A Distributed Control Approach for Heterogeneous Linear Multiagent Systems

TL;DR

This paper introduces a distributed control method for heterogeneous linear multiagent systems, enabling cooperative output regulation over directed graphs, with new problem definitions and control laws supported by theoretical conditions and numerical examples.

Contribution

It proposes a new definition of the cooperative output regulation problem and analyzes its solvability using three distributed control laws for heterogeneous multiagent systems.

Findings

New definition of cooperative output regulation broadening function tracking capabilities.

Conditions for solvability of the problem with different control laws are established.

Numerical examples demonstrate the effectiveness of the proposed control strategies.

Abstract

This paper considers an internal model based distributed control approach to the cooperative output regulation problem of heterogeneous linear time-invariant multiagent systems over fixed directed communication graph topologies. First, a new definition of the linear cooperative output regulation problem is introduced in order to allow a broad class of functions to be tracked and rejected by a network of agents. Second, the solvability of this problem with three distributed control laws, namely dynamic state feedback, dynamic output feedback with local measurement, and dynamic output feedback, is investigated by first considering a global condition and then providing an agent-wise local sufficient condition under standard assumptions. Finally, two numerical examples are provided to illustrate the selected contributions of this paper.

Equations

$$M_1=T\left[\begin{array}{cc}{S}_1& S_2\\ 0& G_1\end{array}\right]T^{-1},M_2=T\left[\begin{array}{c}{S}_3\\ G_2\end{array}\right],M_3=T\left[\begin{array}{c}{S}_4\\ 0\end{array}\right],$$

$$M_1=G_1,M_2=G_2,M_3=0,$$

$$G_1=\mathrm{diag}({\beta }_1,\dots ,{\beta }_p),G_2=\mathrm{diag}({\sigma }_1,\dots ,{\sigma }_p),$$

$${\dot{x}}_i(t)$$

$$y_i(t)$$

$${\dot{x}}_i(t)$$

$$e_i(t)$$

$$e_{\mathrm{v}i}(t)$$

$$u_i(t)$$

$${\dot{z}}_i(t)$$

$$y_{\mathrm{m}i}(t)$$

$$u_i(t)$$

$${\dot{z}}_i(t)$$

$$u_i(t)$$

$${\dot{z}}_i(t)$$

Full text

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**A Distributed Control Approach for Heterogeneous

Linear Multiagent Systems⋆** ††thanks: ⋆  This research was supported by the Dynamics, Control, and Systems Diagnostics Program of the National Science Foundation under Grant CMMI-1657637. ††thanks: ‡  Corresponding Author: Engineering Building C 2209, 4202 East Fowler Avenue, Tampa, Florida 33620, United States of America (Address); +1 813 974 5656 (Phone); yucelen@usf.edu (Email).

Selahattin Burak Sarsilmaz  and  Tansel Yucelen†**

Department of Mechanical Engineering

University of South Florida

Abstract — This paper considers an internal model based distributed control approach to the cooperative output regulation problem of heterogeneous linear time-invariant multiagent systems over fixed directed communication graph topologies. First, a new definition of the linear cooperative output regulation problem is introduced in order to allow a broad class of functions to be tracked and rejected by a network of agents. Second, the solvability of this problem with three distributed control laws, namely dynamic state feedback, dynamic output feedback with local measurement, and dynamic output feedback, is investigated by first considering a global condition and then providing an agent-wise local sufficient condition under standard assumptions. Finally, two numerical examples are provided to illustrate the selected contributions of this paper.

Keywords — Heterogeneous multiagent systems;  distributed control;  cooperative output regulation;  internal model

1 Introduction

Heterogeneous multiagent systems formed by networks of agents having different dynamics and dimensions present a significantly broader class of multiagent systems than their heterogeneous and homogeneous counterparts that consist of networks of agents having different dynamics with the same dimension and identical dynamics, respectively. Therefore, analysis and synthesis of distributed control approaches for this class of multiagent systems that rely on local information exchange has been an attractive research topic in the systems and control field over the last decade.

In particular, the cooperative output regulation problem of heterogeneous (in dynamics and dimension) linear time-invariant multiagent systems, where the output of all agents synchronize to the output of the leader, over general fixed directed communication graph topologies have been recently investigated in [Su_2012, su_huang_2012_b, Huang_C_tac, Li_2015, Farnaz_automatica, Cai_2017, Lu_Liu_TAC_2017]. This problem can be regarded as the generalization of the linear output regulation problem given in, for example, [Huang_book] to multiagent systems. As a consequence, distributed control approaches to this regulation problem can be classified into two categories:

• •

The first category is predicated on feedforward design methodology, where the authors of [Su_2012, su_huang_2012_b, Li_2015, Cai_2017, Lu_Liu_TAC_2017] present contributions. In the presence of plant uncertainties, however, this methodology is known to be not robust since the feedforward gain of each agent relies on the solution of the regulator equations.

