We describe a decentralized formation problem for multiple robots, where an H∞ formation controller is proposed. The network of dynamic agents with external disturbances and uncertainties are discussed in formation problems. We first describe how to design social potential fields to obtain a formation with the shape of a polygon. Then, we provide a formal proof of the asymptotic stability of the system, based on the definition of a proper Lyapunov function and H∞ technique. The advantages of the proposed controller can be listed as robustness to input nonlinearity, external disturbances, and model uncertainties, while applicability on a group of any autonomous systems with n-degrees of freedom. Finally, simulation results are demonstrated for a multiagent formation problem of a group of six robots, illustrating the effective attenuation of approximation error and external disturbances, even in the case of agent failure or leader tracking.
1. Introduction
All around the world, nature presents examples of collective behavior in groups of insects, birds, and fishes. This behavior has produced sophisticated functions of the group that cannot be achieved by individual members [1, 2]. Therefore, the research on the coordination of robotic swarms has attracted considerable attention. Taking the advantages of distributed sensing and actuation, a robotic swarm can perform some cooperative tasks such as moving a large object that is usually not executable by a single robot [3–7]. Applications about the analysis and design of robotic swarms included autonomous unmanned aerial vehicles, congestion control of communication networks, and distributed sensor networks autonomous, and so forth [1, 2, 8–10].
In general, a robotic formation problem is defined as the organization of a swarm of agents into a particular shape in a 2D or 3D space [8]. This kind of control strategy can be applied into several different fields. For example, in the industrial field, this formation control strategy can be applied to a group of Automated Guided Vehicles (AGVs) moving in a warehouse for goods delivery. The main idea is to make a group of AGVs cooperatively deliver a certain amount of goods, moving in a formation. The creation of a formation with the desired shape is useful to precisely constrain the action zone of the AGVs, thus reducing the chance of collisions with other entities (e.g., human guided vehicles).
In the literature, many different approaches to formation control can be found. The main existing approaches can be divided into two categories: centralized [11] and distributed [12]. Because of the intrinsic unreliability of centralized methods, we focus our attention to distributed ones: all the agents are equal, and if one of them stops working, the other ones can still complete their task. Several formation control strategies can be found as potential fields [8], behavior-based [13], leader-following [14–16], graph-theoretic [12], and virtual structure approaches [17, 18].
In recent years, some methods based on potential fields are integrated with some nonlinear control schemes such as feedback linearization method (e.g., Sliding Mode Control (SMC)), which concludes in more robust formation control designs of dynamic agents [15–18]. For example, Takahashi et al. [15] proposed an SMC-based formation control scheme for multiple mobile robots, using the leader-following strategy, in which they defined some performance indexes, so that robots can be controlled according to their ability. Defoort et al. [16] also developed a robust coordinated control scheme based on leader-follower approach to achieve formation maneuvers. They used first- and second-order SMC to address the formation problem of N mobile robots of unicycle type with two driving wheels. Moreover, Cheaha et al. [17] presented a region-based shape controller for a swarm of fully actuated robots, where a linear approximator was used to approximate the unknown dynamic model and an SMC controller integrated with artificial potential functions was used to satisfy a predetermined geometric 2D formation.
Recently, H∞ optimal control techniques have been found to be an effective solution to treat robust stabilization and tracking problems, in presence of external disturbances and system uncertainties [19–24]. In an H∞ control technique, the main design goal is to force the gain from unmodelled dynamics, external disturbances, and approximation errors to be equal or less than a prescribed disturbance attenuation level (H∞ attenuation constraint) [19]. This goal is generally represented as a Linear Matrix Inequality (LMI) problem.
In the traditional H∞ control the exact model of the system must be known. However, in order to propose a robust control method, an integration between this robust scheme with fuzzy logic approximators can propose effective controllers for uncertain dynamic models [25].
Since Zadeh [26] initiated the fuzzy set theory, fuzzy logic systems (FLS) have been widely applied to many real world applications [27–30]. However, fuzzy control has not been viewed as a rigorous science due to a lack of formal synthesis techniques which guarantee the very basic requirements of global stability and acceptable performance. In fact, if the mathematical model of a robot is known, then conventional linear and nonlinear approximation methods should be given higher priority. However, fuzzy control should be useful in situations where (1) there is no acceptable mathematical model for the robot and (2) there are experienced human operators who can satisfactorily approximate the plant and provide qualitative control rules in terms of vague and fuzzy sentences. There are many practical situations where both (1) and (2) are true. Besides, FLS schemes have been widely used in motion control of single robots [31, 32]. Using FLS integrated with H∞ control technique can improve the robustness of controller and ensures the stability [25].
