Iterative LMI Approach to Robust Hierarchical Control of Homogenous Linear Multi-Agent Systems Subject to Polytopic Uncertainty and External Disturbance

This paper investigates design of robust hierarchical control for linear multi-agent systems (MAS) subject to polytopic uncertainty and external disturbance. Each agent of MAS is described by a homogeneous linear time-invariant dynamic model. The control structure has two layers, namely, an upper layer and a lower layer. Local actions are executed in the lower layer. Each agent shares information through an undirected graph with neighboring agents in the upper layer to achieve the global stabilization and disturbance attenuation. We employ a parameter-dependent Lyapunov function to formulate the robust control design of the local and global feedback in terms of bilinear matrix inequalities and incorporate constraints on disturbance attenuation. We propose the sufficient condition, where system matrices and Lyapunov variables are separated. The parameterization of the controller depends on a common slack variable instead of the Lyapunov matrices. We develop an iterative approach based on the coordinated optimization to solve sub-problems over linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness of the proposed robust control designs. It is shown that our proposed robust control designs outperform other robust control designs in terms of achievable disturbance attenuation and maximum admissible uncertainty bound.


I. INTRODUCTION
I N recent years, there have been researches investigating the hierarchical control of multi-agent systems (MAS) because of its wide applications in various areas containing biological networks, energy management systems, transportation networks, etc. The key concept of hierarchical control framework is to attain certain global objectives by using local measurements of subsystems (i.e., agents) and inter-agent cooperation via information exchange. Various classes of network systems were effectively coped with by this framework [1].
In general control systems, uncertainties and external disturbances are known sources of stability and performance degradation [2]. It is thus worth investigating the problem of disturbance attenuation for uncertain MAS. The robustness of MAS against disturbance and uncertainties can be handled in terms of H ∞ and H 2 norms [3]. One common approach for robust analysis and synthesis is Lyapunov theory. Finding a Lyapunov function which guarantees robust stability for all admissible systems is the main task. There are two main types of Lyapunov functions. The first type, referred to as quadratic function, is a common parameter-independent Lyapunov function (PILF) which explicitly depends on time. A sufficient condition to guarantee stability of MAS is derived in terms of linear matrix inequalities and then is called parameter-independent linear matrix inequality (PILMI) approach. This approach gives conservative results due to the following causes [4]. First, it utilizes only a fixed Lyapunov function for stability and performance for the entire uncertain domain. Second, design specifications use a common Lyapunov matrix and it is difficult to search the solution due to the product between the Lyapunov matrix variable and system matrices. The second type is a parameterdependent Lyapunov function (PDLF) whose matrix variable depends on uncertain parameters. PDLF can reduce the conservatism of robust stability and performance [5], [6]. Therefore, this approach has gained notable attractions recently. When utilizing this approach, robust analysis and control conditions are formulated in terms of parameterdependent bilinear matrix inequalities (PDBMIs) involving uncertain parameters [7].
There has been a number of studies dealing with single uncertain and perturbed systems using PDLMIs, e.g., [8]- [12]. However, only a few works have been existed for uncertain and perturbed MAS. For example, [13] investigated the leader-follower consensus of higher order MAS with directed and switching topology. A PDLF sufficient condition for obtaining consensus was formulated as a PDLMI. [14] addressed the stabilization and disturbance attenuation for homogeneous MAS with the norm-bounded uncertainty. The formulation incorporates H ∞ and H 2 design criteria on disturbance attenuation. A quadratic Lyapunov function is employed, and robust design of feedback controllers is given in terms of LMI. The limitation of this approach is a common Lyapunov function for entire uncertainty domain. In addition, [15] addressed the design of robust consensus controllers for homogeneous MAS in which agents contain polytopic uncertainty and external disturbance and interagent communications are subject to uncertainty. Then, a robust H ∞ stabilization was formulated by a PDLF and a sufficient condition was derived as a PDLF. Nevertheless, the controller parameterization depends on the Lyapunov matrix variable and the obtained result may be still conservative. To obtain a robust controller independent of uncertain parameters, it is possible to find a parameter-independent Lyapunov matrix variable utilizing the PILMI approach. An effective way to overcome this conservativeness is to separate the Lyapunov function with system matrices by introducing additional slack variables. This is called the dilated approach.
