability criteria are derived in [23] to judge whether all the roots of quasi-polynomials of neutral type with complex coefficients are in the open left-half complex plane.In this subsection, based on these criteria in [23], we provide some slightly different Hurwitz stability criteria for the quasipolynomialH(λ) and the distribution of roots ofHi(λ) and

Definition 1(Hurwitz stability)[25]: The quasi-polynomialH(λ)is said to be Hurwitz stable if and only if all the roots ofH(λ)=0lie in the open left-half plane of the complex plane.

Fora sufficiently largel∈Z+, let

and

We now state and establish the following results.

Lemma 1: The quasi-polynomialH(λ) is Hurwitz stable if and only if

where ρi(H) is an imaginary signature ofH(λ) defined by

Lemma 2: The quasi-polynomialH(λ) is Hurwitz stable if and only if

where ρr(H) is a real signature ofH(λ) defined by

where

Substituting λ =iz,z∈R, into (12), we denote

Let

B. Necessary Conditions for Hurwitz Stability of H(λ) With Complex Coefficients

According to the preliminary results on Hurwitz stability presented in Section III-A, we derive several necessary conditions for Hurwitz stability of the quasi-polynomialH(λ) in this subsection.

Lemma 7: The necessary condition for the quasi-polynomialH(λ) to be Hurwitz stable is

Lemma 8: The necessary condition for the quasi-polynomialH(λ) to be Hurwitz stable is

RewriteH(λ) as

where

By properties of polynomials, one can rewriteb(λ) as

where λj,j=1,2,...,n, denote the roots ofb(λ).LetLb,Rb,andIbbe the numbers of roots ofb(λ) in the open left-half plane, in the open right-half plane, and on the imaginary axis of the complex plane, respectively.Substituting λ=iz,z∈R,into (16), let

Let

Note that

Moreover, let

Proof: By referring to the process of the proof of Theorem 1 in [24] and using Lemmas 9 and 10, one can conclude here that

Then by Lemma 7 and the expression (22), one can draw the condition (21).■

According to Lemma 7 and the expression (24), one can draw the condition (23).■

Remark 2: It should be pointed out that Theorems 1 and 2 in[24] provide necessary conditions for the quasi-polynomialH(λ)to have some fixed number of roots in the open righthalf plane of the complex plane.In this paper, Theorems 1 and 2 directly present necessary conditions for the quasi-polynomialH(λ) to be Hurwitz stable.

C. Necessary Conditions for Hurwitz Stability of H (λ) With Real Coefficients

As a special case, when all the coefficients ofH(λ) are real,we have the following results.

Lemma 11: The necessary condition for the quasi-polynomialH(λ) with real coefficients to be Hurwitz stable is

whereQRis the number of real roots ofHi(z) over

Proof: When the coefficients ofH(λ) are all real,Hi(0)=0 andHi(z) is an odd function, which implies that the real roots ofHi(z) are all symmetrical about the origin.From (8), (15),(17), and (18),Hi(z) can be rewritten as

Then the real roots ofHi(z) except forz=0 can be determined by

We first consider the result (25) fornbeing even.For the quasi-polynomialH(λ) with real coefficients, it is clear that ϕ1,n=0, which yields

By Lemma 7, the condition (13) is the necessary condition forH(λ) to be Hurwitz stable.From (13) and (27), we have

which leads to the condition (25).

One can obtain the condition (25) fornbeing odd following the same lines as above.■

Then similar to Theorems 1 and 2, we derive the following theorems on necessary conditions for the quasi-polynomialH(λ)with real coefficients to be Hurwitz stable.

Theorem 3: The necessary condition for the quasi-polynomialH(λ) with real coefficients to be Hurwitz stable is

IV.PROTOCOL DESIGN FOR CONSENSUS OF SECOND-ORDER LTI MULTI-AGENT SYSTEMS

In this section, we first derive some result on the PI protocol design for consensus of the second-order multi-agent system (1) under a directed network topology.We then present the result on the PI protocol design for consensus of the system under an undirected network topology.Furthermore, we investigate the effects of delays on consensus of systems of harmonic oscillators/double integrators, which are the special cases of second-order multi-agent systems, under PI control.

A. Consensus Protocol Design Under a Directed Topology

The purpose on the consensus protocol design of the multiagent system (1) is to determine the values of the parametersKpandKifor placing all the roots of the function δk(s) to the left of the lines=-σ in thes-plane.The roadmap on the protocol design is provided as follows.We first give some necessary conditions forKpandKito locate all the roots of the function δk(s) to the left of the lines=-σ, which are provided for the estimate of allowable regions ofKpandKi.Then we provide a necessary and sufficient condition for the fixed values ofKpandKito locate all the roots of the functionδk(s)to the left of the lines=-σ, which is provided for the determination of the effective values ofKpandKiby searching in the allowable regions.The final parameter values for the protocol are determined from the global effective values ofKpandKifor any δk(s),k=2,...,N.