In this paper, a geometric formation is considered as the goal and an artificial potential is defined to guide the agents through this formation. A partially unknown nonlinear dynamic model is adopted to each n-degrees of freedom agent. Therefore, an adaptive interval type-2 fuzzy approximator is combined with H∞ control technique to propose a novel decentralized adaptive fuzzy formation control methodology, with robust characteristics. The main advantage of this control strategy is insensitivity to robot dynamic uncertainties, external disturbances, and input nonlinearities.
Moreover, in existing adaptive nonlinear control methodologies which are based on SMC control (e.g., [17]), each agent approximator needs to know the position and velocity of all other robots to approximate the unknown model dynamics; however, in the current proposed decentralized strategy, only the position and velocity of each robot are enough to be known to its approximator.
The rest of this paper is organized as follows: Section 2 presents the system description, problem formulation, and potential function evaluation. An introduction of interval type-2 fuzzy logics systems is described in Section 3. Design of the proposed controller and stability analysis are discussed in Sections 4 and 5, respectively. Simulation results are included in Section 6 and Section 7 provides the concluding remarks.
2. System Description and Problem Formulation
The major goal in this study is to solve a multiagent formation control problem (i.e., controlling the relative position and orientation of the agents to create a desirable formation). One of the effective solutions for this problem is using an electrostatic-like potential function design which guides the agents through continues smooth paths and avoids agent collisions. Such a potential function design has been discussed in various papers (e.g., [1, 2, 8, 18]). Therefore, in Section 2.1 we will explain a simple potential function design, in order to solve the formation control of a group of N point massless agents, where the kinematic of the ith agent is considered as
(1)z˙i=ui,i∈{1,2,…,n},
in which zi∈Rn is the coordinate matrix (for a robot with n-degrees of freedom) and ui∈Rn denotes the control inputs.
However, one of the main shortcomings of this kinematic model is that it does not correspond to the dynamics of realistic agents. To overcome this shortcoming more general dynamic models like unicycle models [33] or other wheeled vehicle models can be discussed.
In Section 2.2 one of the most general n-degrees of freedom dynamic models of real robots is considered to propose more realistic solutions for formation control of multiagent systems. The main feature of this model is that any agent (robot) with n-degrees of freedom (e.g., Autonomous Underwater Vehicles (AUVs) [34], Unmanned Aerial Vehicles (UAVs) [35, 36], etc.) can be adopted to this model.
2.1. Formation Control Massless Agents
To propose a control law, an artificial potential function is designed. This potential function can be comprised of interagent interactions, environmental effects (e.g., obstacles, goals, etc.), or other exceptional terms.
Consider the pairwise potential fields, which are defined between agents as
(2)Fij=Lij(|zi-zj|),∀i,j∈{1,2,…,n},
where Lij is designed to define a proper interagent potential function. It is assumed that each agent senses the resultant potential of all other agents.
The overall potential function is proposed to be in the form of
(3)F=∑i=1N-1∑j=i+1NLij(|zi-zj|)+∑i=1NQi(|zi|),
where Qi defines the global potential of each agent.
Finally, the following three assumptions for potential function are considered [17, 18].
Assumption 1.
Fis continuously differentiable.
Assumption 2.
F is strictly convex.
Assumption 3.
F is positive definite.
For example, the following potential function can be chosen for a desired polygonal formation in a 2D Space:
(4)F=∑i=1N-1∑j=i+1N(|zi-zj|2-dij)2+∑i=1N(|zi|2-ri)2.
At the first step, to propose a solution for multiagent formation control, the steepest descent direction [8, 17, 18] is chosen as
(5)fi=∂F∂zi,
and the control law
(6)ui=-fi,∀i∈{1,2,…,n}
is proposed.
By substituting (6) in (1) the kinematic model is obtained as
(7)z˙i=-fi=-∂F∂zi,∀i∈{1,2,…,n},
which can be rewritten in the matrix form as Z˙=-∇F where Z=[z1,z2,...,zn] is the overall generalized coordinate vector.
In the next subsection, it is proposed to assume the multiagent system with a general dynamic model. Furthermore, in Section 3 a robust adaptive fuzzy controller using an H∞ approach is used to force the satisfaction of (7). In other words the proposed controller is designed to enforce the speed of each agent along the negative gradient of potential function in (7).
2.2. Formation Control of Robots with Dynamic Models
In this subsection a general dynamic model [37] is addressed to represent any kind of autonomous n-degrees of freedom system. This model has been previously used in some existing works (e.g., [17, 18]).
