An Output-Feedback Control Approach to the H ∞ Consensus Integrated with Transient Performance Improvement Problem
AAn Output-Feedback Control Approach to the H ∞ Consensus Integrated with Transient PerformanceImprovement Problem
Jingyao Wang a , Zhisheng Duan a,b , Jianping Zeng a a Department of Automation, Xiamen University, Xiamen 361000, China b Department of Technology and Engineering Science, Peking University, Beijing 100871,China
Abstract
This paper considers the consensus performance improvement problem ofnetworked general linear agents subject to external disturbances over Marko-vian randomly switching communication topologies. The consensus controllaws can only use its local output information. Firstly, a class of full-orderobserver-based control protocols is proposed to solve this problem, whichdepends solely on the relative outputs of neighbours. Then, to eliminatethe redundancy involved in the full-order observer, a class of reduced-orderobserver-based control protocols is designed. Algorithms to construct bothprotocols are presented, which guarantee that agents can reach consensusin the asymptotic mean square sense when they are not perturbed by dis-turbances, and that they have decent H ∞ performance and transient perfor-mance when the disturbances exist. At the end of this manuscript, numericalsimulations which apply both algorithms to four networked Raptor-90 heli-copters are performed to verify the theoretical results. Keywords:
Distributed control; H ∞ consensus; transient performance;reduced-order observer; Markovian randomly switching topology. Email addresses: [email protected] (Jingyao Wang), [email protected] (Zhisheng Duan), [email protected] (Jianping Zeng) This work was partially supported by the National Nature Science Foundation ofChina under Grant Nos. 61803319, U1713223 and 61673026.
Preprint submitted to System & Control Letters October 16, 2018 a r X i v : . [ m a t h . O C ] O c t . Introduction The coordination control of networked agents, for example, unmannedaerial vehicles (UAVs), has gained increasing attention thanks to its paramountimportance in both civilian applications like monitoring oil fields and pipelines,and applications to homeland security like border patrol. Due to this, nu-merous control protocols have been constructed to address various controlproblems, e.g., [1, 2]. Though these works have provided a theoretical un-derpinning to the application of coordination control, there is still a criticalrequest for coordination control laws which are uncomplicated to implementin practice and designed for hazardous environments.In practice, agents, for example, UAVs, when they are required to con-duct tasks like monitoring oil fields and pipelines, are usually exposed tocomplicated environments. Under such circumstance, the wind in the out-door space and other sources of external disturbances may deteriorate thecontrol performance and even threaten the life of agents. For example, smallfixed-wing UAVs are highly susceptible to wind. Moreover, agents are alsosuffering from disturbances cased by issues like un-modeled high-order dy-namics, which arise quite often in practice and are to be found, for instance, inaircraft control systems and large scale flexible structures for space stations(such as antenna, solar arrays, etc.). Thus, guaranteeing good robustnessperformance of agents is highly valued in the design of control laws. To im-prove the robustness performance of agents, great efforts have been devotedand hence there are a large amount of references on this topic in the publishedliterature. According to the number of agents, results on this topic can beclassified into robust control design for single agent and robust control designfor a group of agents. For the single agent case, the novel publication [2] pro-vides benchmark solutions, like H ∞ optimal control and H optimal control,to deal with various external disturbance attenuation problems. For the net-worked agents case, there are also many researchers who have endeavored toimprove the H ∞ and H consensus performances of multi-agent systems withlinear dynamics. To name a few, [3, 4, 5, 6, 7, 8, 9, 10]. Note that a com-mon assumption is adopted in the above references, i.e., the communicationgraph is time-invariant or fixed and even known previously. However, thisassumption has limited the application of the proposed control algorithmsin [3, 4, 5, 6, 7, 8, 9, 10]. In fact, the communication topologies are usuallyrandomly switching, because of many practical backgrounds including gossipalgorithms and communication patterns (for example, [11, 12]). In addition,2uring the information transmission, packet drop and node failure phenom-ena can be described as random switching graph process. Furthermore, theimportance of the multi-agent consensus with various random graph has at-tracted a lot of interest recently. Results related to this include but notlimited to [13, 14, 15, 16, 17].To the best of our knowledge, there are only a few works investigating therobustness performance of multi-agent systems with the randomly switchingcommunication topologies. In [18], the L - L ∞ containment control problemof second-order multi-agent systems with Markovian switching topologies andexternal disturbances is investigated. In [19], we proposed a class of state-feedback control protocols to improve the H ∞ and transient performances ofagents with linear dynamics over randomly switching topologies. We mainlyfocused on improving the