To our best knowledge, there has not been any work using the dilated approach for the robust hierarchical control of perturbed and uncertain MAS. This motivates us to investigate the dilated BMI for attaining specific disturbance attenuation of uncertain MAS by using an affine PDLF with respect to uncertain parameters. The derived controller is independent of non-common Lyapunov matrix variables but is dependent of a common slack variable [16]. Moreover, robust hierarchical controllers can be designed using the new dilated BMI conditions to improve system performance specified by disturbance attenuation. More specifically, we propose a systematic design approach of hierarchical H ∞ and H 2 controllers for linear MAS subject to polytopic uncertainty using the dilated BMI conditions. Our contributions are as follows.
• New robust design conditions are derived using PDLF, which includes quadratic PILF as a special case. The BMI design criteria are then determined and effectively solved by an iterative approach based on the coordinated optimization of sub-problems over linear matrix inequalities. • We compare the proposed BMI method with the PILMI method and the previous robust control method [14] to show the performance improvement in our proposed design. In particular, our proposed robust controllers give the upper bound of disturbance attenuation less than that obtained by other approaches. In addition, the proposed robust controllers yield stabilization of MAS with a larger maximum admissible bound of uncertainty.
The organization of this paper is as follows. We introduce problem formulation in Section II. Robust hierarchical H ∞ and H 2 state feedback control laws are proposed in Section III. Section IV illustrates numerical examples. Finally, conclusions are given in Section V.

A. MODEL OF AGENTS
The dynamics of agents considered in the current work consists of polytopic uncertainty and external disturbance as follows.
represent the state vector, control input vector, measured output, and external disturbance input of the i-th agent, respectively. Here, δ = δ 1 , . . . , δ M T is the timeinvariant polytopic uncertain vector which belongs to the unit simplex Θ given by where M is the number of vertices. The matrices A(δ) ∈ R n×n , B(δ) ∈ R n×q , C(δ) ∈ R m×n , B d (δ) ∈ R n×l , and D(δ) ∈ R m×q , 0 < q ≤ n denote uncertain system matrix, input matrix, output matrix, disturbance input matrix, and direct transition matrix, respectively. Those uncertain matrices are of polytopic type, which are represented in term of the uncertainty vector δ as follows, where A k , B k , B dk , C k , and D k are known constant matrices. Then the overall homogeneous MAS composing of N identical dynamical agents is represented by whereÃ The notation ⊗ denotes the Kronecker product.
The following reviews some basic concepts of graph theory. An undirected graph G = {V, E} stands for communication topology of MAS where V = {v 1 , v 2 , . . . , v N } and are the sets of vertex and edge, respectively. Next, N i represents the set of neighbors of the i-th agent. The graph G is connected if there exits a path between any two vertices.
The H ∞ and H 2 norms of the closed-loop transfer function G yd (s, δ) from d(t) to y(t) of system (4) [3] are utilized to assess the effect of parametric uncertainty and exogenous disturbance.
