ory [Helmke and Moore, 1994, Chapter 5].
For integers 1≤n≤min(M,N), let
M(n,M×N) ={X∈RM×N|rank(X) =n}
denote the set of real M×N matrices of fixed rankn. The following results will be useful in the analysis of some main results in this chapter.
Lemma 7. M(n,M×N)is a smooth and connected manifold of dimension n(M+N−n),
if max(M,N)>1. The tangent space ofM(n,M×N)at an element X is
TXM(n,M×N) ={∆1X+X∆2|∆1∈RM×M,∆2∈RN×N} (3.26) A matrix differential equation ˙X= F(t,X)evolving on the matrix spaceRM×N is said to berank-preservingif the rank of every solution X(t)is constant as a function of finite t, that is, rank(X(t)) = rank(X(0)) for all t ≥ 0. The following lemma characterizes such rank-preserving flows (cf. Lemma 1.22 in Chapter 5 of [Helmke and Moore, 1994]).
§3.6 Appendix: background on rank-preserving matrix flow 33
Lemma 8. Let I ⊂ R be an interval and let A(t) ∈ RM×M, B(t) ∈ RN×N, t ∈ I, be a
continuous time-varying family of matrices. Then ˙
X(t) = A(t)X(t) +X(t)B(t), X(0)∈RM×N (3.27) is rank-preserving. Conversely, every rank-preserving differential equation on RM×N is of the form(3.27)for matrices A(t)and B(t).
The proof of Lemma 8 is based on the fact that (3.27) defines a time varying vector field on the subset of the tangent space ofM(n,M×N)described by (3.26). The full proof can be found in [Helmke and Moore, 1994, Page 139].
Remark 4. Note that the above lemma on rank-preserving flows also implies that the limit
Chapter4
Exponential stability for formation
control systems with generalized
controllers
Chapter summary
This chapter discusses generalized controllers for distance-based rigid formation shape stabilization and aims to provide a unified approach for the convergence anal- ysis. We consider two types of formation control systems according to different characterizations of target formations: minimally rigid target formations and non- minimally rigid target formations. For the former case, we firstly prove the expo- nential stabilityfor rigid formation systems when using a general form of shape con- troller with certain properties. From this viewpoint, different formation controllers proposed in the previous literature can be included in a unified framework. We then extend the result to the case that the target formation is non-minimally rigid, and show that exponential stability of the formation system is still guaranteed with generalized controllers.
4.1
Introduction
4.1.1 Background and related work
As reviewed in [Oh et al., 2015], the existing results on formation control can be characterized into position-, displacement-, and distance-based control strategies ac- cording to different types of sensed and controlled variables. Among these formation control strategies, distance-based formation control receives particular interest as it does not require all the agents to have the global knowledge of a common coordinate frame and they can measure the relative position to neighboring agents using their own coordinate frames. Thus, in this chapter we focus on distance-based formation control. In particular, we confine our attention toundirected rigid formations, while relevant discussions on directed formation control can be found in e.g. [Mou et al., 2015].
With underpinnings from rigid graph theory (see Chapter 2), it has been estab- lished that the formation shape can be achieved by controlling a certain set of inter- agent distances [Olfati-Saber and Murray, 2002; Anderson et al., 2008b; Krick et al., 2009]. In the rigid formation stabilization problem, one typical controller that has been studied extensively in the literature takes the following form (see e.g. [Krick et al., 2009]):
˙
pi =
∑
j∈Ni(pj−pi)(kpj−pik2−d2ij) (4.1) The above controller (4.1), which is derived from a well-defined potential function shown in (2.9), serves as a standard control law for stabilizing rigid formations. The dynamics of the formation control system (4.1) have been investigated in several suc- ceeding papers, e.g. [Dörfler and Francis, 2010; Belabbas et al., 2012; Anderson and Helmke, 2014]. We mention that alternative kinds of formation controllers other than the one in (4.1) are also available, which have been reported sparsely in the literature (see e.g. [Smith et al., 2006; Anderson et al., 2007; Dimarogonas and Johansson, 2010; Tian and Wang, 2013]).
The main objective of this chapter is to analyze general forms of formation con- trollers to stabilize rigid shapes. The main contributions of this chapter include a unified approach to discuss the convergence and controller performance of gener- alized formation controllers, and the associated exponential stability of general for- mation systems when certain properties of the potential function are satisfied. We show that for a large family of formation control systems which generalizes most existing formation controllers proposed in the literature, the exponential stability of the distance error system can be guaranteed. As is well-known in the control field, exponential stability has a robustness property against small perturbations [Khalil, 2002]. Such a robustness property has been employed in recent papers [Mou et al., 2016; Sun et al., 2015b; Garcia de Marina et al., 2015] to study the behavior of rigid undirected formations when there are mismatches or discrepancies between neigh- bor agents’ distance measurements. Exponential stability ensures that the derived distance error system under controller (4.1) with small distance perturbations still converges to a bounded neighbor set of the origin. Note that in the analysis of the formation robustness issue in [Mou et al., 2016; Sun et al., 2015b; Garcia de Marina et al., 2015], the formation control system is limited to the case of the controller in (4.1). To this end, this chapter shows that formation distance error systems with generalized controllers, which include many special forms of formation controllers studied in the literature, also inherit this robustness property as a consequence of the exponential stability.
In this chapter, we first consider formation stabilization control when the target formation is minimally rigid, which is a common assumption in the literature. We list several requirements of the potential function and its gradient function associated with the distributed control for each agent which render an exponential convergence of the formation system. We give an explicit lower bound on the convergence rate, and also discuss several properties of the generalized formation control systems. By deriving a reduced distance error system via the decomposition principle of a rigid