Abstract

In this paper, the consensus problem for general linear time-invariant (LTI) multi-agent systems (MASs) with a single input is studied in a new optimal control framework. The optimal cooperative control law is designed from a modified linear quadratic regulator (LQR) method and an inverse optimal control formulation. Three cost function terms are constructed to address the consensus, control effort, and cooperative tracking, respectively. Three distinct features of this approach can be achieved. First, the optimal feedback control law is derived analytically without involving any numerical solution. Second, this formulation guarantees both asymptotic stability and optimality. Third, the cooperative control law is distributed and only requires local information based on the communication topology to enable the agents to achieve consensus and track a desired trajectory. The performance of this optimal cooperative control method is demonstrated through an example of attitude synchronization of multiple satellites.

1 Introduction

Cooperative control problem for multi-agent systems (MASs) has received tremendous attention in the last two decades owing to its wide range of applications in mobile robots, unmanned aerial vehicles, autonomous underwater vehicles, spacecraft, and automated highway systems [1]. The distributed multi-agent cooperative systems are enabled by rapid progress in communication, sensing, and actuation. The cooperative teamwork provides much more flexibility and robustness in performance to accomplish certain missions compared to single-agent systems. A comprehensive overview on the progress of multi-agent coordination has been provided by Cao et al. [2].

The technical core of cooperative control of MASs is the concept of consensus. The consensus among agents is equivalent to achieving a common state by neighbor-to-neighbor interaction and information exchange, which can be modeled by graph theory. The classical consensus problem and the associated graph and matrix theories were introduced and extensively studied for single or double integrator systems [37]. The basic consensus algorithm was generalized to the higher order systems in Refs. [810]. Other recent progress on the average consensus algorithms includes developing novel weighting strategies [11] and new classes of fixed-time protocols [12]. Considering the constraints of real applications such as limited communication ranges and limited bandwidth, the concept of switching topology (instead of time-fixed topology) emerged and the conditions on achieving consensus under switching or time-varying topologies were investigated [13,14]. Besides, the leader–follower algorithm as the most straightforward approach for solving the tracking problem has been studied [1517].

Multi-agent cooperative control has also been investigated from the optimization perspective. The main research along this line focused on making the most efficient network configuration to attain the desired property. Kim and Mesbahi [18] derived the fastest convergence rate by maximizing the second smallest non-negative eigenvalue of the Laplacian matrix. Cao and Ren [19] investigated the optimal topology (optimal Laplacian matrix) for a given performance index via the linear quadratic regulator (LQR) method. However, only single-integrator systems are considered. It addressed the consensus problem from an inverse optimal control perspective in the sense that any symmetric Laplacian matrix is optimal with respect to a properly chosen cost function. In Ref. [20], synchronization of identical general linear systems based on the LQR was proposed and an active leader node or control node generated the desired tracking trajectories. The solution to the algebraic Riccati equation (ARE) was used to design the feedback control gain such that the synchronization error dynamics is asymptotically stable. However, the cost function is not clearly defined. Tuna [21] showed that a linear feedback control law obtained from the ARE based on LQR can synchronize agents if the linear systems of identical agents are stabilizable. Similar to Ref. [20], there was no discussion on the optimality and physical meaning of the cost function associated with the ARE. Wang and Xin [22] proposed an inverse optimal control (introduced by Bernstein [23]) for a system of agents to address not only consensus but also obstacle/collision avoidance. This similar approach was applied to cooperative control of multiple autonomous robots [24] and the flocking problem [25]. However, the same simplified double-integrator dynamics is assumed in these works for the MAS model. In Ref. [26], optimal Laplacian matrices can be obtained for second-order systems under independent position and velocity topology using LQR. Recently, many innovations have been made in solving the consensus problem by the optimal control theory. Xie and Lin [27] solved the problem of global optimal consensus for higher order integrators with bounded controls starting from any arbitrary initial states and in the presence of actuator saturation. Dehshalie et al. [28] used optimal control theory to design a fault tolerant control law for MASs with single and multi-input actuators under both directed and undirected communication topologies. A numerical approach was developed by Bailo et al. [29] to solve the consensus problem of the nonlinear multi-agent system of Cucker–Smale type and the first-order optimality conditions were obtained by using the Barzilai–Borwein gradient descent method. Recently, the model predictive control has been used to solve cooperative control problems such as controlling connected and autonomous vehicles in the intelligent transportation system [30].