whereσ(.) indicates the largest singular value of a matrix. The objective of this work is to design a hierarchical feedback controller for the perturbed uncertain MAS (4) such that a given constraint on inter-agent information exchange depicted by graph G is satisfied. The control input of network system consists of two terms, i.e., the local control input u l (t) and the global control input u g (t), where F l ∈ R q×n and F u ∈ R q×n indicate the local and global feedback controller gains, respectively. It is proven from [1], [17] that the information exchange among agents can be assured by using the hierarchical feedback gain F c which belongs to the following class Subsequently the controller structure modified from [1], [17], [18] is exhibited in Figure 1. In this paper, we employ the design approach of the nominal MAS in [1], [17], [18] and propose the robust design to handle the model uncertainty of MAS and the influence of input external disturbances to output performance. In addition, we improve the robust control in [14] which is developed for homogeneous MAS with the norm-bounded uncertainty. The new robust design ensures the robust stabilization and reduces disturbance attenuation for homogeneour MAS with the polytopic uncertainty. The proposed approach constitutes a significant improvement over the previous works [1], [14], [17], [18]. VOLUME 4, 2016 Let Λ = diag{λ 1 , . . . , λ N } with diagonal elements being eigenvalues of L. Since the Laplacian matrix L is positive semi-definite [19], it can be diagonalized with an orthogonal matrix U ∈ R N ×N , i.e., U T LU = Λ. The following changes of variables are made, Next, substituting controller (8) into MAS model (4) and then multiplying the term (U T ⊗ I n ) to both sides of (4) yields . It is obvious that the system (11) is decomposed into N independent subsystems as follows, . It is obvious that performance of (11) can be simultaneously synthesized through those N subsystems (12).

B. DESIGN PROBLEMS
In this research, we investigate two control design problems. First, the robust H ∞ control design problem is described in the following. Problem 1 (robust H ∞ control design): Design a controller (8) such that the closed-loop system (1) satisfies the performance criterion given by Second, the robust H 2 control design problem is presented below. Problem 2 (robust H 2 control design): Let γ 2 > 0 is a given disturbance suppression index, synthesize a controller (8) to assure the closed-loop system (1) performance indicated by ∥G yd (s, δ)∥ 2 < γ 2 , ∀δ ∈ Θ for any disturbance with infinite energy d(t) ̸ = 0, d(t) ∈ R lN .

III. MAIN RESULTS
This section first provides sufficient conditions for robust H ∞ control design and robust H 2 control design and then presents their design procedures.

A. ROBUST H∞ CONTROL DESIGN
Theorem 1: Given γ ∞ > 0, the robust H ∞ control design problem is solved if there exist symmetric positive definite matrices X ik ∈ R n×n , i = 1, . . . , N, k = 1, . . . , M , G 1 , G 2 ∈ R q×n , G ∈ R n×n , and a scalar number T . The local and global controller gains are then computed by Proof: See Appendix A for the proof. Finally, letγ ∞ = γ 2 ∞ . The upper bound for robust H ∞ control can be found by solving the following optimization problem, It is worth noting that because there exist product terms between decision variables α ∞ and (G 1 , G 2 , G), the conditions (14) are BMIs. Thus, global minimizationγ ∞ in (15) cannot be found generally by utilizing convex optimization algorithms. However, it can be observed that the conditions (14) become LMIs in (X ik , G 1 , G 2 , G) if α ∞ is fixed. Furthermore, if (X ik , G 1 , G 2 , G) are fixed, then conditions (14) become LMIs in α ∞ . On the other hand, a possible way to solve this kind of optimization is to use the coordinate optimization [21], [22] The main idea of this technique is solving LMIs optimization problems in each coordinate alternatively to achieve suboptimal controller. Let us explain how to apply the iterative approach for (15) as follows. We first start with a given α ∞ and a correspondingγ ∞ . Then by fixing decision variables either α ∞ or (G 1 , G 2 , G), we alternatively solve minimization problems over LMI constraints. When the number of iterations increases, the value ofγ ∞ is monotonically decreasing. The stopping criteria of this process is either the improvement in the value ofγ ∞ is less than the desired absolute tolerance or the LMIs minimization problems are infeasible. Next, we propose the procedure of the robust H ∞ control design composing of three main steps.
1) Derive Laplacian matrix L ∈ R N ×N of graph G.