Consider a group of N fully autonomous agents. The dynamics of the ith simple agent is strongly nonlinear [37] and can be written in the general form
(8)M(zi)z¨i+C(zi,z˙i)z˙i+g(zi)=ui,
where zi∈Rn is the coordinate matrix (for a robot with n-degrees of freedom); M(zi)∈Rn×n is a symmetric positive definite matrix and represents the inertia coefficients. C(zi,z˙i)∈Rn×n is the matrix of centripetal, Coriolis, damping, and rolling resistance forces; g(zi)∈Rn is an n-vector of gravitational forces and ui∈Rn denotes the control inputs.
In most practical control problems of multiagent systems the inertia matrix M(zi) is a known constant matrix independent of zi. Therefore, the following assumption is considered.
Assumption 4.
M is the inertia matrix of robots, which is assumed to be a known and constant matrix.
Let us rewrite (8) as
(9)Mz¨i+C(zi,z˙i)z˙i+g(zi)=ui.
It is straightforward to rewrite (9) as
(10)z¨i=-M-1C(zi,z˙i)z˙i-M-1g(zi)+M-1ui.
In the next sections, the dynamic of each single agent will be assumed to be in the form of (10).
3. Interval Type-2 Fuzzy Logic System
In this section, the interval type-2 fuzzy set and the inference of the type-2 fuzzy logic system will be presented. A type-2 fuzzy set in universal set X is denoted as A~ which is characterized by a type-2 membership function uA~(x) in (12). The uA~(x) can be referred to as a secondary membership function or referred to as a secondary set, which is a type-1 fuzzy set in [0,1]. In (13), fx(u) is a secondary grade, which is the amplitude of a secondary membership function; that is, 0≤fx(u)≤1. The domain of a secondary membership function is called the primary membership of x. In (13), Jx is the primary membership of x, where u∈Jx⊆[0,1]for all x∈X; μ is a fuzzy set in [0,1], rather than a crisp point in [0,1],
(11)A~=∫x∈XuA~(x)x=∫x∈X[∫u∈Jxfx(u)/u]x,Jx⊆[0,1].
When fx(u)=1,for allu∈Jx⊆[0,1], then the secondary MFs are interval sets such that uA~(x) in (13) can be called an interval type-2 MF. Therefore, the type-2 fuzzy set can be rewritten as
(12)A~=∫x∈XuA~(x)x=∫x∈X[∫u∈Jx1/u]x,Jx⊆[0,1].
Also, a Gaussian primary MF with uncertain mean and fixed standard deviation having an interval type-2 secondary MF can be called an interval type-2 Gaussian MF (13). It can be expressed as
(13)uA~(x)=exp[-12(x-mσ)2],m∈[m1,m2].
It is obvious that the type-2 fuzzy set is in a region, called a footprint of uncertainty (FOU) and is bounded by an upper MF and a lower MF, which are denoted as u¯A~(x) and u_A~(x), respectively. Hence, (13) can be reexpressed as
(14)A~=∫x∈X[∫u∈[u_A~(x),u¯A~(x)]1/u]x.
A type-2 fuzzy logic system (FLS) is very similar to a type-1 FLS as shown in Figure 1; the major structure difference being that the defuzzifier block of a type-1 FLS is replaced by the output processing block in a type-2 FLS which consists of type-reduction followed by defuzzification.
The structure of the type-2 fuzzy logic system.
There are five main parts in a type-2 FLS: fuzzifier, rule base, inference engine, type reducer, and defuzzifier. A type-2 FLS is a mapping f:ℜp→ℜ1. After defuzzification, fuzzy inference, type reduction, and defuzzification, a crisp output can be obtained.
Consider a type-2 FLS having p inputs x1∈X1,…,x1∈Xp and one output y∈Y. The type-2 fuzzy rule base consists of a collection of IF-THEN rules, as in the type-1 case. We assume there are M rules and the rule of a type-2 relation between the input space X1×X2×⋯×Xp and the output space Y can be expressed as
(15)R1:IFx1isF~1land…andxpisF~pl,THENyisG~l,l=1,2,…,M,
where F~jl are antecedent type-2 sets (j=1,2,…,p) and G~ls are consequent type-2 sets.
The inference engine combines rules and gives a mapping from input type-2 fuzzy sets to output type-2 fuzzy sets. To achieve this process, we have to compute unions and intersection of type-2 sets, as well as compositions of type-2 relations. The output of inference engine block is a type-2 set. By using the extension principle of type-1 defuzzification method, type-reduction takes us from type-2 output sets of the FLS to a type-1 set called the “type-reduced set.” This set may then be defuzzified to obtain a single crisp value.