In this study, the main contribution is generalizing the approach in Refs. [22,24], and [25] to solve the consensus problem for general single-input linear time-invariant (LTI) systems. The proposed method is developed from a modified LQR and an inverse optimal control formulation to achieve optimal consensus and trajectory tracking, respectively. The main features of this work can be summarized as follows: (1) the proposed optimal cooperative control can be applied to general LTI systems (rather than mere integrator systems); (2) the control law is guaranteed to be both stabilizing and optimal; (3) the optimal control law is obtained in an analytical form without using any iterative numerical algorithm; and (4) the cooperative control law is a linear function of the Laplacian matrix, and is thus distributed, which only needs neighboring agents' information.

The remainder of this paper is organized as follows: In Sec. 2, some preliminaries and fundamental concepts of graph theory are reviewed. The system is described and the problem is formulated in Sec. 3. Main results are presented in Sec. 4. The performance of the proposed method is demonstrated via an attitude synchronization example in Sec. 5 and conclusion remarks are given in Sec. 6.

2 Preliminaries

Since communication between agents can be modeled by graphs, some basic notions of the graph theory are briefly reviewed [31]. G=(N,E) represents a graph with a nonempty set of nodes N and a nonempty set of edges E where EN×N. A graph is called “directed” if all the pairs are ordered. A directed path is a sequence of ordered edges in the form of (i1,i2),(i2,i3),, where ijN. (i1,i2) represents agent 2 can receive information from agent 1 but not vice versa. In an undirected graph, (i1,i2) means that agent 1 and agent 2 can obtain information from each other. An undirected graph is called “connected” if a path can be found between each pair of nodes. Communication topology is described by the adjacency matrix Ad=[aij(d)]ng×ng in which aii(d)=0 and
(1)
where ng is the number of agents. For an undirected graph, the adjacency matrix is symmetric. The Laplacian matrix of a system of agents is defined as
(2)
where D is the diagonal degree matrix with the entries of dii=j=1ngaij(d). For an undirected graph, the Laplacian matrix L is symmetric and has a simple zero eigenvalue with an associated eigenvector 1ng (a column vector of all ones), and all the other eigenvalues positive if and only if the graph is connected. Every row sum of L is zero, i.e.,
(3)

3 Problem Statement and Formulation

A general controllable LTI system is said to achieve consensus when all agents' states converge to the same common value, (i.e., XiXj0) at the same time [1], where Xi is the state vector of agent i.

Equivalently, consensus is reached if there exists a real-valued function Xcs(t) such that X(t)Xcs(t)0 as t. X is the state vector of the whole system of agents defined as

where Xij represents the ith state of the jth agent.

In this study, it is assumed that each agent has the same dynamics of
(4)
The dynamics of the whole system of agents can be written as
(5)

where A=AgIng=[aij]n×nIng,B=BgIng=[bj]n×1Ing. n is the dimension of the single agent's system and U=[U1U2Ung]T is the control input vector for the agents.

The final consensus state vector should satisfy the system Eq. (5). Since Ucs=0nng×1 when the system achieves consensus, we have
(6)
An error state vector is defined as
(7)
Taking the time derivative of the error state vector yields
(8)

Consensus is said to be reached if the system (8) is asymptotically stable.

For any controllable LTI system, a transformation matrix T can be found to transform the system to a controllable canonical form [32]
(9)
where
(10)
is the controllability matrix and W is called flipped Toeplitz matrix with the form of
(11)

ai's are the coefficients of the characteristic equation of the state matrix Ag.

