Brief paperTwo-time-scale adaptive internal model designs for motion coordination☆
Introduction
The internal model principle (IMP) (Francis & Wonham, 1976) is a fundamental control design tool for rejecting disturbances and tracking signals generated by exogenous systems. Recently, the IMP has been employed to design cooperative control laws for multi-agent systems. In Bai, Arcak, and Wen (2011), a connection between the IMP and passivity was established to achieve adaptive motion coordination. The IMP has also been used in output regulation (Huang, 2015), dynamic average consensus (Bai, Freeman, & Lynch, 2010), distributed Kalman filtering (Bai, Freeman, & Lynch, 2011), and synchronization (De Persis and Jayawardhana, 2012, Shafi and Bai, 2015, Wieland et al., 2011).
One common assumption in using the IMP is that a model of the exogenous system is available for control designs. When the exogenous system generates a sinusoidal signal, this assumption means that the frequency of the sinusoid is known. In certain scenarios, prior knowledge of the exogenous system can be identified or obtained (e.g., in controlled environments) and thus this assumption is valid. However, it restricts the application of the IMP in scenarios where the model of the exogenous system is unknown or changes over time. For example, autonomous agents may update its reference velocity in reaction to dynamic environments. Therefore, the model information of the reference velocity may change over time.
In this paper, we consider a group of agents whose objective is to reach consensus while tracking a leader’s sinusoidal reference velocity subject to a constant bias. A sinusoidal velocity represents the typical motion when a vehicle orbits. For example, when an aircraft makes a coordinated turn, its linear velocity is a sinusoidal function with the frequency being the angular rate of the turn. In the presence of constant disturbances, such as wind, the linear velocity becomes biased sinusoids.
When the frequency of the reference velocity is known, Bai et al. (2011, Chapter 3) provided a passivity-based internal model control design that achieves consensus and tracking of the reference velocity. We extend the design in Bai et al. (2011) to the scenario where the frequency of the sinusoidal reference velocity is unknown to the followers. One approach for this scenario is to parameterize the internal model and estimate the unknown parameters in the internal model (Marino and Tomei, 2003, Nikiforov, 2001, Serrani et al., 2001, Su and Huang, 2013). However, this parameter estimation approach requires re-designing the passivity-based control and does not exploit a passivity property in the internal model.
We take an approach different from the parameter estimation approach. We develop a two-time-scale adaptive internal model control approach that takes advantage of the passivity properties inherent in the internal model, the agent dynamics, and the network, and yields decentralized control algorithms. In particular, the two-time-scale approach consists of a slow system estimating the frequency of the reference velocity and a fast system representing the group dynamics. The stability of the fast system is ensured by the passivity properties when the agents’ estimated frequencies are treated as frozen parameters. As the transients of the group dynamics vanish, the slow system becomes periodic, allowing us to analyze its stability using averaging theory. With the stability of the fast system and the slow system established, we use two-time-scale averaging theory (Sastry & Bodson, 1989) to prove the convergence of the estimated frequency to the true frequency and consensus of the agents. Our approach estimates only the frequency of the sinusoid and does not require re-designing the passivity-based control law.
We consider two scenarios for the two-time-scale approach. In the first scenario, each follower updates its frequency estimate without communicating with other followers. In the second scenario, we allow each follower to communicate its local control signal to its neighbors. We augment the control design in the first scenario with a coupling term that combines the control signals from the neighboring agents. Using numerical examples, we show that the coupling term becomes an additional degree of freedom to tune the system and results in improved transient performance.
The main contribution of this paper is a two-time-scale averaging approach to designing decentralized adaptive internal model controls for multi-agent systems with passive dynamics. This approach enables agents to achieve consensus while tracking a completely unknown sinusoidal reference velocity subject to a constant bias. We note that Li, Liu, Ren, and Xie (2013) studied distributed tracking control with a bounded reference velocity and proposed discontinuous distributed control laws to achieve consensus. However, Li et al. (2013) assume that agents’ dynamics are identical and accurately known for LMI-based control designs. Since our approach employs input–output properties of the multi-agent system, it allows heterogeneous and unknown agent dynamics. We also note that similar frequency estimation approaches for a single system have been discussed in Esbrook, Tan, and Khalil (2011) and Brown and Zhang (2004).