There are many kinds of type-reduction, such as centroid, height, modified weight, and center-of-sets. The center-of-sets type reduction will be used in this paper and can be expressed as:
(16)Ycos(Y1,…,YM,F1,…,FM)=[yl,yr]=∫y1⋯∫yM∫f1⋯∫fM1∑i=1Mfiyi/∑i=1Mfi,
where Ycos is the interval set determined by two end points yl and yr, and fi∈Fi=[f_i,f¯i]. In the meantime, an interval type-2 FLS with singleton fuzzification and meet under minimum or product t-norm f_i and f¯i can be obtained as
(17)fi_=μ_F~1i(x1)*⋯*μ_F~pi(xp),(18)f¯i=μ¯F~1i(x1)*⋯*μ¯F~pi(xp).
Also, yi∈Yi and Yi=[yli,yri] are the centroid of the type-2 interval consequent set G~i. For any value y∈Ycos, y can be expressed as
(19)y=∑i=1Mfiyi∑i=1Mfi,
where y is a monotonic increasing function with respect to yi. Also, yl is the minimum associated only with yli, and yr is the maximum associated only with yri. Note that yl and yr depend only on mixture of f_i or f¯i values. Therefore, the left-most point yl and the right-most point yr can be expressed as a fuzzy basis function (FBF) expansion, that is,
(20)yl=∑i=1Mfliyli∑i=1Mfli=∑i=1Myliξli,yr=∑i=1Mfriyri∑i=1Mfri=∑i=1Myriξri,
respectively, where ξli=fli/∑i=1Mfi and ξri=fri/∑i=1Mfi.
If the FBF vector denoted as ξ_l=[ξl1,ξl2,…,ξlM] and ξ_r=[ξr1,ξr2,…,ξrM], and let y_lT=[yl1,yl2,…,ylM] and y_rT=[yr1,yr2,…,yrM], then (20) can be rewritten as
(21)yl=∑i=1Mfliyli∑i=1Mfli=∑i=1Myliξli=y_lTξ_l,(22)yr=∑i=1Mfriyri∑i=1Mfri=∑i=1Myriξri=y_rTξ_r.
For illustrative purposes, we briefly provide the computation procedure for yr. Without loss of generality, assume the yris are arranged in ascending order, that is, yr1≤yr2≤⋯≤yrM.
Step 1.
Compute yr in (22) by initially setting fri=(f¯i+f_i)/2 for i=1,2,…,M, where f_i and f¯i have been precomputed by (18) and (19) and let yr′=yr.
Step 2.
Find R(1≤R≤M-1) such that yrR≤yr′≤yrR+1.
Step 3.
Compute yr in (22) with fri=f_i for i≤R and fri=f¯i for i>R and let yr′′=yr.
Step 4.
If yr′′≠yr′, then go to Step 5, If yr′′=yr′, then stop and set yr=yr′′.
Step 5.
Set yr′ equal to yr′′ and return to Step 2.
The point to separate two sides by number R can be decided from the above algorithm, one side using lower firing strengths f_i’s and another side using upper firing strengths f¯i’s. Therefore, the yr in (22) can be rewritten as
(23)yr=∑i=1Rf_iyri+∑i=R+1Mf¯iyri∑i=1Rf_i+∑i=R+1Mf¯i=∑i=1Rq_riyri+∑i=R+1Mq¯riyri=[Q_rQ¯r][y_ry¯r]=ξ_rTΘ_r,
where q_ri=f_i/Dr, q¯ri=f¯i/Dr and Dr=(∑i=1Rf_i+∑i=R+1Mf¯i). In the meantime, we have Q_r=[q_r1,q_r2,…,q_rR], Q¯r=[q¯r1,q¯r2,…,q¯rR], ξ_rT=[Q_rQ¯r], and Θ_rT=[y_ry¯r].
The procedure to compute yl is similar to compute yr. Just in Step 2, we determine L(1≤L≤M-1), such that ylL≤yl′≤ylL+1 and in Step 3 let fli=f¯i for i≤L and fli=f_i for i>L. Then yl in (21) can also be rewritten as
(24)yl=∑i=1Lf¯iyli+∑i=L+1Mf_iyli∑i=1Lf¯i+∑i=L+1Mf_i=∑i=1Lq¯liyli+∑i=L+1Mq_liyli=[Q¯lQ_l][y¯ly_l]=ξ_lTΘ_l,
where q_li=f_i/Dl, q¯li=f¯i/Dl and Dl=(∑i=1Lf¯i+∑i=L+1Mf_i). In the meantime, we have Q_l=[q_l1,q_l2,…,q_lR], Q¯l=[q¯l1,q¯l2,…,q¯lR], ξ_lT=[Q¯lQ_l], and Θ_lT=[y¯ly_l].
