Control of large 1D networks of double integrator agents: role of heterogeneity and asymmetry on stability margin
We consider the distributed control of a network of heterogeneous agents with double integrator dynamics to maintain a rigid formation in 1D Euclidean space. The control signal at each vehicle is allowed to use relative position and velocity with its two nearest neighbors. Most of the work on this problem, though extensive, has been limited to homogeneous networks, in which agents have identical mass and controller, and symmetric control, in which information from front and back neighbors are weighted equally. We examine the effect of heterogeneity and asymmetry on the closed loop stability margin, which is measured by the real part of the least stable pole of the closed-loop system. By using a PDE (partial differential equation) approximation in the limit of large number of vehicles, we show that heterogeneity has little effect while asymmetry has a significant effect on the stability margin. When control is symmetric, the stability margin decays to as , where is the number of agents, even when the agents are heterogeneous in their masses and control gains. In contrast, we show that arbitrarily small amount of asymmetry in the velocity feedback gains can improve the decay of the stability margin to . Poor design of such asymmetry makes the closed loop unstable for sufficiently large . Moreover, if there is equal amount of asymmetry in both position and velocity feedback gains, the stability margin of the network can be bounded away from , uniformly in . This results thus eliminates the degradation of closed-loop stability margin with increasing , but its sensitivity to external disturbances becomes much worse than symmetric control. Numerical computations are provided to corroborate the analysis.
In this paper we examine the closed loop dynamics of a network consisting of interacting agents arranged in a line, where the agents are modeled as double integrators and each agent interacts with its two nearest neighbors (one on either side) through its local control action. This is a problem that is of primary interest to formation control applications, especially to platoons of vehicles, where the vehicles are modeled as point masses. An extensive literature exists on 1-D automated platoons; see [2, 3, 4, 5, 6, 7] and references therein. In the vehicular platoon problem, the formation try to track a desired trajectory while maintaining a rigid formation geometry. The desired trajectory of the entire vehicular platoon is given in terms of trajectory of a fictitious reference vehicle, and the desired formation geometry is specified in terms of constant inter-vehicle spacings.
Although significant amount of research has been conducted on robustness-to-disturbance and stability issues of double integrator networks with decentralized control, most investigations consider the homogeneous case in which each agent has the same mass and employs the same controller (exceptions include [8, 9, 10]). In addition, only symmetric control laws are considered in which the information from both the neighboring agents are weighted equally, with [11, 7] being exceptions. Khatir et. al. proposes heterogeneous control gains to improve string stability (sensitivity to disturbance) at the expense of control gains increasing without bound as increases . Middleton et. al. considers both unidirectional and bidirectional control, and concludes heterogeneity has little effect on the string stability under certain conditions on the high frequency behavior and integral absolute error . On the other hand,  examines the effect of asymmetry (but not heterogeneity) on the response of the platoon as a result of sinusoidal disturbances in the lead vehicle, and concludes the asymmetry makes sensitivity to such disturbances worse.
In this paper we analyze the case when the agents are heterogeneous in their masses and control laws used, and also allow asymmetry in the use of front and back information. A decentralized bidirectional control law is considered that uses only relative position and relative velocity information from the nearest neighbors. We examine the effect of heterogeneity and asymmetry on the stability margin of the closed loop, which is measured by the absolute value of the real part of the least stable pole. The stability margin determines the decay rate of initial formation keeping errors. Such errors arise from poor initial arrangement of the agents. The main result of the paper is that in a decentralized bidirectional control strategy, heterogeneity has little effect on the stability margin of the overall closed loop, while even small asymmetry can have a significant impact. In particular, we show that in the symmetric case, the stability margin decays to as , where is the number of agents. We also show that the asymptotic scaling trend of stability margin is not changed by agent-to-agent heterogeneity as long as the control gains do not have front-back asymmetry. On the other hand, arbitrary small amount of asymmetry in the way the local controllers use front and back information can improve the stability margin by a considerable amount. When each agent weighs the relative velocity information from its front neighbor more heavily than the one behind it, the stability margin scaling trend can be improved from to . In contrast, if more weight is given to the relative velocity information with the neighbor behind it, the closed loop becomes unstable if is sufficiently large. In addition, when there is equal amount of asymmetry in position and velocity feedback gains, the closed-loop is exponential stable for arbitrary finite , and the stability margin can be uniformly bounded with the size of the network. This result makes it possible to design the control gains so that the stability margin of the system satisfies a pre-specified value irrespective of how many vehicles are in the formation. However, in this special case, the sensitivity to disturbance becomes much worse than symmetric control. In contrast, with judicious asymmetry in velocity feedback alone improves the sensitivity to external disturbance.