Using the transformation matrix (9), the state vector and the final consensus vector can be expressed as
(12a)
(12b)
Subtracting Eq. (12b) from Eq. (12a) leads to
(13)
Using Eq. (13), the system (8) can be converted to a controllable canonical form as
(14)
where
(15)
The last row of A¯ contains the coefficients of the characteristic equation of the system state matrix Ag and
(16)
Now, the system (14) can be rewritten as
(17)
where
(18)
and
(19)
The new equivalent system has the form of
(20)
where
(21)

The consensus problem becomes finding a feedback control input U2 such that the system (20) is asymptotically stable.

In this paper, the cooperative control problem is formulated in an optimal control framework as follows:
(22)
J1 is the cost for state deviations or consensus cost given by
(23)

where R1 is a (nng×nng) diagonal and positive semidefinite matrix. An approach for constructing this matrix will be shown in Sec. 4.

J2 is the control effort cost given by
(24)

where R2=wc2Ing is positive definite and wc is the weighting parameter.

J3 is the cost for tracking and has the form of
(25)

where h(X¯̂) is constructed by an inverse optimal control approach and contains the tracking penalty function, which will be described in Sec. 4.

4 Main Results

4.1 Optimal Control Solution.

The following Lemma is used in this paper to prove the asymptotic stability and optimality of the proposed cooperative control law.

Lemma 4.1 [23]. Consider the nonlinear dynamical system
(26)
with f(0,0)=0 and a cost functional given by
(27)
where U(·) is an admissible control. Let Dn be an open set and Ωm. Assume that there exists a continuously differentiable function V:D and a control law ϕ:DΩ such that
(28)
(29)
(30)
(31)
(32)
(33)

where H(X¯̂,U)T̂(X¯̂,U)+V(X¯̂)f(X¯̂,U) is the Hamiltonian function. The superscript denotes partial differentiation with respect to X¯̂.

Then, with the feedback control
(34)
the solution X¯̂0 of the closed-loop system is locally asymptotically stable and there exists a neighborhood of the origin D0D such that
(35)
In addition, if X¯̂0D0, then the feedback control (34) minimizes J(X¯̂0,U(·)) in the sense that
(36)
where S(X¯̂0) denotes the set of asymptotically stabilizing controllers for each initial condition X¯̂0D. Finally, if D=n,Ω=m and
(37)

The solution X¯̂(0)0 of the closed-loop system is globally asymptotically stable.

Proof. Refer to Ref. [23].▪

Before presenting the main theorem, we define the tracking penalty function
(38)
(39)
where G is a weighting matrix with tunable elements wdi constructed by
(40)

and X¯D is the reference trajectory along which the system is expected to follow. G will be shown to be positive semidefinite after Lemma 4.2. Combining this tracking penalty function with the consensus cost function J1 enables the system of agents to follow a specified desired trajectory consensually. Note that only one agent having access to the reference is sufficient to guarantee that the entire system follows the desired trajectory if the communication topology is connected.

In order to investigate the eigenvalues of G, the following lemma from exterior algebra is required.

Lemma 4.2 [33,34]. LetRbe a commutative ring andmbe a positive integer. The characteristic polynomial of anyGMm(R)can be written as
(41)

where ck(G)=(1)ktr(Λk(G)). Λk(G) is the kth exterior power of G and Mm(R) is the set of all linear mapping with rank m. Furthermore, Λ1(G)=G and Λk(G)=0 for k>r, where r is the rank of G.

Proof. Refer to Refs. [33] and [34].▪

According to Lemma 4.2, the characteristic polynomial of G can be written as
(42)
The matrix G is symmetric and its rank is r=1 since each column is a product of any other column and a constant. Therefore, Λk(G)=0for anyk2. Since Λ1(G)=G, Eq. (42) is reduced to
(43)

which implies that the set of eigenvalues contains (m1) zeros and i=1m(wdi)2; therefore, G is positive semidefinite.

Before providing the main results, the following two Lemmas are introduced.

Lemma 4.3. L2is positive semidefinite andL21ng×1=0ng×1if the graph is undirected and connected.