In order to provide a more general result, we define

where

Let

whereλ ∈C and σ ∈R+.Then we have

Proposition 1: The numbers of roots of the function δk(s) to the left of the lines=-σ , to the right of the lines=-σ, and on the lines=-σ are the same as the numbers of roots of the functionHk(λ) in the open left-half plane, in the open righthalf plane, and on the imaginary axis, respectively.

Denote bynDthe order ofD(λ/τ-σ) in λ.It is clear thatnD=3.RewriteD(λ/τ-σ) as

where λ1∈C, λ2∈C and λ3=τσ are the roots ofD(λ/τ-σ).LetRDbe the number of roots ofD(λ/τ-σ) in the open righthalf plane of the complex plane.Let λ=iz,z∈R.Then we define

We further define

Let λ =iz,z∈R.Then we obtain

where

and

Let

Let

Denote byz˜ ∈R the largest root and byz˜ ∈R the smallest root of (40), (41), and

Let λ =iz,z∈R.Then we obtain

From (30) and (60), we have

Compared (67) with (11), the corresponding expressions are given as follows

By Lemma 3, one can draw the conclusion of this proposition.

Let Ωkbe the effective parameter set ofKp-Kifor satisfying the requirement for the distribution of roots of the function δk(s).Then one can determine the effective parameter setΩ ofKp-Kifor consensus of the second-order multi-agent system (1) as

Example 1: Consider a multi-agent system with a directed topology whose adjacency matrix and corresponding Laplacian matrix are given by

The network topology is given by the example in [21].There are five agents in the entire system.The agent is a second-order system described by

Fig.1.The effective values of K p and K i.

Fig.2.The dynamic responses with τ =0.07 s and K p=4, K i=5.2.

Fig.3.Outputs of the protocols u k(t), k =1,...,5.

Since all the eigenvalues of the characteristic functionδ(s)are placed to the left of the lines=-0.8 in the complex plane when settingKp=4 andKi=5.2, the system has some consensus margin for the parameter variation.Here we will show consensus of the multi-agent system (69) with different communication delays under the fixed protocol parametersKp=4 andKi=5.2.

On the one hand, we consider the time delay is τ=0.15 s,which is identical and constant.Then the position error and the velocity error dynamic responses of the system are shown in Figs.4(a) and 4(b), respectively.On the other hand, we take into account that the multi-agent system (69) experiences heterogenous communication delays around τ=0.07 s.Specifically, the consensus protocoluk(t),k=1,...,5 contains time delay τk, where τ1=0.08 s, τ2=0.06 s, τ3= 0.07 s , τ4=0.05 s , and τ5=0.09 s.Then the position error and the velocity error dynamic responses of the system are shown in Figs.5(a) and 5(b), respectively.The results given in Figs.4

Fig.4.The dynamic responses with τ =0.15 s and K p=4, K i=5.2.

Fig.5.The dynamic responses with heterogeneous delays and ,.

Kp=4Ki=5.2 and 5 illustrate that the performance has been improved in addition to achieving consensus of the multi-agent system(69).

B. Consensus Protocol Design Under an Undirected Topology

Consider a multi-agent system described by (1), where the network topology is undirected.In this case, the eigenvalues of the Laplacian matrix are all real, which leads to the fact that δk(s) withk=2,...,Nare the functions with real coefficients.One can apply the results in Section IV-A to the system (1)with an undirected network topology because the real number is the special case of the complex number.Since the coefficients of the corresponding functions are real coefficients, the results on the protocol design for consensus of the multi-agent system under an undirected network topology can be simplified.

The allowable region ofKpcan be determined by the following proposition.

Proposition 5: The necessary condition for the parameterKpto place all the roots of the function δk(s) to the left of the lines=-σis

Proof: The proof of this proposition can be completed by following the same lines as the proof of Proposition 2 and considering Theorems 3 and 4.

The allowable region ofKican be determined by the following proposition.

Proposition 6: The necessary condition for the parameterKito place all the roots of the function δk(s) to the left of the lines=-σis

The following result is provided for determining the effective values ofKpandKi.

Proposition 7: For the fixed values ofKpandKi, all the roots of the function δk(s) are placed to the left of the lines=-σ in thes-plane if and only if

C. Harmonic Oscillators and Double Integrators

For the system(1), whenc1=0 andc0∈R+, it becomes the system of harmonic oscillators as follows

wherepk(t)∈R is the position state,vk(t)∈R is the velocity state, anduk(t)∈R is the consensus protocol described by (2)with τ ≥0.

Following the results derived in Sections IV-A and IV-B,one can design the PI protocol with delayed position information to achieve consensus of the system (72).

We now consider a special case.There is no communication delay, i.e., τ=0, between agentkand agentjin the network.The corresponding PI protocol is given by

It is shown in [15] that the system of harmonic oscillators cannot achieve consensus under a proportional (P) protocol.In what follows, we show that the system cannot achieve consensus by the PI protocol (73).

Proposition 8

: Consensus of the system (72) cannot be achieved by the PI protocol (73).