In this paper, we propose a PDE approximation to the coupled system of ODEs that model the closed loop dynamics of the network. This is inspired by the work  that examined stability margin of 1-D vehicular platoons in a similar framework. Compared to , this paper makes two novel contributions. First, we consider heterogeneous agents (the mass and control gains vary from agent to agent), whereas  consider only homogeneous agents. Secondly,  considered the scenario in which the desired trajectory of the platoon was one with a constant velocity, and moreover, every agent knew this desired velocity. In contrast, the control law we consider requires agents to know only the desired inter-agent separation; the overall trajectory information is made available only to agent . This makes the model more applicable to practical formation control applications. It was shown in  for the homogeneous formation that asymmetry in the position feedback can improve the stability margin from to while the absolute velocity feedback gain did not affect the asymptotic trend. In contrast, we show in this paper that with relative position and relative velocity feedback, asymmetry in the velocity feedback gain alone and in both position and velocity feedback gains are very important. The stability margin can be improved considerably by a judicious choice of asymmetry.
Although the PDE approximation is valid only in the limit , numerical comparisons with the original state-space model shows that the PDE model provides accurate results even for small ( to ). PDE approximation is quite common in many-particle systems analysis in statistical physics and traffic-dynamics (see the article  for an extensive review.). The usefulness of PDE approximation in analyzing multi-agent coordination problems has been recognized also by researchers the controls community; see [13, 14, 7, 15] for examples. A similar but distinct framework based on partial difference equations has been developed by Ferrari-Trecate et. al. .
The rest of this paper is organized as follows. Section II presents the problem statement and the main results of this paper. Section III describes the PDE model of the network of agents. Analysis and control design results together with their numerical corroboration appear in Sections IV and V, respectively. The paper ends with a summary in Section VI.
Ii Problem statement and main results
Ii-a Problem statement
We consider the formation control of heterogeneous agents which are moving in 1D Euclidean space, as shown in Figure 1 (a). The position and mass of each agent are denoted by and respectively. The mass of each agent is bounded, for all , where and are constants. The dynamics of each agent are modeled as a double integrator:
where is the control input (acceleration or deceleration command). This is a commonly used model for vehicle dynamics in studying vehicular formations, which results from feedback linearization of actual non-linear vehicle dynamics [3, 17].
The desired trajectory of the formation is given in terms of a fictitious reference agent with index whose trajectory is denoted by . Since we are interested in translational maneuvers of the formation, we assume the desired trajectory is a constant-velocity type, i.e. for some constants and . The information on the desired trajectory of the network is provided only to agent . The desired geometry of the formation is specified by the desired gaps for , where is the desired value of . The control objective is to maintain a rigid formation, i.e., to make neighboring agents maintain their pre-specified desired gaps and to make agent follow its desired trajectory . Since we are only interested in maintaining rigid formations that do not change shape over time, ’s are positive constants.
In this paper, we consider the following decentralized control law, whereby the control action at the -th agent depends on i) the relative position measurements ii) the relative velocity measurements with its immediate neighbors in the formation:
where , are the front and back position gains and are the front and back velocity gains respectively. For the agent with index which does not have an agent behind it, the control law is slightly different:
Each agent knows the desired gaps and , while only agent knows the desired trajectory of the fictitious reference agent.
where The dynamics of the -th agent are obtained by combining (1) and (3), which are slightly different from (4). The desired trajectory of the -th agent is . To facilitate analysis, we define the following tracking error:
By defining the state , the closed loop dynamics of the network can now be written compactly from (6) as:
where is the closed-loop state matrix and we have used the fact that since the trajectory of the reference agent is equal to its desired trajectory.
Ii-B Main results
The first two results rely on the analysis of the following PDE (partial differential equation) model of the network, which is seen as a continuum approximation of the closed-loop dynamics (6). The details of derivation of the PDE model are given in Section III. The PDE is given by
with boundary condition:
where and are defined as follows:
and are respectively the continuum approximation of of each agent with the following stipulation:
We formally define symmetric control and stability margin before stating the first main result, i.e. the role of heterogeneity on the stability margin of the network.