The defuzzified crisp value from an interval type-2 FLS is obtained as
(25)y(x_)=yl+yr2=12(ξ_rTΘ_r+ξ_lTΘ_l)=12[ξ_rTξ_lT][Θ_rΘ_l]=ξ_TΘ_,
where (1/2)[ξ_rTξ_lT]=ξ_T and [Θ_rTΘ_lT]=Θ_T.
4. Controller Design Methodology
In this section a novel formation error based on the integral of formation gradient (5) will be proposed. Then, a robust H∞ controller will be designed and a fuzzy logic system will be utilized to approximate the unknown parts of dynamic models. The main feature of the proposed novel control scheme is its decentralized characteristic, robustness to external disturbances, input nonlinearities, and measurement noises. Besides, by using the proposed controller, the formation can be achieved from any initial conditions.
Consider, the novel formation error for the ith robot as
(26)e_i(t)=zi(t)+∫0tfi(τ)dτ,
where e_i∈Rn, zi represents the coordinate vector of ith robot in (10) and fi is the gradient of potential function defined in (7). It is straightforward to write the first and second derivatives of (26) as
(27)e˙_i(t)=z˙i(t)+fi,e¨_i(t)=z¨i(t)+f˙i.
Our design goal is to propose an adaptive fuzzy controller so that
(28)e¨_i+k1e˙_i+k2e_i=0
is achieved, where k1 and k2 are chosen to make (28) asymptotically stable.
To design the controller, consider the control law proposed as
(29)ui=M(Hi(zi,z˙i)-f˙i-k1e˙_i-k2e_i),
where
(30)Hi(zi,z˙i)=M-1Ci(zi,z˙i)z˙i+M-1g(zi).
In order to use this control law, which is designed based on the feedback linearization control method, the function Hi(·) (i.e., C(·) and g(·)) must be known. However, in practice these matrices may be unknown for most of real dynamical robots. To overcome this, we make use of an adaptive fuzzy logic system H^i(·) to approximate Hi(·).
Therefore, using the singleton fuzzifier, product inference, and weighted average defuzzifier [38], the output of the fuzzy model can be expressed as
(31)H^i(zi,z˙i∣θ_i)=12(ζ_ilT(zi,z˙i)θ_il+ζ_irT(zi,z˙i)θ_ir),
where
(32)ζ_il=[ζ_1ilT0⋯00ζ_2ilT⋯0⋮⋮⋱⋮00⋯ζ_nilT],θ_il=[θ_1ilθ_2il⋮θ_nil],ζ_ir=[ζ_1irT0⋯00ζ_2irT⋯0⋮⋮⋱⋮00⋯ζ_nirT],θ_ir=[θ_1irθ_2ir⋮θ_nir].
Equation (31) suggests to us to rewrite the overall control law (29) as
(33)ui=M(H^i(zi,z˙i∣θ_i)-f˙i-kTe_i-u_ai),
where uai is engaged to attenuate the fuzzy logic approximation error and external disturbances.
A block diagram of the proposed control methodology is shown in Figure 2.
Block diagram of the proposed adaptive fuzzy H∞ control scheme.
Remark 1 (input nonlinearity).
In what follows, another main property of the proposed H∞ is introduced. It will be shown that the formation problem of multiagent system can be achieved even in the presence of dead-zone nonlinearities of the control actuators. Let us modify the dynamic model (9) as
(34)Mz¨i+C(zi,z˙i)z˙i+g(zi)=Φ(ui)+di(t)
where
(35)Φ(ui)=[ϕ(ui1)ϕ(ui2)⋮ϕ(uin)]
and ϕ(·):R→R represents the dead-zone function and can be expressed as
(36)ϕ(u)={m(u-b)u≥b,0-b<u<bm(u+b)u≤-b,,
where b is the width of the dead-zone and m is the slope of dead-zone line.
The dead-zone parameters b and m are assumed to be bounded and the bounds of m and b are known as b∈[bmin,bmax] and m∈[mmin,mmax]. Therefore (36) can be rewritten as
(37)ϕ(u)=mu+ν(u),ν(u)={-mbu≥b,-mu-b<u<b,mbu≤-b.