Proof. Refer to Ref. [22].▪

Lemma 4.4. αkL2+βkLis positive definite and(αkL2+βkL)1ng×1=0ng×1if the graph is undirected and connected and
(44)

where αk and βk are constant numbers, and ei is the ith eigenvalue of L.

Proof. Since L is the Laplacian matrix of an undirected and connected graph, there exist two matrices Q and Λ [31] such that
(45)
where Q is the matrix of eigenvectors of L and Λ is a diagonal matrix whose entries are the eigenvalues of L. For L2, one can write
(46)
Using the same approach, αkL2+βkL can be written as:
(47)

where ei is the ith eigenvalue of L. Since L is the Laplacian matrix for a connected and undirected graph, it is positive semidefinite and ei0. Therefore, αkL2+βkL is positive definite if αkei2+βkei>0.

In addition
(48)

The main result of this paper is presented in the following theorem.

Theorem 4.1. For ng identical agents with the same controllable LTI dynamics (4), and connected and undirected communication topology, the feedback control law
(49)
makes the system (5) achieve consensus and track the reference trajectory, while minimizing the cost functional (22). X(n)=[Xn1Xn2Xnng]T is the vector of the last states of the agents. K¯ in Eq. (49) is defined by
(50)
where K1 is defined in Eq. (19) and P is the solution of the ARE that will be given in the proof. The cost function h(X¯̂) in J3 in Eq. (25) is constructed as
(51)

where S=B¯R21B¯T and /X¯̂(n)g(X¯̂) is the derivative of the tracking penalty function with respect to the last states of the agents.

Proof. According to the definition of the cost functional (22) and Lemma 4.1, T̂(X¯̂,U2) and f(X¯̂,U2) are defined as
(52)
(53)
A Lyapunov function is chosen as
(54)

For V(X¯̂) to be a valid Lyapunov function, it should be continuously differentiable with respect to X¯̂, which is obvious from the definition of g(X¯̂).

The Hamiltonian function is constructed as
(55)
Taking the derivate with respect to U2, the optimal control is determined as
(56)
Next, all the conditions from Eq. (28) to Eq. (33) must be verified. Substituting U2* in the second term of the Hamiltonian function (55) yields
(57)
where SB¯R21B¯T. Note that the term X¯̂T(2PA¯2)X¯̂ in Eq. (55) is a scalar and can be rewritten as X¯̂T(A¯2TP+PA¯2)X¯̂ in Eq. (57). The Hamiltonian function with the optimal control ϕ(X¯̂) becomes
(58)
In order to satisfy the condition (32), Eq. (58) should be zero, which requires that
(59)
and
(60)

The importance of Eq. (60) is that it allows us to determine h(X¯̂) and thus the cost function term J3.

Equation (59) is an ARE. In order to solve the ARE, one needs to construct R1 such that it is positive semidefinite and P is positive definite. Here, R1 is constructed as
(61)
where wi's are the tunable weights for the ith state and P(i1)i,1in are the entries above the main diagonal of the matrix P (the solution of the ARE (59)). These terms are subtracted from the diagonal entries of R1 to guarantee that the Laplacian matrix L appears in the solution P of the ARE. Expanding (59) leads to
(62)

The system of algebraic equations (62) allows us to solve the ARE analytically. P is determined by the following steps.

  1. The equations for the entries on the main diagonal are first solved
    (63)
    and yield
    (64)
  2. The equations for each row in the system (62) are solved one by one. For example, the first row leads to the following equation:
    (65)

All the row equations are solved in a similar way to obtain all the entries of P. Note that all the entries are solved analytically without any numerical approximation and iteration.

From the above solutions, it is seen that all the entries of P can be written as a linear combination of L and L2 (i.e., αkL2+βkL, where both αk and βk are functions of state and control weighting parameters). Therefore, according to Lemma 4.4, one can always choose weighting parameters such that R1 is positive semidefinite and P is positive definite.

Since P is positive definite and G is positive semidefinite, V(X¯̂)>0,X¯̂D,X¯̂0 and the condition (29) is satisfied.