The rest of the paper is organized as follows. In Section 2, we review a passivity-based design when the frequency of the reference velocity is available to all the agents. In Section 3, we consider the scenario where the frequency of the reference velocity is available only to the leader and present our two-time-scale adaptive design. In Section 4, we analyze the stability of the proposed adaptive design. In Section 5, we introduce an augmented adaptive control design when the followers are allowed to communicate their local control signals. Section 6 presents two simulation examples to illustrate the effectiveness of the designs. Conclusions and future work are discussed in Section 7.
Notation: The vectors and represent the by 1 vectors with all entries 1 and 0, respectively. The set of real numbers is denoted by . The set of by real matrices is denoted by . Let be the identity matrix. The Kronecker product of matrices and is denoted by . The notation denotes a block diagonal matrix with on the diagonal, where can be a matrix or a scalar. The transpose of a real matrix is denoted by . We denote by the entry in the th row and the th column of a matrix .
Section snippets
A cooperative system
We consider a cooperative system of agents, where the state of each agent , , is represented by . To simplify the notation we consider the scalar case . However, the results in this paper extend to with the use of Kronecker algebra. The information flow between the agents is described by a connected and bidirectional graph . Let denote the set of neighbors of agent . We define a weighted graph Laplacian matrix of , whose elements are given by
A two-time-scale adaptive design
The design in (11), (12) assumes that is available to the followers. We now relax this assumption and consider the scenario where only the leader has the information of .
We replace in by , an estimate of , for agent , , and update according to where is a small positive constant and is the th element of . As shown in Section 4, and converge to two sinusoids and . Thus, the normalization term
Stability analysis
We make use of the two-time-scale averaging theory in Sastry and Bodson (1989, Section 4.4) to prove that the origin of is exponentially stable. Towards this end, we convert the equilibrium of the closed-loop system (11), (8), (13), (14) to the origin of . We define and note from (9), (16) that
The concatenated form of (11), (8) is given by
An augmented update law
The adaptive design in (13), (14) employs only local signals and to update . We next show that if the agents exchange ’s, the result in Theorem 1 still holds. In Section 6, we use one numerical example to illustrate that exchanging can lead to improved transient performance.
We define a directed graph of nodes that describes the information flow of , . Let be the set of neighbors of the th agent in , . We define a weighted graph Laplacian matrix
Simulation examples
In this section, we present two simulation examples to illustrate the effectiveness of the proposed adaptive designs.
Conclusions and future work
We consider a motion coordination problem where a group of agents is required to achieve consensus while tracking a leader’s sinusoidal reference velocity (subject to a constant bias). Only the leader has the frequency information of the reference velocity. We propose two decentralized two-time-scale internal model designs to estimate the unknown frequency information and achieve consensus. In particular, we introduce a coupling term in the second design that leads to improved transient
He Bai is an Assistant Professor in Mechanical and Aerospace Engineering at Oklahoma State University. He received his B.Eng. degree from the Department of Automation at the University of Science and Technology of China, Hefei, China, in 2005, and the M.S. and Ph.D. degrees in Electrical Engineering from Rensselaer Polytechnic Institute in 2007 and 2009, respectively. From 2009 to 2010, he was a Post-doctoral Researcher at the Northwestern University, Evanston, IL. Before joining Oklahoma State
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Cited by (0)
He Bai is an Assistant Professor in Mechanical and Aerospace Engineering at Oklahoma State University. He received his B.Eng. degree from the Department of Automation at the University of Science and Technology of China, Hefei, China, in 2005, and the M.S. and Ph.D. degrees in Electrical Engineering from Rensselaer Polytechnic Institute in 2007 and 2009, respectively. From 2009 to 2010, he was a Post-doctoral Researcher at the Northwestern University, Evanston, IL. Before joining Oklahoma State University in 2015, he was a Senior Research and Development Scientist at UtopiaCompression Corporation, working with the US Air Force and DARPA on Unmanned Aircraft Systems (UAS) research and technologies. He was the Principal Investigator (PI) or co-PI for a number of research projects on UAS sense-and-avoid, cooperative target tracking, and target handoff in GPS-denied environments. He published a research monograph “Cooperative control design: a systematic passivity-based approach” in Springer in 2011. His research interests include multi-agent systems, nonlinear estimation and sensor fusion, path planning, intelligent control, and GPS-denied navigation for UAS applications.
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The material in this paper was partially presented at the 2015 American Control Conference, July 1–3, 2015, Chicago, IL, USA. This paper was recommended for publication in revised form by Associate Editor Martin Guay under the direction of Editor Miroslav Krstic.