transient performance of agents in networks, i.e.,reducing the big overshoot and large oscillation, which are caused by theinitial-state uncertainty. The reason of improving the transient performanceis that, as pointed out in [20], because of the initial-state uncertainty, thetransient performance might be unacceptable or might deteriorate the distur-bance rejection level of H ∞ control. Following this pattern, we also studiedthe target containment control problem of multi-agent systems under theMarkovian randomly switching topologies in [21]. It is worth noting thatthe control laws in [18, 19, 21] rely on the relative states of neighbouringagents. In fact, the state information might be unavailable in many circum-stances. Instead, each agent can access the local output information, i.e., theoutput of itself and the outputs from its neighbours. Moreover, the relativeoutputs of neighbours can be measured by sensor networks, i.e., a variety ofsonar sensors. Therefore, designing control laws which solely depend on localoutput information is more practical.Motivated by the aforementioned works, we study the H ∞ consensus con-trol integrated with transient performance problem of general linear multi-agent systems with Markovian randomly switching interconnections. Here,we first propose a class of distributed control protocols which uses an observerto estimate the entire state information. The main advantage of this classof control protocols is that it solely requires the relative output informationof neighbouring agents. However, the main drawback of these protocols isthat estimating the entire state information may involve a certain degree ofredundancy, since actually part of the state information is available in theoutput. To eliminate the redundancy, we design a class of reduced-orderobserver to estimate the missing state information and then design the con-3ensus control laws based on the relative states of observers and the relativeoutputs of neighbouring agents. The main difficulty of constructing the abovetwo classes of control laws lies in that each possible communication graphis directed, which implies that the Laplacian matrices of these graphs aregenerally asymmetric, rendering the constructing of the consensus protocolsand the selection of appropriate Lyapunov function far from being easy.The reminder of this paper is as follows. Section II presents some pre-liminary results which are key to solve the concerned problems. SectionsIII and IV are dedicated to address the H ∞ stochastic consensus integratedwith transient performance problem. Section V shows the effectiveness ofthe proposed control laws with a simulation example. Notation: let R n × n be the set of n × n real matrices. Let L r [0 , ∞ )denote the space of square integrable vector functions over [0 , ∞ ), which areof r dimension. And let I N denote the identity matrix of order N . Representthe L norm of the corresponding function with (cid:107) · (cid:107) .
2. Preliminaries
Denote by σ ( t ) , t ∈ R + , a right-continuous time-homogeneous Markovprocess based on the probability space. σ ( t ) takes values in a finite state space S = { , , . . . , s } . Its infinitesimal generator matrix is denoted by Q = [ q ij ],where q ij denotes the transition rate from state i to state j with q ij ≥ i (cid:54) = j ; q ii = N (cid:80) j (cid:54) = i q ij otherwise. Thus, matrix Q is a transition matrix with rowsummation being zero and all off-diagonal elements non-negative.At time t , denote the communication graph among all agents (nodes)by G σ ( t ) (cid:44) ( V , E σ ( t ) ), where V (cid:44) { , , . . . , N } and E σ ( t ) are respectivelythe node set and the edge set. The associated adjacency matrix is A ( σ ( t )) (cid:44) [ a ij ( σ ( t ))] ∈ R N × N , where a ij ( σ ( t )) > j, i ) ∈ E σ ( t ) ; a ij ( σ ( t )) = 0 otherwise.The Laplacian matrix associated with G σ ( t ) can thus be defined, i.e., L σ ( t ) (cid:44) [ l ij ( σ ( t ))] ∈ R N × N , where l ij ( σ ( t )) = − a ij ( σ ( t )) if i (cid:54) = j ; l ii ( σ ( t )) = (cid:80) Nj (cid:54) = i a ij ( σ ( t ))otherwise. We say the communication graph at time t is balanced, if eachnode’s in-degree equals to its out-degree.Represent the union of the s graphs G i = ( V , E i ) , i = 1 , , . . . , s, by G un (cid:44) ∪ si =1 G i = ( V , ∪ si =1 E i ). The Laplacian matrix associated with the union graph G un is denoted by L un , which is the sum of L i , i = 1 , , . . . , s .4 emma 1. For any vectors p, q ∈ R n and positive definite matrix Φ ∈ R n × n ,the following matrix inequality holds: p T q ≤ p T Φ p + q T Φ − q. (1)
3. Problem Formulation and Full-Order Observer-Based ControlProtocol
This section focuses on formulating the H ∞ consensus control integratedwith transient performance improvement problem and then constructing thedistributed control protocol to deal with it. Consider the following N agents:˙ x i ( t ) = Ax i ( t )+ Bu i ( t )+ Dω i ( t ) ,y i ( t ) = C x i ( t ) , i = 1 , , . . . , N, (2)where x i ( t ) ∈ R n , u i ( t ) ∈ R m , ω i ( t ) ∈ L (cid:96) [0 , ∞ ] and y i ( t ) ∈ R q are the state,the control input, the external disturbance and the measured output of the i -th agent, respectively; A ∈ R n × n , B ∈ R n × m , C ∈ R q × n and D ∈ R n × (cid:96) are constant matrices.In this paper, we consider the network of agents described by (2) withMarkovian switching communication graph G σ ( t ) . The switching process isdefined in Section 2 and satisfies the following assumption: Assumption 1.