The condition (33) can be shown to hold as follows
(66)
Since ϕ=R21B¯TPX¯̂12R21B¯Tg(X¯̂) from Eq. (56), one has
(67)
Using Eq. (67), Eq. (66) becomes
(68)

Therefore, Eq. (68) verifies the condition (33), i.e., H(X¯̂,U2,VT(X¯̂))0

Using Eqs. (59) and (60), the condition (31) can be shown to hold as follows:
(69)
Since X¯̂TR1X¯̂ is positive semidefinite and (X¯̂TP+12gT(X¯̂))S(PX¯̂+12g(X¯̂)) is positive definite, VT(X¯̂)f(X¯̂,ϕ(X¯̂))0 if h(X¯̂)0. From Eq. (60), h(X¯̂) is derived as
(70)

where X¯̂(k) is the vector of the kth state of all the agents. Equation (70) shows that one can always find proper weights such that h(X¯̂)0. Specifically, for a given set of state weighting parameters (i.e., w1,w2,,wn), the control weight parameter wc can be chosen small enough such that the positive term 14wc2X¯̂(n)g(X¯̂) is always greater than the other terms that are sign-indefinite.

The conditions (28) and (30) are satisfied if g(X¯̂)=0 and g(X¯̂)=0 when X¯̂=0. Note that after reaching consensus, when the agents go along a desired trajectory, we have: X¯cs=X¯D, which indicates X¯=X¯D when X¯̂=0. Therefore, according to the definition of g(X¯̂), V(X¯̂)=0 when X¯̂=0, and the condition (28) holds.

The second term in the optimal control (56) becomes
(71)

It is obvious that when all the agents go along the desired trajectory, Eq. (71) is zero and the condition (30) holds.

Now, all the conditions (28)(33) in Lemma 4.1 are verified and therefore, the control law ϕ(X¯̂) is an optimal control to minimize the cost in Eq. (22). In the meanwhile, the closed-loop system (20) is asymptotically stable. Furthermore, it is evident that the Lyapunov function (54) satisfies V(X¯̂) as X¯̂. Thus, the closed-loop system is globally asymptotically stable.

The optimal control law for system (14) can be obtained from Eq. (21) as
(72)

where g(X¯̂)=[gX¯̂(1)gX¯̂(2)gX¯̂(n)]T and X(i)=[Xi1Xi2Xing]T.

Since R21=1wc2In and B¯=[0001]1×nTIng, the second term of the controller (72) can be written as
(73)
Equation (72) is the optimal control for the system (14), which is equivalent to the system (20). The optimal cooperative control law can be written in a more compact form as
(74)

where K¯ is defined by Eq. (50).

Applying the control (74) into Eq. (14) leads to
(75)
Replacing X̂ with XXcs yields
(76)
According to Eq. (6), X˙cs=AXcs and Ucs=K¯T1Xcs=0nng×1. Thus, Eq. (76) is reduced to
(77)
gX¯̂(n) is calculated in Eq. (71). Since XD is known to the agent i that has access to the desired trajectory, X¯̂(n)g(X¯̂)=X¯(n)g(X¯). Now, the vector (X¯iX¯D) can be converted to the original state in
(78)
Using Eq. (78), the optimal control law becomes (49)

Remark 4.1. In the conventional optimal control approach, the cost functional is given a priori and the optimal control law is derived by minimizing it. It can be seen that the approach described in Theorem 4.1 is an inverse optimal control approach. A Lyapunov function V(X¯̂) is constructed first based on the stability conditions (28), (29), and (31) and then the optimal control law ϕ(X¯̂) is derived from the optimality condition HU=0. By satisfying the optimality condition (32), h(X¯̂) in the cost functional J3 is constructed while satisfying the stability condition (31). In other words, for this approach, the cost function h(X¯̂) is not specified a priori as the conventional optimal control design. It is constructed inversely from the stability and optimality conditions (28) to (33). The benefit of this design is that the resulting control law is guaranteed to be both stabilizing and optimal. In addition, according to Lemma 4.1, the minimum cost made by the derived control law ϕ(X¯̂) is equal to the introduced Lyapunov function as shown in Eq. (35).

