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Observer-Based Control of Second-Order Multi-vehicle Systems in Bearing-Persistently Exciting Formations

Zhiqi Tang, Baris Fidan, Karl H. Johansson, Jonas Martensson, Tarek Hamel

Abstract

This paper proposes an observer-based formation tracking control approach for multi-vehicle systems with second-order motion dynamics, assuming that vehicles' relative or global position and velocity measurements are unavailable. It is assumed that all vehicles are equipped with sensors capable of sensing the bearings relative to neighboring vehicles and only one leader vehicle has access to its global position. Each vehicle estimates its absolute position and velocity using relative bearing measurements and the estimates of neighboring vehicles received over a communication network. A distributed observer-based controller is designed, relying only on bearing and acceleration measurements. This work further explores the concept of the \textit{Bearing Persistently Exciting} (BPE) formation by proposing new algorithms for bearing-based localization and state estimation of second-order systems in centralized and decentralized manners. It also examines conditions on the desired formation to guarantee the exponential stability of distributed observer-based formation tracking controllers. In support of our theoretical results, some simulation results are presented to illustrate the performance of the proposed observers as well as the observer-based tracking controllers.

Observer-Based Control of Second-Order Multi-vehicle Systems in Bearing-Persistently Exciting Formations

Abstract

This paper proposes an observer-based formation tracking control approach for multi-vehicle systems with second-order motion dynamics, assuming that vehicles' relative or global position and velocity measurements are unavailable. It is assumed that all vehicles are equipped with sensors capable of sensing the bearings relative to neighboring vehicles and only one leader vehicle has access to its global position. Each vehicle estimates its absolute position and velocity using relative bearing measurements and the estimates of neighboring vehicles received over a communication network. A distributed observer-based controller is designed, relying only on bearing and acceleration measurements. This work further explores the concept of the \textit{Bearing Persistently Exciting} (BPE) formation by proposing new algorithms for bearing-based localization and state estimation of second-order systems in centralized and decentralized manners. It also examines conditions on the desired formation to guarantee the exponential stability of distributed observer-based formation tracking controllers. In support of our theoretical results, some simulation results are presented to illustrate the performance of the proposed observers as well as the observer-based tracking controllers.
Paper Structure (11 sections, 8 theorems, 39 equations, 4 figures)

This paper contains 11 sections, 8 theorems, 39 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Persistence of excitation
  4. Graph Theory
  5. Bearing Rigidity and Bearing Persistently Exciting Formation
  6. Problem formulation
  7. Bearing-based localization and velocity estimation
  8. Decentralized localization and velocity estimation
  9. Observer-based control
  10. Simulation Results
  11. Conclusion

Key Result

Lemma 1

Let $Q:=\sum_{i=1}^{l}\pi_{y_i}$. Then the matrix $Q$ satisfies the PE condition in Definition def:pe of matrix, if one of the following conditions is satisfied:

Figures (4)

  • Figure 1: Eamples of BPE formations in two (a1-b3) and three-dimensional space (c1-d3). Red lines represent edges for which the corresponding bearing vectors are PE, and blue lines represent edges for which the corresponding bearing vectors are not necessarily PE.
  • Figure 2: Evolution of the estimation error $(\boldsymbol \delta_p,\boldsymbol \delta_v)$ under the centralized observer with measurement noises. Gains are chosen as: $\kappa=10,\ M(0)=I_{2dn},\ Q=10I_{dn}, \ S=0.01I_{2dn}$
  • Figure 3: Evolution of the estimation error $(\boldsymbol \delta_p,\boldsymbol \delta_v)$ under the decentralized observer with measurement noises. Gains are chosen as: $\kappa_k=10,\ M_k(0)=100I_{2d},\ Q_k=10I_{d}, \ S_k=0.01I_{2d}, \ \kappa_{o1}=10, \ \kappa_{o2}=5$.
  • Figure 4: Evolution of the estimation and control errors $(\boldsymbol \delta_p,\boldsymbol \delta_v)$, $(\tilde{\boldsymbol p},\tilde{\boldsymbol v})$. Gains for the decentralized observer are the same as in Fig. \ref{['fig:decen']}. Gains for controller: $k_{p_i}=5$ and $k_{p_i}=2$.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Lemma 2
  • Theorem 1: Centralized state estimation
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 8 more