The continuous-time Markov process with a transition ratematrix Q governing the switching process of the communication topologies isergoic. As a result of Assumption 1, for the time-homogeneous Markov process,its invariant distribution is unique π = [ π π . . . π s ] T , where π i ≥
0, forall i ∈ S ; thus the distribution of σ ( t ) equals to π for all t ∈ R + .Herein the union G un of the s graphs satisfies the following assumption: Assumption 2.
There exists a directed spanning tree in the union graph G un of all the possible topologies, and each possible topology is directed andbalanced. x i ( t ) = A ˆ x i ( t )+ Bu i ( t )+ L ( C ˆ x i ( t ) − N (cid:88) j =1 a ij ( σ ( t ))( y i ( t ) − y j ( t ))) , i = 1 , , . . . , N, (3)where L ∈ R n × q is the observer gain to be designed.Now let z tr,i ( t ) = C ( x i ( t ) − N N (cid:80) j =1 x j ( t )) , i = 1 , , . . . , N, be the transientpart of the concerned states, where C ∈ R q × n is a constant matrix. Withthe above notation, we are ready to define the H ∞ consensus integrated withtransient performance improvement problem. Definition 1.
The H ∞ consensus integrated with transient performance im-provement problem is to design a class of controllers for the networked agentsdescribed by (2) such that when agents are not perturbed by external disturbance, i.e., w i ( t ) ≡ , i = 1 , , . . . , N , all agents can reach consensus, i.e., E [ (cid:107) x i ( t ) − x j ( t ) (cid:107) ] = 0 , i, j = 1 , , . . . , N ; given γ > , when the external disturbance exists, i.e., w i ( t ) (cid:54) = 0 , i =1 , , . . . , N , the following measure index is less than γ , which mea-sures the disturbance attenuation performance of the concerned output z tr,i ( t ) , i = 1 , , . . . , N : J tr (cid:44) sup (cid:107) ω ( t ) (cid:107) + x T ¯ Rx +ˆ x T ¯ R ˆ x (cid:54) =0 E [ z Ttr ( t ) z tr ( t )] (cid:107) ω ( t ) (cid:107) + x T ¯ Rx + ˆ x T ¯ R ˆ x , (4) where z tr ( t ) = [ z Ttr, ( t ) z Ttr, ( t ) ... z Ttr,N ( t ) ] T , ω ( t ) = [ ω T ( t ) ω T ( t ) ... ω TN ( t ) ] T , x =[ x T (0) x T (0) ... x TN (0) ] T and ˆ x = [ ˆ x T (0) ˆ x T (0) ... ˆ x TN (0) ] T ; ¯ R (cid:44) I N ⊗ R , where ⊗ denotes the Kronect product and R is a previously given positive-definite matrix. Based on the observed state information, we introduce a class of con-trollers for each agent, which is in the following form: u i ( t ) = − τ K ˆ x i ( t ) , i = 1 , , . . . , N, (5)6here τ ∈ R and K ∈ R m × n are to be designed. Hereafter, we call controller(5) the full-state observer-based consensus protocol, since it is constructedbased on the observer (3). Remark 1.
The full-state observer-based consensus protocol (5) solely relieson the relative outputs of neighbouring agents; hence it is distributed.
Let x ( t ) (cid:44) [ x ( t ) x ( t ) ... x N ( t ) ] and ˆ x ( t ) (cid:44) [ ˆ x ( t ) ˆ x ( t ) ... ˆ x N ( t ) ]. Substituting(5) into (2) and rewriting them in a compact form yield˙ x ( t ) = ( I N ⊗ A ) x ( t ) − τ ( I N ⊗ BK )ˆ x ( t ) + ( I N ⊗ D ) ω ( t ) . (6)Similarly, we have the trajectory of ˆ x ( t ) yielding˙ˆ x ( t ) = ( I N ⊗ ( A + LC − τ BK ))ˆ x − ( L σ ( t ) ⊗ LC ) x. (7)Now, we introduce the consensus error variable ζ i ( t ) (cid:44) x i ( t ) − N N (cid:80) j =1 x j ( t )for each agent i = 1 , , . . . , N , which denotes the distance of agent i ’s statesfrom the average of all agents’. Note that ζ ( t ) (cid:44) [ ζ T ( t ) ζ T ( t ) ... ζ TN ( t ) ] T =( M ⊗ I n ) x ( t ), where M (cid:44) I N − N T . Also, let δ i ( t ) (cid:44) (ˆ x i ( t ) − x i ( t )) − N N (cid:80) j =1 (ˆ x j ( t ) − x j ( t )) for each agent i = 1 , , . . . , N . We also have δ ( t ) (cid:44) [ δ T ( t ) δ T ( t ) ... δ TN ( t ) ] T = ( M ⊗ I n )(ˆ x ( t ) − x ( t )).Write ζ ( t ) and δ ( t ) in a column vector and denote it by e ( t ) (cid:44) [ ζ T ( t ) δ T ( t ) ] T .With equations (6) and (7), we can write the evolution trajectory of e ( t ) intoa compact form, i.e., ˙ e ( t ) = F ( t ) e ( t ) + ¯ Dω ( t ) (8)where F ( t ) = (cid:20) I N ⊗ ( A − τ BK ) − τ I N ⊗ ( BK )( I N − τ L σ ( t ) ) ⊗ ( LC ) I N ⊗ ( A + LC ) (cid:21) , and ¯ D = (cid:20) I N ⊗ D (cid:21) . To address the H ∞ consensus control integrated with transient perfor-mance improvement problem, we design the following algorithm to constructthe observer (3) and the distributed controller (5):7 lgorithm 1. Step 1 Given γ > , solve the following inequalities to geta pair of feasible solution ( P , r ) P A T + AP − r BB T D P C T P C T D T − γ I (cid:96) C P − ρ (1+¯ π λ max ) I q C P − I q < , (9) (cid:20) γ κ R I n I n P (cid:21) > , (10) r > , (11) where λ max is the largest eigenvalue of matrix L T un L un ; κ denotes thelargest eigenvalue of matrix M . Step 2
Get ( P , Q, r ) satisfying the following inequalities: A T P + P A + C T Q T + QC + r P − BB T P − − P D Q T − D T P − γ I (cid:96) Q ρ I n < , (12) P < γ κ R, (13) r > , (14) where P has been obtained from Step . Step 3
Choose a small enough positive constant ρ such that ρ < . If r − ρ < ρr , choose the coupling strength τ to satisfy r − ρ < τ < ρr , and let K = B T P − and L = P − Q , otherwise go back to Step 1. Next we will show that the full-order observer-based consensus protocol(5) designed according to Algorithm 1 enables all agents to solve the H ∞ consensus control integrated with transient performance improvement prob-lem. Theorem 1.
Under Assumptions and , given a constant γ > , the full-order observer-based consensus protocol (5) enables the group of agents de-scribed by (2) to solve the H ∞ consensus control integrated with transientperformance improvement problem if the control gains K, L and the cou-pling gain τ are chosen as in Algorithm . roof. Consider the following Lyapunov function candidate V ( t ) = s (cid:88) i =1 V i ( t ) (15)where V i ( t ) = E [ e T ( t )diag( I N ⊗ P − , I N ⊗ P ) e ( t )1 σ ( t )= i ], in which matrices P and P are positive-definite.Given i = 1 , , . . . , s , taking the derivative of function V i ( t ) along thetrajectory (8) yieldsd V i ( t ) = E [d e T ( t )diag( I N ⊗ P − , I N ⊗ P ) e ( t )1 σ ( t )= i ]+ s (cid:88) i =1 E [ e T ( t )diag( I N ⊗ P − , I N ⊗ P )d e ( t )1 σ ( t )= i ]+ s (cid:88) j =1 q ji V j ( t )+ o (d t ) . (16)Since σ ( t ) has a unique invariant distribution π = [ π π . . . π s ] T and π i ≥ ¯ π , for all i ∈ S , we have˙ V ( t ) ≤ E [( (cid:20) I N ⊗ ( A − τ BK ) − τ I N ⊗ ( BK )( I N − τ ¯ π L un ) ⊗ ( LC ) I N ⊗ ( A + LC ) (cid:21) (cid:20) ζ ( t ) δ ( t ) (cid:21) + (cid:20) I N ⊗ D (cid:21) ω ( t )) T (cid:20) I N ⊗ P − I N ⊗ P (cid:21) (cid:20) ζ ( t ) δ ( t ) (cid:21) + (cid:20) ζ ( t ) δ ( t ) (cid:21) T (cid:20) I N ⊗ P − I N ⊗ P (cid:21) ( (cid:20) I N ⊗ ( A − τ BK ) − τ I N ⊗ ( BK )( I N − τ ¯ π L un ) ⊗ ( LC ) I N ⊗ ( A + LC ) (cid:21) (cid:20) ζ ( t ) δ ( t ) (cid:21) + (cid:20) I N ⊗ D (cid:21) ω ( t ))]= E [ ζ T ( t )( I N ⊗ ( A T P − + P − A − τ P − BB T P − )) ζ ( t )+ 2 ζ T ( t )( I N ⊗ ( − τ P − BB T P − + C T L T P )+ ¯ π L T un ⊗ ( − C T L T P )) δ ( t )+ δ T ( t )( I N ⊗ (( A + LC ) T P + P ( A + LC ))) δ ( t )+ 2 ζ T ( t )( M ⊗ ( P − D )) ω ( t )+ 2 δ T ( t )( M ⊗ ( P D )) ω ( t )] , (17)9here the last equality holds by letting K = B T P − .By using Lemma 1, it gives that2 ζ T ( t )( I N ⊗ ( − τ P − BB T P − ) δ ( t )= 2 τ ζ T ( t )( I N ⊗ ( − P − B ))( I N ⊗ ( B T P − )) δ ( t ) ≤ ρτ ζ T ( t )( I N ⊗ ( P − BB T P − )) ζ ( t )+ 4 ρ τ δ T ( t )( I N ⊗ ( P − BB T P − )) δ ( t ) , (18)where ρ is a positive constant satisfying ρ <