4.2 Distributed Feature of the Optimal Control Law.

It is necessary to show that the optimal cooperative control law (49) is a distributed control law. To this end, one can first investigate the first term of the optimal cooperative control U* in Eq. (49), which can be expanded as the following equation:
(79)

where t¯ij are the entries of T1. Note that B¯ is in the form of [0001]Ing and thus the product of R21B¯TP in K¯ (Eq. (50)) only keeps the last column of P, i.e., Pin, which is calculated by Eq. (64). Since Pin's are always linear functions of the Laplacian matrix L as shown in Eq. (64), i.e., Pin=wiwcL,1in, when they are multiplied by the state vector X in Eq. (79), the feedback information exchange to implement the optimal control occurs only between the agent and its neighbors with whom it has the communication links defined by the Laplacian matrix L. The other terms consisting of ai's (coefficients of the characteristic equation of Ag) and identity matrices in Eq. (79) relate to the agent's own state. The second term of U* in Eq. (49) only relates to the state of the agent that has access to the reference. Therefore, from the above discussions, the entire optimal cooperative control law is distributed since each agent's control law only needs local information from its own and its communicating neighbors.

Remark 4.2. According to Eq. (79), only the last column of P calculated by Eq. (64) is required in order to calculate the optimal control law.

4.3 Discussion of Optimal Cooperative Control Design.

In this section, the principle of the proposed optimal cooperative control design including the motivation of the inverse optimal control is recapitulated. The optimal control formulation includes three cost functions J1=0(X¯̂TR1X¯̂)dt (Eq. (23)), J2=0(U2TR2U2)dt (Eq. (24)), and J3=0h(X¯̂)dt (Eq. (25)). Note that the first two cost functions are based on the normal LQR formulation to achieve the optimal consensus objective while only the third cost function is designed using the inverse optimal control approach to achieve the optimal tracking. This formulation is based on the following objectives: (1) the control law is in an analytical form without applying iterative numerical algorithms; (2) the designed control law must be distributed so that each agent only needs information from its neighbors according to the communication topology; (3) the control law must guarantee both stability and optimality; and (4) the agents can consensually track a designated desired trajectory.

The mission to achieve the above objectives is accomplished in two parts. In the first part, consensus is reached among agents and in the second part, tracking is achieved by the agents who have access to the reference trajectory communicating with the agents who do not in order to make the entire agent system track the desired trajectory. The first cost function J1 is responsible for the first consensus objective while the third cost function J3 accounts for the second tracking objective by including the tracking penalty function (38) in the construction of h(X¯̂) in J3. Note that in the absence of the tracking penalty function or minimizing J3, only consensus can be achieved without tracking capability. It is worth mentioning that as t, XcsXD but during the mission XD is only known to a subgroup of the agents and Xcs is not known a priori.

The motivation of adopting the inverse optimal control to construct the cost function J3 can be recapitulated as follows: (1) optimizing both consensus and tracking in order to achieve the above four objectives is difficult to formulate with one unified cost function. In the formulation, the difference between the two forms of the error vectors, i.e., Eq. (7) for consensus and X¯iX¯D for tracking, makes it hard to combine the consensus cost and the tracking cost into one unified cost function and then use the conventional optimal control approach to derive an analytical distributed cooperative control law. (2) A distributed cooperative control law requires that it should be a function of the Laplacian matrix L that specifies the communication topology, and LX should appear in the feedback term of the control law. The specially constructed weighting matrix R1 (Eq. (61)) in the consensus cost function J1 enables the solution to the ARE, i.e., P, to contain the Laplacian matrix L. Specifically, Pin=wiwcL,1in (Eq. (64)) allows LX to appear in the final control law, as discussed in Sec. 4.2. However, to achieve the tracking objective, if including the tracking penalty (38) in the consensus cost J1 in a conventional way, the control law will not have LX in the feedback term, and this crucial distributed property will be destroyed. (3) The inverse optimal control formulation can guarantee both stability and optimality as shown in Theorem 4.1. The Lyapunov function guaranteeing closed-loop stability is the optimal cost for the optimal controlled system, which is shown in Eq. (35) and is the main benefit of this design. (4) The inverse optimal control enables us to derive an analytical optimal control law that does not need any numerical iterations. Normally, for the conventional optimal control, numerical methods are needed to solve the ARE. With the proposed approach, constructing R1 in a novel way of Eq. (61) allows one to solve ARE analytically, which is shown in Eqs. (62)(65). If the conventional optimal control approach is applied with the tracking penalty function directly incorporated into the cost function, the above discussed benefits would not be achieved.