• •

The second category is predicated on internal model principle, where the authors of [Huang_C_tac, Farnaz_automatica] present contributions. While this methodology is robust with respect to small variations of the plant parameters as compared to feedforward design methodology, it cannot be applied when the transmission zero condition does not hold.

The common denominator of these papers is that an exosystem, which has an unforced linear time-invariant dynamics, generates both a reference trajectory and external disturbances to be tracked and rejected by networks of agents. Specifically, the system matrix of the exosystem is explicitly used by controllers of all agents in [Su_2012, su_huang_2012_b, Li_2015, Lu_Liu_TAC_2017] and a proper subset of agents in [Cai_2017]; or each agent incorporates a $p$-copy internal model of this matrix in its controller [Huang_C_tac, Farnaz_automatica].

1.1 Contributions

Considering applications of the distributed control approaches in [Su_2012, su_huang_2012_b, Huang_C_tac, Li_2015, Farnaz_automatica, Cai_2017, Lu_Liu_TAC_2017], it can be a challenge to precisely know the system matrix of the exosystem, even the dynamical structure of the exosystem; especially, when an external leader interacts with the network of agents or a control designer simply injects optimized trajectory commands to the network based on, for example, an online path planning algorithm. In order to guarantee ultimately bounded tracking error in such cases, a new, generalized definition for the cooperative output regulation problem is needed.

This paper focuses on heterogeneous (in dynamics and dimension) linear time-invariant multiagent systems over general fixed directed communication graph topologies. First, we present the generalized definition for the linear cooperative output regulation problem. Second, we investigate the solvability of this problem for internal model based distributed dynamic state feedback, output feedback with local measurement, and output feedback control laws. To this end, we not only consider global conditions but also provide agent-wise local sufficient conditions under standard assumptions. Considering large-scale applications of multiagent systems, the agent-wise local sufficient conditions are primarily important for independent controller design of each agent (i.e., without depending on the dynamics of other agents ).

The system-theoretical approach presented in this paper111  Although they are not completely related, [sarsilmaz_yucelen_asme_dscc_2017, sarsilmaz_yucelen_acc_2018] may be regarded as preliminary works of this paper. is relevant to the studies in [Huang_C_tac, Farnaz_automatica], where they also focus on the linear cooperative output regulation problem with an internal model based distributed dynamic state feedback control law. Specifically, [Farnaz_automatica] extends the approach in [Huang_C_tac] to an output feedback control under an output feedback stabilizability condition. In addition to the generalized definition of the linear cooperative output regulation problem, the contribution of this paper differs from the studies in [Huang_C_tac, Farnaz_automatica] based on the following points:

• •

First, we note that the theoretical contribution of this paper covers not only the dynamic state feedback problem but also the dynamic output feedback problem with local measurement as well as the dynamic output feedback problem. Unlike the results presented in [Farnaz_automatica], this paper does not assume the output feedback stabilizability for the dynamic output feedback problem with local measurement. With regard to the dynamic output feedback problem, the results of this paper does not require agents to access their own states or outputs.

• •

To prove the existence of a unique solution to the matrix equations that are crucial for the solvability of the problem, Section III in [Huang_C_tac] (Theorem 4 in [Farnaz_automatica]) decomposes these matrix equations, which consist of the overall dynamics of the multiagent system, into matrix equations, which deal with the dynamics of each agent separately. In contrast, we do not decompose these matrix equations; see the sixth paragraph of Appendix A for the advantage. In particular, Lemma LABEL:Lemma3 of this paper, which is also applicable to dynamic output feedback cases, guarantees that these matrix equations have a unique solution without requiring their decompositions.

• •

A considerable number of gaps in the related results of [Huang_C_tac, Farnaz_automatica] is illustrated by counterexamples in Appendices and fixed in Appendices as well as in Section LABEL:first_controller.

1.2 Organization

The rest of the paper is organized as follows. Section 2 presents the notation and the essential mathematical preliminaries. Section 3 formulates the linear cooperative output regulation problem considered in this paper. The solvability of this problem is investigated in Section LABEL:DSF and two illustrative numerical examples are presented in Section LABEL:INE. Finally, Section LABEL:con concludes the paper.