From the aforementioned assumption on bounds of m and b, ν(u) can be assumed bounded, (i.e., ν(u)≤ρ), where ρ is the known upper bound that can be chosen as ρ=mbmax. By considering d-i(t)=di(t)+Υ(ui) and u-i=Mui, where
(38)Υ(ui)=[ν1(ui1)ν2(ui2)⋮νn(uin)],M=mIn×n
then (34) can be rewritten as
(39)Mz¨i+C(zi,z˙i)z˙i+g(zi)=u-i+d-i(t),
which is the same as (9). Therefore, we have proved that the proposed H∞ feedback controller is also robust to dead-zone input nonlinearities (36).
5. Stability Analysis
This section presents the stability proof of the proposed novel adaptive fuzzy controller in (33). A Lyapunov candidate will be proposed and then an adaptation law and a robust compensator control input will be derived to satisfy the H∞ tracking performance in (50).
To derive the adaptive law for adjusting θ_i, we first define the optimal parameter vector θ_i* as
(40)θ_i*=argminθ_i∈Ω[sup∥H^i(z,z˙∣θi)-Hi(z,z˙)∥],
and the minimum approximation error is defined as
(41)w_i=Hi(zi,z˙i)-H^i(zi,z˙i∣θ_i*),
where it can be assumed that w_i∈L∞ [38].
By choosing the control input as [39] after some manipulations, (10) can be rewritten as
(42)z¨+f˙i=(H^i(zi,z˙i∣θ_i)-Hi(zi,z˙i))+k1e˙_i+k2e_i-u_ai,
and the formation error dynamic can be expressed as
(43)e¨_i=(H^i(zi,z˙i∣θ_i)-Hi(zi,z˙i))+k1e˙_i+k2e_i-u_ai.
Moreover by defining E_i=[e1i,e˙1i,e2i,e˙2i,…,eni,e˙ni] it is straightforward to write
(44)E˙_i=AE_i+Bu_ai+B(Hi(zi,z˙i)-H^i(zi,z˙i∣θ_i)),
where
(45)A=In×n⊗[01-k2-k1]2×2,B=In×n⊗[01]T.
Based on (31), (40), and (41), the matrix form of formation error in (44) can be rewritten as
(46)E˙_i=AE_i+Bu_ai+Bζ_iT(zi,z˙i)θ~_i+Bw_i,
where θ~_i=θ_i-θ_i*.
In the following theorem, it will be shown that the proposed control law (33) guarantees the stability and robustness of formation problem.
Theorem 2.
Consider a group of N fully autonomous agents with the dynamic represented in (8) and with the control law in (33). The robust compensator of ith robot u_ai and the fuzzy adaptation law are chosen as
(47)u_ai=-1rBTPE_i,(48)θ˙_il=-γζ_il(zi,z˙i)BTPE_i,θ˙_ir=-γζ_ir(zi,z˙i)BTPE_i,
where r and γ are positive constants and P is the positive semidefinite solution of following Riccati-like equation:
(49)PA+ATP+Q-2rPBBTP+1ρ2PBBTP=0,
where Q is a positive semidefinite matrix and 2ρ2≥r.
Therefore, the H∞ tracking performance
(50)∑i=1N[-∫0TE_iTQE_idt]≤∑i=1N[E_i(0)TPE_i(0)+12γθ~_(0)ilTθ~_il(0)+12γθ~_(0)irTθ~_ir(0)]+∑i=1N[ρ2∫0Tw_iTw_idt]
can be achieved for a prescribed attenuation level ρ and all the variables of closed loop system are bounded.
In order to derive the adaptive law for adjusting θ_i, the Lyapunov candidate is chosen as
(51)V=∑i=1N[12E_iTPE_i+14γθ~_ilTθ~_il+14γθ~_irTθ~_ir].
Using (46), the time derivative of V is
(52)V˙=12∑i=1N[E˙_iTPE_i+E_iTPE˙_i+12γθ~˙_ilTθ~_il+12γθ~_ilTθ~˙_il+12γθ~˙_irTθ~_ir+12γθ~_irTθ~˙_ir]=12∑i=1N[{E_iTPBζ_irT(zi,z˙i)θ~_ir}E_iTATPE_i+u_aiTBTPE_i+12θ~_ilTζ_il(zi,z˙i)BTPE_i+12θ~_irTζ_ir(zi,z˙i)BTPE_i+w_iTBTPE_i+E_iTPAE_i+E_iTPBu_ai+12E_iTPBζ_ilT(zi,z˙i)θ~_il+E_iTPBζ_irT(zi,z˙i)θ~_ir+E_iTPBw_i]+14∑i=1N[1γθ~˙_ilTθ~_il+1γθ~_ilTθ~˙_il+1γθ~˙_irTθ~_ir+1γθ~_irTθ~˙_ir].