5 An Illustrative Example

In this section, the performance of the optimal cooperative control design is demonstrated through an example of attitude synchronization of a group of five identical satellites. A sketch of the satellite attitude control system and the model are shown in Fig. 1 [35,36]. The system can be considered as two separate masses (a large mass called “the body” and one “attached mass”) that are connected. The system can be modeled by a mass-spring-damper system.

Fig. 1
The satellite attitude control system [35,36]
Fig. 1
The satellite attitude control system [35,36]
Close modal
The equations of motion can be described by
(80)

where TC is the control torque, J1 and J2 are moments of inertia, k is the spring constant, and d is the viscous damping constant.

Assume J1=J2=1,k=0.09,d=0.022 and the state vector is Xi=[θ2,θ˙2,θ1,θ˙1]T where θi and θ˙i are the rotational angle and angular rate of the mass i, respectively. The system state and input matrices constructed based on the governing equations are
(81)

The communication topology is shown in Fig. 2. It is assumed that only satellite 1 has access to the reference trajectory. Satellites 4 and 5 receive information from 1 but the other two satellites do not have direct communication with Satellite 1. We assume that the angular rates are expected to reach a constant value of 0.35 rad/s. Therefore, the angles must increase along the ramp θ=0.35t rad after reaching consensus.

Fig. 2
Communication topology of the satellites
Fig. 2
Communication topology of the satellites
Close modal
The state and control weighting parameters are selected as: w1=10,w2=4,w3=30,w4=2, and wc=0.1. The tunable weights for the tracking penalty function are wd1=10,wd2=50,wd3=50,wd4=0.5. The initial conditions are chosen as follows:
(82)
The corresponding Laplacian matrix is
(83)

Figures 37 show the results. Figures 3 and 5 show rotational angles of the attached masses and bodies, respectively. It can be seen that the rotational angles converge to the reference trajectory consensually and increase along the ramp of θ=0.35t. Figures 4 and 6 show angular rates of the attached masses and bodies, respectively. The angular rates converge to the constant value of 0.35 rad/s. It is worth noting that the responses of the satellite 1 are very smooth and do not have oscillations because it has access to the reference trajectory. Figure 7 presents the control torque responses, which shows that the optimal control does not require a large control effort to achieve consensus and tracking.

Fig. 3
Rotational angles of the attached masses (the first states)
Fig. 3
Rotational angles of the attached masses (the first states)
Close modal
Fig. 4
Angular rates of the attached masses (the second states)
Fig. 4
Angular rates of the attached masses (the second states)
Close modal
Fig. 5
Rotational angles of the bodies (the third states)
Fig. 5
Rotational angles of the bodies (the third states)
Close modal
Fig. 6
Angular rates of the bodies (the fourth states)
Fig. 6
Angular rates of the bodies (the fourth states)
Close modal
Fig. 7

6 Conclusions

In this paper, the multi-agent consensus tracking problem for a class of general linear time-invariant systems is investigated in an optimal control framework. The consensus is reached by a modified LQR formulation, and cooperative tracking is achieved by applying an inverse optimal control design to derive a proper cost function. The optimal control law has an analytical form and is a linear function of the Laplacian matrix such that the control implementation is distributed in that it only needs local information of the agent's own state and its neighbors with communication links. Both optimality and stability of the control law are proved. An attitude synchronization example is utilized to illustrate the effectiveness of the proposed optimal cooperative control.

Funding Data

  • USDA NIFA (Grant No. 2019-67021-28993; Funder ID: 10.13039/100005825).

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