2) Solve the optimization problem (15) by utilizing Algorithm 1 to obtain solution in terms of (γ ∞ , G 1 , G 2 , G, α ∞ , X ik ) in robust H ∞ control design. 3) Compute hierarchical state feedback controller (8). Remark 1: We will compare the proposed RPHIC with a robust quadratic H ∞ control (RQHIC) that guarantees the . . , M , Laplacian matrix L, and chosen absolute tolerance ϵ ∞ . (12). Design of RQHIC is based on a common quadratic Lyapunov function. Denote γ ∞,q be the disturbance attenuation of RQHIC. Let γ ∞,q = γ 2 ∞,q . We find the Lyapunov function candidate We adopt similar steps in the proposed robust H ∞ control design to determine minimization of γ ∞,q which can be found by solving the following optimization problem. Let . The state-feedback gain of RQHIC has the form where F l∞,q = G 1 X −1 and F u∞,q = G 2 X −1 . It is noted that our design conditions using the parameterdependent Lyapunov function is less conservative than that using a common quadratic Lyapunov function for the entire uncertain domain.

B. ROBUST H2 CONTROL DESIGN
The following theorem provides a sufficient condition to design of robust H 2 control design.
. . , M , and α 2 > 0 satisfying the following LMIs The hierarchical feedback gains are calculated by Proof: See Appendix B for the proof. Lastly, denoteγ 2 ≜ γ 2 2 , then the smallest disturbance attenuation level can be found by solving the following minimization problem, is satisfied. (19) Note that the conditions (18a) are BMI constraints due to the existence of product terms including decision variables α 2 and (X ik , G 1 , G 2 , G). Nevertheless, it can be seen that the conditions (18) It then allows us to apply iterative approach [21], [22] for minimization problems (19) with LMI constraints to attain desired sub-optimal controller. Let us provide the following algorithm.
Next, a procedure of robust H 2 control design is given, which contains three main steps.
2) Solve the optimization problem (19) using Algorithm 2 to attain solution in terms of (γ 2 , Remark 2: We will compare the proposed RPH2C with a robust quadratic H 2 control (RQH2C). Design of RQH2C is based on a common matrix variable that guarantees the H 2 performance of the uncertain MAS (12). We find the Lyapunov function candidate Algorithm 2: Iterative LMI of Robust Parameter-Dependent H 2 Control (RPH2C) Input: the system matrices (A k , B k , B dk , C k , D k ), k = 1, . . . , M , Laplacian matrix L, and chosen absolute tolerance ϵ 2 .
are fixed at previous most value. Then set α 2,j = α 2 ; (iii) solve (19) to determine (X ik , G 1 , G 2 , G, E ik ) andγ 2 where α 2 is fixed at α 2,j . Then setγ 2,j = γ 2 ; Until either |γ 2,j −γ 2,j−1 | ≤ ϵ 2 or LMIs minimization problem in (ii) or (iii) is infeasible. end The following optimization problem is proposed to find the minimum disturbance attenuation obtained by RQH2C. A comparison in terms of minimum disturbance attenuation γ 2 to illustrate the improvement of our proposed design is given in the numerical example. Denoteγ 2,q = γ 2 2,q . min γ2,q,G1,G2,X,E ikγ 2,q , Then the nominal hierarchical statefeedback gain of RQH2C is in the form where F l2,q = G 1X −1 , and F u2,q = G 2X −1 . It is noted that our proposed synthesis conditions use the parameter-dependent Lyapunov function which leads to PDBMI conditions. It allows us to use multiple Lyapunov matrix variables for different vertices of uncertain agents.
Hence, PDBMI design approach is less conservative than that using a common quadratic Lyapunov function for entire uncertain domain.

IV. NUMERICAL EXAMPLES
In this section, numerical examples are given to show effectiveness of the proposed robust control designs. We make comparison between results obtained from our proposed designs with those attained from other robust control designs.