2 Mathematical Preliminaries

A standard notation is used in this paper. Specifically, $\mathbb{R}$, $\mathbb{R}^{n}$, and $\mathbb{R}^{n\times m}$ respectively denote the sets of all real numbers, $n\times 1$ real column vectors, and $n\times m$ real matrices222  In this paper, all real matrices are defined over the field of complex numbers.; $\textbf{1}_{n}$ and $I_{n}$ respectively denote the $n\times 1$ vector of all ones and the $n\times n$ identity matrix; and “$\triangleq$” denotes equality by definition. We write $(\cdot)^{\mathrm{T}}$ for the transpose and $\|\cdot\|_{2}$ for the induced two norm of a matrix; $\sigma(\cdot)$ for the spectrum333  We follow Definition 4.4.4 in [Bernstein_book]. and $\rho(\cdot)$ for the spectral radius of a square matrix; $(\cdot)^{-1}$ for the inverse of a nonsingular matrix; and $\otimes$ for the Kronecker product. We also write $A\leq B$ for $A\in\mathbb{R}^{n\times m}$, $B\in\mathbb{R}^{n\times m}$ if entries $a_{ij}\leq b_{ij}$ for all ordered pairs $(i,j)$. Finally, $\text{diag}(A_{1},\ldots,A_{n})$ is a block-diagonal matrix with matrix entries $A_{1},\dots,A_{n}$ on its diagonal.

We now concisely state the graph theoretical notation used in this paper, which is based on [Lewis_book]. In particular, consider a fixed (i.e., time-invariant) directed graph ${\mathcal{G}}=(\mathcal{V},\mathcal{E})$, where \mathcal{V}=\big{\{}v_{1},\ldots,v_{N}\big{\}} is a nonempty finite set of $N$ nodes and $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is a set of edges. Each node in $\mathcal{V}$ corresponds to a follower agent. There is an edge rooted at node $v_{j}$ and ended at $v_{i}$ (i.e., $(v_{j},v_{i})\in\mathcal{E}$) if and only if $v_{i}$ receives information from $v_{j}$. $\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ denotes the adjacency matrix, which describes the graph structure; that is, $a_{ij}>0\Leftrightarrow(v_{j},v_{i})\in\mathcal{E}$ and $a_{ij}=0$ otherwise. Repeated edges and self loops are not allowed; that is, $a_{ii}=0,\ \forall i\in{\mathcal{N}}$ with {\mathcal{N}}=\big{\{}1,\ldots,N\big{\}}. The set of neighbors of node $v_{i}$ is denoted as N_{i}=\big{\{}j\ |\ (v_{j},v_{i})\in\mathcal{E}\big{\}}. In-degree matrix is defined by $\mathcal{D}=\mathrm{diag}(d_{1},\ldots,d_{N})$ with $d_{i}=\sum_{j\in N_{i}}a_{ij}$. A directed path from node $v_{i}$ to node $v_{j}$ is a sequence of successive edges in the form \big{\{}(v_{i},v_{p}),(v_{p},v_{q}),\ldots,(v_{r},v_{j})\big{\}}. If $v_{i}=v_{j}$, then the directed path is called a loop. A directed graph is said to have a spanning tree if there is a root node such that it has directed paths to all other nodes in the graph. A fixed augmented directed graph is defined as ${\mathcal{\bar{G}}}=(\mathcal{\bar{V}},\mathcal{\bar{E}})$, where \mathcal{\bar{V}}=\big{\{}v_{0},v_{1},\ldots,v_{N}\big{\}} is the set of $N+1$ nodes, including leader node $v_{0}$ and all nodes in $\mathcal{V}$, and $\mathcal{\bar{E}}=\mathcal{E}\cup\mathcal{E^{\prime}}$ is the set of edges with $\mathcal{E^{\prime}}$ consisting of some edges in the form of $(v_{0},v_{i})$, $i\in\mathcal{N}$.

The concept of internal model introduced next slightly modifies Definition 1.22 and Remark 1.24 in [Huang_book].

Definition 1

Given any square matrix $A_{0}$, a triple of matrices $(M_{1},M_{2},M_{3})$ is said to incorporate a $p$-copy internal model of the matrix $A_{0}$ if

[TABLE]

or

[TABLE]

where $S_{l},\ l=1,2,3,4$, is any matrix with an appropriate dimension, $T$ is any nonsingular matrix with an appropriate dimension, the zero matrix in $M_{3}$ has as many rows as those of $G_{1}$, and

[TABLE]

where for $l=1,\dots,p$, $\beta_{l}\in\mathbb{R}^{s_{l}\times s_{l}}$ and $\sigma_{l}\in R^{s_{l}}$ satisfy the following conditions:

$a)$* The pair $(\beta_{l},\sigma_{l})$ is controllable.