2. Similarly, we have2 ζ T ( t )( I N ⊗ ( C T L T P )) δ ( t ) ≤ ρζ T ( t )( I N ⊗ ( C T C )) ζ ( t )+ 4 ρ δ T ( t )( I N ⊗ ( P LL T P )) δ ( t ) , (19)and 2¯ πζ T ( t )( L T un ⊗ ( − C T L T P ) δ ( t ) ≤ ρ ¯ π ζ T ( t )(( L T un L un ) ⊗ ( C T C )) ζ ( t )+ 4 ρ δ T ( t )( I N ⊗ ( P LL T P )) δ ( t ) . (20)Substituting inequalities (18), (19) and (20) into (17) yields˙ V ( t ) ≤ E [ ζ T ( t )( I N ⊗ ( A T P − + P − A − τ (2 − ρ ) P − BB T P − + ρ (1 + ¯ π λ max ) C T C )) ζ ( t )+ δ T ( t )( I N ⊗ (( A + LC ) T P + P ( A + LC )+ 4 ρ τ P − BB T P − + 8 ρ P LL T P )) δ ( t )+ 2 ζ T ( t )( M ⊗ ( P − D )) ω ( t )+ 2 δ T ( t )( M ⊗ ( P D )) ω ( t )] . (21)Let ˜ ζ ( t ) = ( U T ⊗ I n ) ζ ( t ) , ˜ δ ( t ) = ( U T ⊗ I n ) δ ( t ) and ˜ ω ( t ) = ( U T ⊗ I n ) ω ( t ),where U is a unitary matrix of matrices M such that U T M U = diag { , , . . . , } ,where the first column of U is set as √ N . With the above variable changes,10e have ˙ V ( t ) ≤ N (cid:88) i =2 E [ ˜ ζ Ti ( t )( A T P − + P − A − τ (2 − ρ ) P − BB T P − + ρ (1 + ¯ π λ max ) C T C ) ˜ ζ i ( t )+ ˜ δ Ti ( t )(( A + LC ) T P + P ( A + LC )+ 4 ρ τ P − BB T P − + 8 ρ P LL T P )˜ δ i ( t )+ 2 ˜ ζ Ti ( t ) P − D ˜ ω i ( t )+2˜ δ Ti ( t ) P D ˜ ω i ( t )] , (22)where to get the last inequality, we have used the fact that ˜ ζ ( t ) = T √ N δ ( t ) ≡ λ max (cid:44) max i =1 ,...,N λ i ( L T un L un ).When ω ( t ) ≡
0, also ˜ ω ( t ) ≡
0, we have˙ V ( t ) ≤ N (cid:88) i =2 E [ ˜ ζ Ti ( t )( A T P − + P − A − τ (2 − ρ ) P − BB T P − + ρ (1 + ¯ π λ max ) C T C ) ˜ ζ i ( t )+ ˜ δ Ti ( t )(( A + LC ) T P + P ( A + LC )+ 4 ρ τ P − BB T P − + 8 ρ P LL T P )˜ δ i ( t )] . (23)If A T P − + P − A − τ (2 − ρ ) P − BB T P − + ρ (1 + ¯ π λ max ) C T C < A + LC ) T P + P ( A + LC )+ 4 ρ τ P − BB T P − + 8 ρ P LL T P < , (25)we have ˙ V ( t ) ≤
0, which implies that all agents can reach consensus whenthe external disturbance does not exist. Next we will prove that Algorithm1 guarantees that inequalities (24) and (25) are always true. In light of Step1 in Algorithm 1, we get P A T + AP − r BB T + ρ (1 + ¯ π λ max ) P C T C P < . (26)11ultiplying both sides of (26) by P − , considering that r < τ (2 − ρ ), andusing Schur’s Complement Lemma we have inequality (24) holds. Then,substituting Q = P L into (12) and using the assertion that (10) and r > ρ τ yield that (25) is true. Thus, the first condition in Definition 1 is satisfied,which implies that the full-order observer-based control protocol enables theagents described by (2) to reach consensus in the asymptotic mean squaresense when they are not perturbed by external disturbances.When ω ( t ) (cid:54) = 0, we note that the last inequality of (22) still holds byadding N (cid:80) i =2 E [˜ z Ttr,i ( t )˜ z tr,i ( t )] − γ N (cid:80) i =2 E [˜ ω Ti ( t )˜ ω i ( t )] to the right-hand side of itand then subtracting it. In light of this, we can get the following inequalityafter some mathematical calculations:˙ V ( t ) ≤ N (cid:88) i =2 E [ ˜ ξ Ti ( t )Σ ˜ ξ i ( t )+ ˜ δ Ti ( t )Σ ˜ δ i ( t )] − N (cid:88) i =2 E [˜ z Ttr,i ( t )˜ z tr,i ( t )]+ γ N (cid:88) i =2 E [˜ ω Ti ( t )˜ ω i ( t )] , where ˜ z tr ( t ) = [ ˜ z Ttr, ( t ) ˜ z Ttr, ( t ) ... ˜ z Ttr,N ( t ) ], ˜ z tr ( t ) = ( U T ⊗ I n ) z tr ( t ), and ˜ z tr, ( t ) ≡ is (cid:20) A T P − + P − A − τ (2 − ρ ) P − BB T P − + ρ (1 + ¯ π λ max ) C T C + C T C P − DD T P − − γ I (cid:96) (cid:21) ;(27)matrix Σ is (cid:20) ( A + LC ) T P + P ( A + LC )+ ρ τ P − BB T P − + ρ P LL T P − P D − D T P − γ I (cid:96) (cid:21) . (28)If the following inequalities hold,Σ < , (29)Σ < , (30)we have ˙ V ( t ) ≤ − N (cid:88) i =2 E [˜ z Ttr,i ( t )˜ z tr,i ( t )]+ N (cid:88) i =2 E [˜ ω Ti ( t )˜ ω i ( t )] . (31)12aking similar steps to prove inequalities (24) and (25), we can verify thatinequalities (29) and (30) are satisfied under Algorithm 1.Integrating both sides of inequality (31) from zero to infinity yields N (cid:88) i =2 E [ (cid:107) ˜ z i ( t ) (cid:107) ] ≤ ζ T (0)( I N ⊗ P − ) ζ (0)+ δ T (0)( I N ⊗ P ) δ (0)+ γ N (cid:88) i =2 E [˜ ω Ti ( t )˜ ω i ( t )]= ζ T (0)( M ⊗ P − ) ζ (0)+ δ T (0)( M ⊗ P ) δ (0)+ γ N (cid:88) i =2 E [˜ ω Ti ( t )˜ ω i ( t )] ≤ ζ T (0)( M ⊗ P − ) ζ (0)+ δ T (0)( M ⊗ P ) δ (0)+ γ N (cid:88) i =1 E [˜ ω Ti ( t )˜ ω i ( t )]= ζ T (0)( M ⊗ P − ) ζ (0)+ δ T (0)( M ⊗ P ) δ (0)+ γ (cid:107) ˜ ω ( t ) (cid:107) ≤ γ x T (0)( I N ⊗ R ) x (0)+ γ ˆ x T (0)( I N ⊗ R )ˆ x (0)+ γ (cid:107) ˜ ω ( t ) (cid:107) , (32)where the last equality is true because the external disturbance consideredherein is deterministic and thus N (cid:80) i =1 E [˜ ω Ti ( t )˜ ω i ( t )] = (cid:107) ˜ ω ( t ) (cid:107) ; the last inequal-ity also holds since κP − < γ R, (33)and κP < γ R < γ R. (34)Since ˜ z tr, ( t ) ≡
0, we have N (cid:80) i =2 E [ (cid:107) ˜ z tr,i ( t ) (cid:107) ] = N (cid:80) i =1 E [ (cid:107) ˜ z tr,i ( t ) (cid:107) ] = E [ (cid:107) z tr ( t ) (cid:107) ].It follows from (32) that J tr < γ , (35)which means that the second condition in Definition 1 is satisfied. Thatis, the H ∞ consensus integrated with transient performance improvementproblem is solved. The proof is completed.13 emark 2. The main advantages of the full-order observer-based consensusprotocol (5) are two-fold. Firstly, as mentioned in Remark , it solely requirethe relative output of neighbours which are measurable by a set of sensor net-works in practice. Note that consensus protocols in [7, 10], which deal withthe H ∞ consensus for general linear multi-agent systems over fixed communi-cation graph, need the exchange of the protocol’s internal states or virtual out-puts, apart from the relative outputs, to maintain “the separation principle”.In fact, the states of the observers are internal information for the agents,which have to be transmitted via some communication networks. Thus, com-pared with protocols in [7, 10], the control law (5) can reduce or even removethe communication burden imposed on the communication networks. Sec-ondly, to the best of our knowledge, most of the references [18, 16, 19, 21]related to the consensus control of multi-agent systems over Markovian ran-domly switching graphs assume that each possible graph is balanced. If werelax this assumption and let each possible graph be strongly connected, thenthere is no guarantee that protocols in [18, 16, 19, 21] can serve their pur-pose. However, the control law (5) can be used to deal with this problem byslightly modifying the consensus error variable.3.3. Reduced-Order Observer-Based Controller Though the full-order observer-based consensus protocol (5) is effectiveto solve the H ∞ consensus integrated with transient performance problem, itstill possesses a certain degree of redundancy, which is caused by the fact thatthe entire state is estimated by the observer (3), but actually part of the stateinformation is available in the output. In this subsection, we will eliminatethe redundancy and thus reduce the dimension of the protocol, especiallyfor the case that agents are described by multi-input multi-output dynamics.To solve the problem mentioned in Section 3.1, we introduce a reduced-orderobserver-based consensus protocol which is based on the relative informationbetween neighbouring nodes. The reduced-order observer-based consensusprotocol is given by˙ v i ( t ) = ¯ F v i ( t )+ Gy i ( t )+ T Bu i ( t )+ T Dω i ( t ) ,u i ( t ) = − τ K [ N (cid:88) j =1 a ij ( σ ( t ))( Q ( y i ( t ) − y j ( t ))+ Q ( v i ( t ) − v j ( t )))] ,i = 1 , , . . . , N, (36)14here ¯ F is a Hurwitz matrix and has no eigenvalues in common with thoseof matrix A . Matrices G and T are the unique solution to the followingSylvester equation T A − ¯ F T = GC (37)which further satisfies that matrix [ C T T T ] T is non-singular. Q and Q aregiven by [ Q Q ] = [ C T T T ] T .Now we can present the algorithm to design the reduced-order observer-based consensus protocol (36). Algorithm 2. Step 1