For reference in the numerical results, we shall refer the robust H ∞ control of polytopic MAS as robust parameterdependent H ∞ control (RPHIC) and robust H 2 control of polytopic MAS as robust parameter-dependent H 2 control (RPH2C). Let us refer the previous robust H ∞ control and robust H 2 control of MAS subject to norm-bounded uncertainty in [14] as RHHIC and RHH2C, respectively.
We shall compare the design results between RPHIC and RHHIC and between RPH2C and RHH2C in the first and second examples, respectively. We first describe the dynamic model of MAS subject to norm-bounded uncertainty and show the equivalent MAS subject to polytopic uncertainty. Then we apply the proposed RPHIC and RPH2C for the polytopic models and compare the computed upper bound of disturbance attenuation and maximum admissible bound of uncertainty. Subsequently, we will compare the design results between RPHIC and RQHIC and between RPH2C and RQH2C in the third and fourth examples.
We utilize a notebook computer with Intel Core i7 2.6 GHz processor, 16 GB of RAM, and 64-bit operating system to solve Example 1-3. The design in Example 4 employs a desktop computer with Intel Core i7 3.20GHz processor and 32GB of RAM. Example 1: This example aims to compare maximum uncertainty bound of two design methods, RPHIC and RHHIC. Three homogeneous dynamical agents having norm-bounded uncertainty are given [14].
where δ k , k = 1, 2 satisfy |δ k | ≤ θ, in which θ is normbounded uncertain parameter. An equivalent polytopic uncertainty form of this dynamic model is as follows. The inter-agent communication described by triangle graph G and its Laplacian matrix L are depicted in Figure 2.
The external disturbances d i (t) are any finite-energy signals, e.g., band-limited signals. We will compare the disturbance attenuation between RPHIC and RHHIC.   19.05] for both RPHIC and RHHIC. It can be seen that when θ is larger, γ ∞ increases. The smaller value of γ ∞ means better performance. Obviously, our proposed RPHIC attains significantly lower disturbance attenuation than that of RHHIC with the same value of θ. In addition, our proposed design conditions of RPHIC are still feasible with θ = 19.05 whereas the design condition of RHHIC is feasible up to the maximum θ = 13.436. Table 1 summarizes the upper bound of γ ∞ from various design conditions. It is shown in Figure 3 and Table 1 that the proposed design condition RPHIC in Theorem 1 allows for a wider range of maximum admissible uncertainty bound θ max and provides smaller disturbance attenuation γ ∞ than that of RHHIC [14]. Especially, θ max is increased by 41.78%, and γ ∞ is decreased by 89.85% when θ = 13. Therefore, design condition of RPHIC gives less conservative design results than the RHHIC in [14].
Note that the optimization problem (15) whose constraints are BMIs has the higher computational complexity when solved by Algorithm 1. This means that the achievement of less conservativeness is attained at the higher computational complexity. This example aims to compare the disturbance attenuation and the maximum achievable uncertainty bound between RPH2C and RHH2C. The MAS composes of three homogeneous dynamical agents with the following dynamic model [14] in which δ k , k = 1, 2 satisfy |δ k | ≤ θ where θ denotes norm-bounded uncertain parameter.
The model of MAS can be expressed by MAS subject to polytopic uncertainty as follows.   Figure  4. It is observed that γ 2 of RPH2C is significantly smaller than that of RHH2C when θ is gradually enlarged. With a fixed θ, our proposed RPH2C yields a much lower disturbance attenuation than that of the RHH2C. Furthermore, with θ = 15, our proposed RPH2C design still find robust state feedback, whereas the synthesis condition of RHHIC becomes infeasible. In fact, RHHIC can work with maximum uncertainty bound θ = 5.781. The upper bound of γ 2 from different design conditions are given in Table 2. Figure 4 and Table 2 indicate that the proposed design condition RPH2C VOLUME 4, 2016 in Theorem 2 allows for a broader variety of maximum admissible uncertainty bound θ max and provides lower value of disturbance attenuation γ ∞ than those of RHH2C in [14]. In particular, θ max is improved by 159.47%, and γ 2 is decreased by 97.36% as θ = 5. As a result, RPH2C approach effectively finds γ 2 for a wider parametric uncertainty range than the existing design condition and hence makes improvement over the RHH2C in [14].