$b)$  The minimal polynomial of $A_{0}$ is equal to the characteristic polynomial of $\beta_{l}$.*

3 Problem Formulation

Consider a system of $N$ (follower) agents with heterogeneous linear time-invariant dynamics subject to external disturbances over a fixed directed communication graph topology $\mathcal{G}$. The dynamics of agent $i\in\mathcal{N}$ is given by

[TABLE]

with state $x_{i}(t)\in\mathbb{R}^{n_{i}}$, input $u_{i}(t)\in\mathbb{R}^{m_{i}}$, output $y_{i}(t)\in\mathbb{R}^{p}$, and external disturbance $\delta_{i}(t)=E_{\delta_{i}}\delta(t)\in\mathbb{R}^{n_{i}}$, where $\delta(t)\in\mathbb{R}^{q_{\delta}}$ is a solution to the unknown disturbance dynamics with an initial condition. In addition, the reference trajectory to be tracked is denoted by $y_{0}(t)=R_{\mathrm{r}}r_{0}(t)\in\mathbb{R}^{p}$, where $r_{0}(t)\in\mathbb{R}^{q_{\mathrm{r}}}$ is a solution to the unknown leader dynamics with an initial condition.

Let $\omega(t)\triangleq[r_{0}^{\mathrm{T}}(t),\delta^{\mathrm{T}}(t)]^{\mathrm{T}}$ $\in\mathbb{R}^{q}$ be the solution of the unknown exosystem, where $q=q_{\mathrm{r}}+q_{\delta}$. Instead of assuming that the exosystem has an unforced linear time-invariant dynamics with a known system matrix (e.g., see [Su_2012, Huang_C_tac, Farnaz_automatica]), we consider that the exosystem has an unknown dynamics. From this perspective, the exosystem can represent any (e.g., linear or nonlinear) dynamics provided that its solution is unique and satisfies the conditions given later in Assumptions LABEL:A1 and LABEL:A2.

Define $E_{i}\triangleq[0\ E_{\delta_{i}}]$ and $R\triangleq[R_{\mathrm{r}}\ 0]$. Furthermore, let $e_{i}(t)\triangleq y_{i}(t)-y_{0}(t)$ be the tracking error. We can then write the dynamics of each agent and its tracking error as

[TABLE]

In this paper, the tracking error $e_{i}(t)$ is available to a nonempty proper subset of agents444  If all agents observe the leader, decentralized controllers can be designed for each agent even though the distributed controllers proposed here are still applicable.. In particular, if node $v_{i}$ observes the leader node $v_{0}$, then there exists an edge $(v_{0},v_{i})$ with weighting gain $k_{i}>0$; otherwise $k_{i}=0$. Each agent has also access to the relative output error; that is, $y_{i}(t)-y_{j}(t)$ for all $j\in N_{i}$. Similar to [Farnaz_automatica], the local virtual tracking error can be defined as

[TABLE]

Now, we define three classes of distributed control laws based on additional available information to each agent:

$1)$** Dynamic State Feedback.*** If each agent has full access to its own state $x_{i}(t)$, then the dynamic state feedback control law is given by*

[TABLE]

where $z_{i}(t)\in\mathbb{R}^{n_{z_{1i}}}$ is the controller state and the quadruple $(K_{1i},K_{2i},G_{1i},G_{2i})$ is specified in Section LABEL:first_controller.

$2)$** Dynamic Output Feedback with Local Measurement.*** If each agent has local measurement output $y_{\mathrm{m}i}(t)\in\mathbb{R}^{p_{i}}$ of the form*

[TABLE]

then the dynamic output feedback control law with local measurement is given by

[TABLE]

where $z_{i}(t)\in\mathbb{R}^{n_{z_{2i}}}$ is the controller state and the quadruple $(\bar{K}_{i},M_{1i},M_{2i},M_{3i})$ is specified in Section LABEL:second_controller.

$3)$** Dynamic Output Feedback.*** If each agent does not have additional information; that is, the local virtual tracking error $e_{\mathrm{v}i}(t)$ is the only available information to it, then the dynamic output feedback control law is given by*

[TABLE]

where $z_{i}(t)\in\mathbb{R}^{n_{z_{2i}}}$ is the controller state and the triple $(\bar{K}_{i},M_{1i},M_{2i})$ is specified in Section LABEL:third_controller.

We now introduce the first and the second assumptions before defining the problem.