The control law (2) is symmetric if each agent uses the same front and back control gains: and , for all .
The stability margin of a closed-loop system, which is denoted by , is the absolute value of the real part of the least stable pole.
Consider the PDE model (8) of the network with boundary condition (9), where the mass and the control gain profiles satisfy , and for all where and are positive constants, and denotes the percent of heterogeneity. With symmetric control, the stability margin of the network satisfies the following:
The result above is also provable for an arbitrary (not necessarily small) when the position gain is proportional to the velocity gain using standard results of Sturm-Liouville theory [18, Chapter 5]. For that case, the result is given in the following lemma and its proof is given in the end of the Appendix.
The main implication of the result above is that heterogeneity of masses and control gains plays no role in the asymptotic trend of the stability margin with as long as the control gains are symmetric. Note that the decay of the stability margin described above has been shown for homogeneous platoons (all agents have the same mass and use the same control gains) independently in , although the dynamics of the last vehicle are slightly different from ours. A similar result for homogeneous platoons with relative position and absolute velocity feedback was also established in .
The second main result of this work is that the stability margin can be greatly improved by introducing front-back asymmetry in the velocity-feedback gains. We call the resulting design mistuning-based design because it relies on small changes from the nominal symmetric gain . In addition, a poor choice of such asymmetry can also make the closed loop unstable. Since heterogeneity is seen to have little effect, and for ease of analysis, we let in the sequel.
For an -agent network with PDE model (8) and boundary condition (9). Let for all , consider the problem of maximizing the stability margin by choosing the control gains with the constraint , where is a positive constant, and . If , the optimal velocity gains are
which result in the stability margin
The formula is asymptotic in the sense that it holds for large and small . In contrast, for the following choice of asymmetry
where is an small positive constant, the closed loop becomes unstable for sufficiently large .
The theorem says that with arbitrary small change in the front-back asymmetry, so that velocity information from the front is weighted more heavily than the one from the back, the stability margin can be improved significantly over symmetric control. On the other hand, if velocity information from the back is weighted more heavily than that from the front, the closed loop will become unstable if the network is large enough. It is interesting to note that the optimal gains turn out to be homogeneous, which again indicates that heterogeneity has little effect on the stability margin.
The astute reader may inquire at this point what are the effects of introducing asymmetry in the position-feedback gains while keeping velocity gains symmetric, or introducing asymmetry in both position and velocity feedback gains. It turns out when equal asymmetry in both position and velocity feedback gains are introduced, the closed loop is exponentially stable for arbitrary . Moreover, the stability margin scaling trend can be uniformly bounded below in when more weights are given to the information from its front neighbor. We state the result in the next theorem.
This asymmetric design therefore makes the resulting control law highly scalable; it eliminates the degradation of closed-loop stability margin with increasing . It is now possible to design the control gains so that the stability margin of the system satisfies a pre-specified value irrespective of how many vehicles are in the formation. The result above is for equal amount of asymmetry in the position feedback and velocity feedback gains. This constraint of equal asymmetry in position and velocity feedback is imposed in order to make the analysis tractable. The analysis of the stability margin in the following cases are open problems: (i) unequal asymmetry in position and velocity feedback, (ii) velocity feedback gains are kept at their nominal symmetric values and asymmetry is introduced in the position feedback gains only.
Iii PDE model of the closed-loop dynamics
In this paper, all the analysis and design is performed using a PDE model, whose results are validated by numerical computations using the state-space model (7). We now derive a continuum approximation of the coupled-ODEs (6) in the limit of large , by following the steps involved in a finite-difference discretization in reverse. We define
Substituting these into (6), we have
To facilitate analysis, we redraw the graph of the 1D network, so that each vehicle in the new graph is drawn in the interval , irrespective of the number of agents. The -th agent in the “original” graph, is now drawn at position in the new graph. Figure 1 shows an example.