Since the optimization problem (19) solved by using Algorithm 2 is subjected to BMI constraints, it has the higher computational complexity than RHH2C. Thus, at the higher cost of computational complexity, the less conservative result is achieved.
We will compare RPHIC with robust quadratic H ∞ control (RQHIC) using a common Lyapunov variable. Figure  2 demonstrates the inter-agent communication described by triangle graph G and its associated Laplacian matrix L.
State-feedback gains of RPHIC are computed by Theorem 1 using Algorithm 1. The initial parameters α ∞,0 is set to be 0.01. The desired absolute tolerance ϵ ∞ is selected to be 10 −6 . Then, we yield feasible solutions after 18 iterations. Table 3 gives state feedback gains and obtained disturbance attenuation of both aforementioned controllers. Obviously, comparing the computed γ ∞ , our proposed RPHIC provides significantly lower disturbance attenuation than that of RQHIC. However, RPHIC spends more computational time than RHHIC.
Subsequently, we design RPH2C and compare with RQH2C. By Theorem 2, state-feedback gains of RPH2C are computed using Algorithm 2. The initial parameters α 2,0 is set to be 0.01. The tolerance ϵ 2 is selected to be 10 −6 . Then, feasible solutions are found after 40 iterations. These results are summarized in Table 4. It is observed that our proposed RPH2C yields a much lower disturbance attenuation than that of the RQH2C. RPH2C spends more computational time than RQH2C.
Example 4 (vehicle suspension systems): The objective of this example is to illustrate effectiveness of proposed robust control design for a homogeneous MAS consisting of four active vehicle suspension systems.
Consider a vehicle suspension composing of one-fourth of the vehicle body mass, suspension components, and one wheel. This model with two degrees of freedom (2DOF) involving only the vertical motion of sprung and unsprung mass caused by the vertical ground displacement z r arising from road irregularities [23], [24] captures many essential features of a real suspension system. Note that the sprung mass m s represents the chassis at one corner of the vehicle. Unsprung mass m u corresponds to the wheel and axle assembly at one corner of the vehicle. The suspension stiffness k s and damping rate of suspension c s stand for passive spring and shock absorber. k t is tyre stiffness. z s and z u are vertical displacement of the sprung mass and unsprung mass. The actuator force f s severs as a control input to eliminate the vibration of the vehicle chassis. The dynamic equation on vertical motion of a vehicle suspension model is obtained by Newton's second law as follows.
The following assumptions are used.
• Damping of tyres and dynamic of the hydraulic actuator can be ignored. • States z s (t) and z u (t) are measurable. • Tyres of vehicle continuously contacts with road.