Proof: For the system (72), one can obtain the error system described by (4) withc1=0 andc0∈R+.The corresponding characteristic function of such an error system is given by

Following the same lines as the analysis in Section II, one can conclude that consensus of the system (72) can be achieved if and only if all the roots of δk(s) are placed in the open left-halfs-plane.

It is clear that the function δk(s) has three roots in the entires-plane.Lets=sr+isiand µk=µkr+iµki.When δk(s)=0, it follows that

Note that the coefficients of (74) and (75) are all real.Equations (74) and (75) both have three real roots.Denote bythe roots of (74).From the elementary symmetric of polynomials, we have

which leads to the fact that eitherroots are positive.Thus, δk(s) has some roots on the imaginary axis or in the open right-halfs-plane.Therefore, one can or some conclude that consensus of harmonic oscillators cannot be achieved by the PI consensus protocol (73).■

Based on the analysis in [15] and by Proposition 8, one can conclude that the time delay plays a positive role in the design of a P protocol or a PI protocol for consensus of the system of harmonic oscillators.

For the system (1), whenc1=c0=0, it becomes the system of double integrators as follows

wherepk(t)∈R is the position state,vk(t)∈R is the velocity state, anduk(t)∈R is the consensus protocol described by (2)with τ ≥0.

We now show that consensus of the system of double integrators cannot be achieved by a PI protocol with and without delayed state information.

Proposition 9: For any delay τ ≥0, consensus of the system (76) cannot be achieved by the PI consensus protocol (2).

Proof: For the system (76), one can obtain the error system described by (4) withc1=c0=0.The corresponding characteristic function of such an error system is given by

Following the same lines as the analysis in Section II, consensus of the system (76) can be achieved if and only if all the roots of the function δk(s) are placed in the open left-halfsplane.

When τ =0, one can conclude that consensus of the system of double integrators cannot be achieved by using the PI consensus protocol following the same lines as the proof of Proposition 8.

We now consider the case of τ >0.From (30), we have the

following expression with σ =0.

By Proposition 1, one can conclude that the numbers of roots of the function δk(s) in (77) in the open right-half plane,in the open left-half plane, and on the imaginary axis are the same as the numbers of roots of the functionHk(λ) in (78) in the open right-half plane, in the open left-half plane, and on the imaginary axis, respectively.Then using (35), we obtain

where

From (80), t he real roots of(z) can be determined by the following equation

V.CONSENSUS PROTOCOL DESIGN ON HIGH-ORDER MULTIAGENT SYSTEMS

In this section, we consider the high-order LTI multi-agent withNidentical dynamics described by

withk=1,2,...,N, wherex1,k(t),...,xn,k(t)∈R are the state variables,uk(t)∈R is the consensus protocol, andc0,...,cn-1,d0,...,dn-1,∈R.The consensus protocoluk(t) is presented in the PI form of

Then following the same analysis as the case of secondorder multi-agent systems in Section II, the corresponding characteristic function of the error system for the system (82)with the consensus protocol (83) can be described by

In this section, we present the results on the consensus protocol design to place all the roots of the function δk(s) to the left of the lines=-σ, which can achieve consensus of the system (82) and further improve the dynamic performance.

Define

where

We further define

where λ1,λ2,...,λn+1∈C are the roots ofD(λ/τ-σ).

Denote byRDthe number of roots ofD(λ/τ-σ) in the open right-half plane.Substitute λ=iz,z∈R , intoD(λ/τ-σ) and let

Then we have

and

In order to obtain the allowable region ofKp, let

Substitute λ =iz,z∈R , intoHˆk(λ) and let

Let

In order to obtain the allowable region ofKi, let

By substitutin g λ =izintoH˜k(λ), we have

Let

By using the functions ξ (z), φ (z), φ ˜k(z),H˜ki(z), and the numberRDof this section in Proposition 3, one can determine the allowable region ofKifor consensus of the system (82).

Moreover, let

Substitute λ =iz,z∈R , intoHk(λ) and D (λ) and let

and

Furthermore, let

Thus, we have

and

For the fixed values ofKpandKiin the allowable regions,one can determine the effective parameter values by using Proposition 4 and (68), whereHki(z),Hkr(z), D(iz), N (iz), and the numbernDare provided above in this section.

VI.CONCLUSION

This paper presents some Hurwitz stability criteria and derives several necessary conditions for Hurwitz stability for a class of quasi-polynomials.Based on the derived results, a PI consensus protocol is designed for a second-order LTI multiagent system with delayed state information via assigning eigenvalues, where allowable regions of consensus protocol parameters are first estimated by using necessary conditions for Hurwitz stability of the quasi-polynomials and the effective values of consensus protocol parameters are then determined by applying Hurwitz stability criteria for the quasipolynomials.Moreover, the effects of delays on consensus of systems of harmonic oscillators/double integrators under a PI consensus protocol are investigated.The results on the second-order LTI multi-agent system are extended to the case of a class of high-order LTI multi-agent systems.

In this paper, we design consensus protocols for LTI multiagent systems with an identical communication delay in the PI form.In our future work, we will focus on consensus protocols for LTI multi-agent systems with multiple communication delays in the PI form or other forms.