Choose matrices ¯ F , G, T, Q and Q to satisfy therequirements given above. Step 2
Given a positive constant γ , solve the following inequalities to get apair of feasible solution ( P , r ) (cid:20) P A T + AP − r BB T D P C T D T − γ I (cid:96) C P − I q (cid:21) < , (38) (cid:20) γ κ R I n I n P (cid:21) > , (39) r > . (40) Step 3
Get ( P , r ) from the following inequalities: ¯ F T P + P ¯ F + r Q P − BB T P − Q T < , (41) P < γ κ R, (42) r > , (43) where P has been obtained from Step . Step 4
Choose a small enough positive constant ρ such that ρ < λ min , where λ min is the second smallest eigenvalue of matrix ˜ L un = L T un + L un . If r ¯ π ( λ min − ρ ) < ρr πλ max , let r ¯ π ( λ min − ρ ) < τ < ρr πλ max , and K = B T P − , where λ max is the largest eigenvalue of matrix L T un L un ; otherwise go back to Step 1. Remark 3.
A necessary and sufficient condition for the feasibility of Algo-rithms and is that ( A, B, C ) is stabilizable and detectable. heorem 2. Under Assumptions and , given a constant γ > , thereduced-order observer-based consensus controller (36) enables agents de-scribed by (2) to solve the H ∞ consensus integrated with transient perfor-mance improvement problem if the control gain K and the coupling gain τ are chosen as in Algorithm . Proof.
Theorem 2 can be proved by taking similar steps as in the proof ofTheorem 1.
4. Simulation Results
In this section, we illustrate the effectiveness of Algorithms 1 and 2 withthe following example.Consider a network of four Raptor-90 helicopters whose dynamics aregiven by (2) where A is given by (44) and A = − . − . − . − . . . − . − . . . . − . . − . − − . . − − . − . . . − . − . − . − . − . − . . − . (44) B = . . . . − . , D = − . − . − . − . . − . ,C = C = . able 1: Helicopter variables. Variablesq Pitch rate in the body frame componentsp Roll rate in the body frame componentsr Yaw rate in the body frame componentsU Velocity along the body frame x-axisV Velocity along the body frame y-axisW Velocity along the body frame z-axis θ Pitch angle φ Roll angle a s Longitudinal blade angle r fb Yaw rate feedback b s Lateral blade angleThis linear state-space model at a hovering point is established by Chen in[22]. The state vector is x = (cid:104) U T V T p T q T φ T θ T a Ts b Ts W T r T r Tfb (cid:105) T . All theabove variables are described in Table 1.The communication topology randomly switches between graphs G and G shown in Figs. 1a and 1b, respectively, both of which are directed andbalanced. Fig. 1c shows the union of G and G . Thus, Assumption 2 issatisfied since the graph G un contains a directed spanning tree. Moreover, letthe generator matrix of the continuous-time Markov process be Q = (cid:20) − − (cid:21) . (45)We assume that the Markov process satisfies Assumption 1 and has a uniqueinvariant distribution π = [ 2 / / L = − . − . − . − . . − . . . − . − . − . − . . . − . − . . . . − . − . . . . − . − . − . . − . . . − . − . − . − . . − . − . − . . − . − . . . − . . . − . − . − . . . . − . − . (47)17
24 3 (a) G (b) G (c) G un Figure 1: The communication graphs: (a) G ; (b) G ; (c) G un = G ∪ G . K l = . . . − . . − . . . . − . − . − . . . . . . . . − . . . − . − . . − . − . . − . . . − . − . (46)¯ F = − . − . − . − . − . − (48) G = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (49) K r = . . . − . . − . . . . − . − . − . . . . . . . . − . . . − . − . . − . − . . − . . . − . − . (50) Q = − . .