, where x 1 (t) stands for the suspension deflection, x 2 (t) represents tyre deflection, x 3 (t) and x 4 (t) denote sprung mass speed and unsprung mass speed, respectively, and u(t) is control input. The ride comfort relates to the vertical accelerationz s (t) encountered by the vehicle body. To reduce the vertical acceleration of the vehicle chassis, the vehicle's structural characteristics impose stiff limit on suspension deflection x 1 (t). Hitting the deflection limit causes uncomfortable for passengers and increases vehicle wear [25], [26]. To guarantee driving safety, the vehicle wheels need to contact continuously with road. A good road holding has good correction with tire deflection z 2 (t). Thus, the output performance which satisfies the objectives can be defined by The state-space model of each vehicle suspension system can be written bẏ In this example, we consider 4 active suspension systems. Each active suspension system is cast as an agent connecting to its neighbors by a square graph. When carrying out experiments, we vary the number of uncertainty parameters in affine model in 5 cases as given in Table 5. The uncertainty bound is caused by discrepancy around the nominal value and specified by the percentage of the nominal value. The parameters of active suspension system are assumed in Table  6.    are synthesized with varying the number of vertices M and initial α ∞,0 . We choose ϵ ∞ = 10 −5 . To this end, we select the best obtained γ ∞ . Table 7 and Table 8 summarize the results. Figure 6 illustrates the γ ∞ obtained by the proposed RPHIC and the RQHIC when varying the number of vertices M from 2 to 32. Overall, γ ∞ is increased as M increases. Obviously, RQHIC gives larger disturbance attenuation γ ∞ than that of RPHIC. The smaller value of γ ∞ is, the better performance will be. Therefore, RPHIC outperforms RQHIC.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.   Figure 7 shows CPU time of RPHIC and RQHIC designs with varying number of vertices M . It is seen that RPHIC spends more time than those of RQHIC. At M = 4, CPU time of RPHIC is higher than that of RQHIC since it requires many iterations. Nevertheless, the proposed RPHIC gives lower γ ∞ as illustrated by Figure 6.
Next, we design RPH2C for the active suspension systems. By Theorem 2, state-feedback gains of RPH2C can be computed using Algorithm 2. The initial parameter α 2,0 is varied and ϵ 2 is selected to be 10 −5 . The design results are summarized in Table 4. It is observed that our proposed RPH2C yields much lower disturbance attenuation gains than that of RQH2C. However, RPH2C requires more computational time than RHH2C.
We employ Theorem 2 to design state-feedback controller by Algorithm 2. These controller gains and γ 2 are computed for various M by varying α 2,0 . The results are given in Table  9 and Table 10.  Figure 8 shows the value of γ 2 obtained by the proposed RPH2C and the RQH2C when varying the number of vertices M . In general, RPH2C gives a smaller value of γ 2 than RQH2C. The condition of RPH2C design is capable of finding γ 2 for all various value of M , whereas RQH2C design is infeasible at the value of M of 16 and 32. When M increases, γ 2 of RPH2C is increased. In contrast, γ 2 of RQH2C is increased for M = 4 and then decreased for M = 8. The RQH2C has a drawback due to a common Lyapunov matrix when designing robust control for the entire uncertainty domain. RPH2C gives a smaller value of γ 2 than RQH2C. Thus, it can be concluded that the RPH2C is less conservative than the RQH2C. CPU time used by both the proposed RPH2C and RQH2C when varying the number of vertices M is shown in Figure  9. It is observed that RPH2C requires more time than those of RQHIC. There is no data of CPU time utilized by RQH2C at M of 16 and 32 since the RQH2C condition is infeasible. RPH2C is still capable of finding γ 2 for M = 16, 32. Note that the proposed RPH2C gives significantly lower γ 2 as depicted by Figure 8.

V. CONCLUSIONS
We present the designs of robust hierarchical H ∞ and H 2 control for multi-agent systems subject to polytopic uncertainty and external disturbance. The design criteria are based on the novel sufficient conditions in terms of BMI with specified upper bound of disturbance attenuation. We develop the iterative method of the coordinated optimization to solve subproblems over LMIs to determine the robust state feedback gains. The numerical results reveal that the novel robust control designs significantly give lower disturbance attenuation than that of the other robust controllers. In addition, the proposed designs guarantee the robust stability of MAS for a larger maximum admissible bound of uncertainty. The ongoing work is to extend the robust design of hierarchical control for heterogeneous polytopic uncertain MAS.
Appendix A Note that G + G T ≻ 0 in (14) guarantees the existence of G −1 .
Substituting M k=1 δ k = 1 into the right side of (42) yields Then isolate all term involving δ k in (43). It is sufficient to guarantee (43) by checking the sign of the finite number of trace(E ik )) − 1) < 0, k = 1, . . . , M .