Substituting (47) in (52) and using the fact that θ~˙_il=θ˙_il,θ~˙_ir=θ˙_ir, we get
(53)V˙=12∑i=1N[E_iTATPE_i-1rE_iTPBBTPE_i+12θ~_ilTζ_il(zi,z˙i)BTPE_i+12θ~_irTζ_ir(zi,z˙i)BTPE_i+w_iTBTPE_i+E_iTPAE_i-1rE_iTPBBTPE_i+12E_iTPBζ_ilT(zi,z˙i)θ~_il+12E_iTPBζ_irT(zi,z˙i)θ~_ir+E_iTPBw_i]+14∑i=1N[1γθ˙_ilTθ~_il+1γθ~_ilTθ˙_il+1γθ˙_irTθ~_ir+1γθ~_irTθ˙_ir]×12∑i=1N[E_iT(ATP+PA-2rP_BBTP)E_i]+14∑i=1N[(E_iTPBζ_ilT(zi,z˙i)+1γθ˙_ilT)θ~_il]+14∑i=1N[(E_iTPBζ_irT(zi,z˙i)+1γθ˙_irT)θ~_ir]+12∑i=1N[w_iTBTPE_i+E_iTPBw_i].
Using adaptation law (48) and the Riccati-like equation (49), the above equation becomes
(54)V˙=12∑i=1N[-E_iTQE_i-1ρ2E_iTPBBTPE_i]+12∑i=1N[w_iTBTPE_i+E_iTPBw_i]×12∑i=1N[-E_iTQE_i-(1ρBTPE_i-ρw_i)T(1ρBTPE_i-ρw_i)]+12∑i=1N[ρ2w_iTw_i]≤12∑i=1N[-E_iTQE_i+ρ2w_iTw_i].
Integrating the above inequality from t=0 to T yields to
(55)V(T)-V(0)≤12∑i=1N[-∫0TE_iTQE_idt+ρ2∫0Tw_iTw_idt].
Using the fact that V(T)≥0 and from (49), the inequality
(56)12∑i=1N[-∫0TE_iTQE_idt]≤12∑i=1N[E_i(0)TPE_i(0)+12γθ~_(0)ilTθ~_il(0)+12γθ~_(0)irTθ~_ir(0)]+12∑i=1N[ρ2∫0Tw_iTw_idt]
is obtained.
Therefore, the H∞ tracking equation (50) can be achieved and the proof is completed.
6. Simulation Results
This section presents four simulation examples to illustrate the effectiveness of the proposed control scheme. In the first example, a group of six agents with known dynamics as in (8) is considered. The second example presents the hexagonal formation of six partially unknown agents and an adaptive fuzzy logic system is used to approximate the unknown dynamics. This example proves the system stability under the proposed novel controller. In order to prove the controller robustness, in the third example a white Gaussian noise is applied to all measured data and one of the agents is forced to be stationary and still the formation maintains its stabilizing performance. In the forth example one of the agents is chosen as the leader with a constant velocity. It is shown that proposed controller is able to form a dynamic 2D moving hexagon which tracks the leader. All the simulation results are implemented in MATLAB with 0.01 secs as the stepsize.
The unique formation problem used in all five simulation examples is a 2D hexagon with unit radius defined by
(57)F=∑i=15∑j=i+16(|zi-zj|2-dij)2,
where dij is specified in Table 1.
Parameter specifications of hexagonal formation.
|i-j|=1
|i-j|=2
|i-j|=3
|i-j|=4
|i-j|=5
dij
1.0
1.7
2.0
1.7
1.0
In addition six random points in the 2D space are chosen to be the initial positions for six agents. These points are assumed to be fixed in all five numerical simulations (Table 2).
Agents initial positions.
Agent number
1
2
3
4
5
6
x0
−2.5
+2.0
−1.0
+1.0
+2.0
+2.5
y0
+1.0
−2.5
+1.0
−1.0
+2.5
−1.0
6.1. Example I (Six Agents with Known Dynamics)
Consider a group of six mobile agents with known dynamic models. Based on general model represented in (8), the nonlinear dynamic of the ith robot is considered as
(58)[0.6000.6][x¨iy¨i]+[0.15000.15][x˙iy˙i]+[0.2000.2][sgn(x˙i)sgn(y˙i)]=ui,
and after some manipulations we get
(59)[x¨iy¨i]=-[0.25000.25][x˙iy˙i]-[0.33000.33][sgn(x˙i)sgn(y˙i)]+[1.66001.66]ui.