015 1 . − . − . − . . − . . . . . . . − . . − . − . . . . − . − . . . . . . − . − . − . − . − . . . − . . . − . − . . − . − . . . . . . − . − . − . − . − . − . − . (52)18 = . − . − . − . . × . × . × . × . × − . . − . . . . − . × − . − . − . × − . × . − . . − . − . − . . × . . . × . × − . . − . . . . − . × − . − . − . × − . × . − . . − . − . − . . × . . . × . × − . . − . . . . − . − . − . − . − . . − . (51) i=1i=2i=3i=4 Figure 2: The roll angles of four agents under the full-state observer-based consensusprotocol (5). Q = − . − . − . − . − . . . . . − . − . − . − . − . . . . − . . . . . . − . . . . . . . − . − . − . . . − . . . − . − . − . . − . − . − . − . − . − . . . . . . − . − . − . − . − . − . − . . . . − − . . (53)Choose R = I and let γ = 4. Using the Yalmip toolbox, we can obtainthe control gain matrices K l and L of the full-state observer-based consensusprotocol through Algorithm 1, given by (46) and (47), respectively. SinceAlgorithm 1 has the feature of decoupling the design of the agent dynam-ics from the communication topologies, we can select the coupling gain byfollowing Step 3 of Algorithm 1. And we set it as 1 . R and γ , we can get the control protocol gainmatrices ¯ F , G , K r , T , Q , and Q of the reduced-order observer-based con-sensus protocol through Algorithm 2, which are presented in (48), (49), (50),(51), (52), and (53), respectively. The coupling gain is also taken as 1 . i=1i=2i=3i=4 Figure 3: The pitch angles of four agents under the full-state observer-based consensusprotocol (5). achieve consensus when agents are perturbed by external disturbances in theform of square wave with period 2 π . Figs. 2 and 3 show the roll angles andpitch angles of these agents driven by the full-state observer-based consensusprotocol with control gain matrices described by (46) and (47). And Figs. 4and 5 present the corresponding trajectories of four agents driven by thereduced-order observer-based consensus protocol with control gain matricesgiven by (48), (49), (50), (51), (52), and (53). It can be seen from thesefigures that all agents can reach consensus though they are perturbed by theexternal disturbances.Now we make a comparison of the performance of grouped agents underthe above two protocols. Fist, we are interested in the consensus time re-quired by the grouped agents, which also reflects the convergence rate. Itfollows from Figs. 2 and 3 that agents need almost 4 seconds to reach con-sensus driven by the full-state observer-based consensus protocol, while theycost almost 6 seconds under the reduced-order observer-based consensus pro-tocol, which can be seen from Figs. 4 and 5. Next, we move our attentionto the transient performance of agents under both protocols. We draw thetrajectories of the same state of some agent in one picture to achieve our goal.We choose the first agent’s roll angle and the fourth agent’s pitch angle toillustrate that reduced-order observer-based consensus protocol allow agentsto have better transient behaviour, which are respectively shown in Figs. 6and 7. It can be seen from Fig. 7 that the overshoot of the forth agent’s pitchangle under the full-state observer-based consensus protocol is almost threetimes larger than that under the reduced-order observer-based consensus pro-20 i=1i=2i=3i=4 Figure 4: The roll angles of four agents under the reduced-order observer-based consensusprotocol (36). tocol. What is more, the trajectory of the pitch angle under the full-stateobserver-based consensus protocol is much more oscillatory than that underthe reduced-order observer-based consensus protocol. Precisely, the full-stateobserver-based consensus protocol makes the agent to eliminate the oscilla-tion in almost 4 seconds, while the reduced-order observer-based consensusprotocol just costs the agent almost 1 second. Figs. 6 shows that the firstagent behaves similarly. Thus, we can conclude that compared with the full-state observer-based consensus protocol, the reduced-order observer-basedconsensus protocol brings better transient performance to agents, but con-verges in a relatively slower rate. By analysing the poles of the closed-loopsystem, we find that the reduced-order observer-based consensus protocolplaces the poles closer from the real axis in the left half s-plane, which re-sults in less oscillation cycles. Therefore, we can conclude that both protocolsguarantee robustness against external disturbance, while the reduced-orderobserver-based consensus protocol constructed here performs better in termsof the transient performance.
5. Conclusions
This paper has considered the H ∞ consensus control integrated with tran-sient performance improvement problem for agents with general linear dy-namics. Two classes of observer-based protocols, which have been proposedto deal with this problem, solely require the output information of nodes.Thus, they are immune to the absence of entire state information. Algo-21 i=1i=2i=3i=4 Figure 5: The pitch angles of four agents under the reduced-order observer-based consensusprotocol (36).
Full-order observer-basedReduced-order observer-based
Figure 6: The roll angles of the fist agent under the full-state observer-based consensusprotocol (blue line) and the reduced-order observer-based control protocol (red line). Full-order observer-basedReduced-order observer-based
Figure 7: The pitch angles of the forth agent under the full-state observer-based consensusprotocol (blue line) and the reduced-order observer-based control protocol (red line). rithms have been designed to construct these protocols, which are applicablein large-scale networks. At the end of this manuscript, numerical simula-tions have been presented to show the effectiveness of these algorithms. Inthe future work, we will investigate the robust stochastic consensus controlproblem of networked agents with uncertainties in their dynamics.
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