The starting point for the PDE derivation is to consider a function that satisfies:
such that functions that are defined at discrete points will be approximated by functions that are defined everywhere in . The original functions are thought of as samples of their continuous approximations. We formally introduce the following scalar functions and defined according to the stipulation:
In addition, we define functions , , , as
Due to (17), these satisfy
To obtain a PDE model from (III), we first rewrite it as
Using the following finite difference approximations:
For large , Eq. (III) can be seen as a finite difference discretization of the following PDE:
The boundary conditions of the above PDE depend on the arrangement of reference agent in the redrawn graph of the network. For our case, the boundary condition is of Dirichlet type at where the reference agent is, and of Neumann type at :
Iv Role of heterogeneity on stability margin
The starting point of our analysis is the investigation of the homogeneous and symmetric case: for some positive constants , where . The analysis leading to the proof of Theorem 1 is carried out using the PDE model derived in the previous section. In the homogeneous and symmetric control case, using the notation introduced earlier, we get
The PDE (8) simplifies to:
This is wave equation with Kelvin-Voigt damping. Due to the linearity and homogeneity of the above PDE and boundary conditions, we are able to apply the method of separation of variables. We assume a solution of the form . Substituting it into PDE (19), we obtain the following time-domain ODE
where solves the following boundary value problem
with the following boundary condition, which comes from (9):
Following straightforward algebra, the eigenvalues and eigenfunction of the above boundary value problem is given by (see  for a BVP example)
The eigenvalues of the PDE (19) are now given by
For small and large so that , the discriminant is negative, making the real part of the eigenvalues equal to . The least stable eigenvalue, the one closest to the imaginary axis, is obtained with :
where is an imaginary number.
We are now ready to present the proof of Theorem 1.
Recall that in case of symmetric control we have
In this case, using the notation introduced earlier, we have
The PDE (8) is simplified to:
The proof proceeds by a perturbation method. To be consistent with the bounds of the mass and control gains of each agent, let
where is a small positive number, denoting the amount of heterogeneity and are the perturbation profiles. Take Laplace transform to both sides of PDE (26) with respect to , we have
Let the perturbed eigenvalue be the Laplace transform of be , where and correspond to the unperturbed PDE (19), i.e.
Eq. (24) provides the formula for (actually, ), and is the solution to above equation, which is given by , where is the Laplace transform of given in (20). Plugging the expressions for and into (27), and doing an balance leads to the eigenvalue equation for the unperturbed PDE, which is exactly Eq. (28):
Next we do an balance, which leads to:
For a solution to exist, must lie in the range space of the operator . Since is self-adjoint, its range space is orthogonal to its null space. Thus, we have,
where is also the basis vector of the null space of operator . We now have the following equation:
Following straightforward manipulations, we got:
where is an imaginary number when is large (). Using this, and substituting the equation above into , and setting , we obtain the stability margin of the heterogeneous network:
Plugging the bounds and , we obtain the desired result.
Iv-a Numerical comparison
We now present numerical computations that corroborates the PDE-based analysis. We consider the following mass and control gain profile:
In the associated PDE model (26), this corresponds to , , . The eigenvalues of the PDE, that are computed numerically using a Galerkin method with Fourier basis, are compared with that of the state space model to check how well the PDE model captures the closed loop dynamics. Figure 2 depicts the comparison of eigenvalues of the state-space model and the PDE model. It shows the eigenvalues of the state-space model is accurately approximated by the PDE model, especially the ones close to the imaginary axis. We see from Figure 3 that the closed-loop stability margin of the controlled formation is well captured by the PDE model. In addition, the plot corroborates the predicted bound (11).
V Role of asymmetry on stability margin
In this paper, we consider two scenarios of asymmetric control, we will first present the results when there is asymmetry in the velocity feedback alone (Theorem 2). The results when there is equal asymmetry in both position and velocity feedback will follow immediately (Theorem 3).
V-a Asymmetric velocity feedback
With symmetric control, one obtains an scaling law for the stability margin because the coefficient of the term in the PDE (26) is and the coefficient of the term is . Any asymmetry between the forward and the backward velocity gains will lead to non-zero and a presence of term as coefficient of . By a judicious choice of asymmetry, there is thus a potential to improve the stability margin from to . A poor choice of control asymmetry may lead to instability, as we’ll show in the sequel.
We begin by considering the forward and backward feedback gain profiles
where is a small parameter signifying the percent of asymmetry and , are functions defined over that capture velocity gain perturbation from the nominal value . Define