To give a solution for the formation problem, formation error is defined as (26) and the control law is designed based on (29), where k1=15 and k2=4. Figure 3(a) shows the formation trajectory of six robots starting from initial conditions (Table 2) to the final unit hexagon (57) in 30 secs and Figure 3(b) shows the potential value.
Hexagonal formation of six agents with known dynamics. (a) Formation trajectory. (b) Formation potential.
The first subfigure (Figure 3(a)) shows how smooth the controller guides all the agents to form the desired hexagon. This geometric formation does not have any fixed position or direction, and it will only be determined by the agents initial position.
The second sub-figure (Figure 3(b)) illustrates the potential decrement through the time. It is shown that the potential is forced to get stabilized in less than 20 secs and it will be shown that the settling time for next simulation examples will more than 20 secs.
6.2. Example II (Six Agents with Partially Unknown Dynamics)
To verify the effectiveness of proposed method the same novel formation error and (26) are chosen, respectively. Consider a group of six agents with the same dynamic models as (59). However, to design the control law, the dynamic model of agents is assumed to be partially unknown (i.e., C(·) and g(·) in (8) are unknown).
Therefore, six fuzzy logic approximators are designed to approximate the unknown dynamic, where each agent approximator just needs the current position and velocity of itself. Three Gaussian membership functions with unit variance are defined and all θ_s are initialized from zero vectors. The learning rate in (48) is set to γ=15 and the output of the fuzzy system is achieved by choosing singleton fuzzification, center average defuzzification, Mamdani implication in the rule base, and product inference engine [38].
Simulation results of the proposed adaptive fuzzy H∞ technique with agents initial positions as shown in Table 2 are shown as following. The motion trajectory in the first 30 secs is illustrated in Figure 4(a) and the formation potential (57) is shown to be stabilized in Figure 4(b).
Hexagonal formation of six agents with partially unknown dynamics. (a) Formation trajectory. (b) Formation potential.
6.3. Example III (Formation Problem in Presence of Measurement Noise and Agent Failure)
In this example the robustness of proposed controller in presence of measurement noise and agent failure will be proved. The proposed potential function (3) and gradient-based method proposed in Section 2 are able to obtain the exact formation even in the case of one agent failure. Therefore, in this example it will be shown that when Agent #3 (x3(0)=-1,y3(0)=+1) is forced to be stationary with zero velocity, other agents move toward this agent to achieve the hexagon formation. In addition a white Gaussian noise with SNR=20 db is applied to all the measured data. All of the model characteristics and controller designs are the same as previous example in Section 6.2. Motion trajectory and formation potential (57) of the first 90 secs of simulation are shown in Figures 5(a) and 5(b), respectively.
Hexagonal formation in presence of 20 db noise and one agent failure. (a) Formation trajectory. (b) Formation potential.
6.4. Example IV (Formation Problem While Tracking the Leader)
Previous example illustrated the good performance of formation stabilization, while agent failure (i.e., one agent remains stationary). However, the structure of potential function explained in (3) suggests to exempt one agent from the control law designed in (33), and let it move freely as the leader [8]. Therefore, to run a more general simulation than previous example where one agent was stationary, here one of the agents is chosen as the leader and moves with a constant speed to a predefined direction. Then it is anticipated that, after some transient formation, the agents position achieves the hexagon form in (57). All the problem parameters and controller design are the same as previous examples (i.e., (57), (59) and (42)). Agent #4 (x4(0)=+1,y4(0)=-1) is chosen as the leader, with constant velocity as
(60)x˙4=+0.030,y˙4=-0.005.
The motion trajectory and formation potential (57) are shown in Figures 6(a) and 6(b), respectively.
Moving hexagonal formation while tracking the leader. (a) Formation trajectory. (b) Formation potential.
Simulation results prove that by using the same control law as (33) even the moving formation can be achieved. It can be seen that the formation is achieved in about 70 secs; however, this numerical simulation contains negligible steady state error (Figure 6(b)).
7. Conclusion
In this paper, the formation control problem of a class of multiagent systems with partially unknown dynamics was investigated. On the basis of the Lyapunov stability theory, a novel decentralized adaptive fuzzy controller with corresponding parameter update law was developed and the stability of the system was proved even in the case of external disturbances and input nonlinearities. All the theoretical results were verified by simulation examples and good performance of the proposed controller was shown even in the case of agent failure, presence of measurement noise, and even moving formations.
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