Robust feedback stabilization of interacting multi-agent systems under uncertainty

2025-04-15 0 0 1.19MB 27 页 10玖币

侵权投诉

Robust feedback stabilization of interacting multi-agent

systems under uncertainty

Giacomo Albi∗

, Michael Herty†

, and Chiara Segala‡

Abstract

We consider control strategies for large-scale interacting agent systems under un-

certainty. The particular focus is on the design of robust controls that allow to bound

the variance of the controlled system over time. To this end we consider H∞control

strategies on the agent and mean ﬁeld description of the system. We show a bound

on the H∞norm for a stabilizing controller independent on the number of agents.

Furthermore, we compare the new control with existing approaches to treat uncer-

tainty by generalized polynomial chaos expansion. Numerical results are presented for

one-dimensional and two-dimensional agent systems.

Keywords. Agent-based dynamics, mean-ﬁeld equations, uncertainty quantiﬁcation, stochas-

tic Galerkin, H∞control

AMS Classiﬁcation. 49K15, 49M25, 93-10, 93D09, 35Q70, 35Q93

1 Introduction

We consider the mathematical modelling and control of phenomena of collective dynamics

under uncertainties. These phenomena have been studied in several ﬁelds such as socio-

economy, biology, and robotics where systems of interacting particles are given by self-

propelled particles, such as animals and robots, see e.g. [1, 7, 15, 24, 32]. Those particles

interact according to a possibly nonlinear model, encoding various social rules as attraction,

∗Department of Computer Science, University of Verona, Str. Le Grazie 15, Verona, I-37134, Italy

†IGPM, RWTH Aachen University, Templergraben, 55, D-52062 Aachen, Germany

‡IGPM, RWTH Aachen University, Templergraben, 55, D-52062 Aachen, Germany

MH and CS thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for the

ﬁnancial support through 320021702/GRK2326, 333849990/IRTG-2379, HE5386/18-1,19-2,22-1,23-1 and

under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612. GA thanks the Italian

Ministry of Instruction, University and Research (MIUR) to support this research with funds coming from

PRIN Project 2017 (No. 2017KKJP4X entitled “Innovative numerical methods for evolutionary partial

diﬀerential equations and applications”).

arXiv:2210.01699v2 [math.OC] 5 Oct 2022

repulsion, and alignment. A particular feature of such models is their rich dynamical struc-

ture, which includes diﬀerent types of emerging patterns, including consensus, ﬂocking, and

milling [17, 23, 29, 45, 50]. Understanding the impact of control inputs in such complex sys-

tems is of great relevance for applications. Results in this direction allow to design optimized

actions such as collision-avoidance protocols for swarm robotics [14, 26, 46, 48], pedestrian

evacuation in crowd dynamics [16, 22], supply chain policies [18, 34], the quantiﬁcation of

interventions in traﬃc management [31, 49, 51] or in opinion dynamics [27, 28]. Further,

the introduction of uncertainty in the mathematical modelling of real-world phenomena

seems to be unavoidable for applications, since often at most statistical information of the

modelling parameters is available. The latter has typically been estimated from experiments

or derived from heuristic observations [5, 8, 37]. To produce eﬀective predictions and to de-

scribe and understand physical phenomena, we may incorporate parameters reﬂecting the

uncertainty in the interaction rules, and/or external disturbances [13].

Here, we are concerned with the robustness of controls inﬂuencing the evolution of a col-

lective motion of an interacting agent system. The controls we are considering are aimed to

stabilize the system’s dynamic under external uncertainty. From a mathematical point of

view, a description of self-organized models is provided by complex system theory, where

the overall dynamics are depicted by a large-scale system of ordinary diﬀerential equations

(ODEs).

More precisely, we consider the control of high-dimensional dynamics accounting Nagents

with state vi(t, θ)∈Rd, i = 1, . . . , N, evolving according to

dtvi(t, θ) =

j=1

aij (vj(t, θ)−vi(t, θ)) + ui(t, θ) +

k=1

θk, vi(0) = v0

i,(1.1)

where A= [aij ]∈RN×Ndeﬁnes the nature of pairwise interaction among agents, and θ=

(θ1, . . . , θZ)>∈RZ×dis a random input vector with a given probability density distribution

on Zas ρ≡ρ1⊗. . . ⊗ρZ. The control signal ui(t, θ)∈Rdis designed to stabilize the state

toward a target state ¯v∈RN×d, and its action is inﬂuenced by the random parameter

θ. This is also due to the fact, that later we will be interested in closed–loop or feedback

controls on the state (v1. . . . , vN)that in turn dependent on the unknown parameter θ.

Of particular interest will be controls designed via minimization of linear quadratic (para-

metric) regulator functional such as

min

u(·,θ)J(u;v0) := Z+∞

exp(−rτ)hv>Qv +νu>Ruidτ, (1.2)

with Qpositive semi-deﬁnite matrix of order N,Rpositive deﬁnite matrix of order N

and ris a discount factor. In this case, the linear quadratic dynamics allow for an optimal

control u∗stabilising the desired state vd= 0, expressed in feedback form, and obtained

by solving the associated matrix Riccati -equations. Those aspects will be also addressed

in more detail below.

In order to assess the performances of controls, and quantify their robustness we propose

estimates using the concept of H∞control. In this setting diﬀerent approaches have been

studied in the context of H∞control and applied to ﬁrst-order and higher-order multiagent

systems, see e.g. [40, 41, 42, 43, 44], in particular for an interpretation of H∞as dynamic

games we refer to [6]. Here we will study an approach based on the derivation of suﬃcient

conditions in terms of linear matrix inequalities (LMIs) for the H∞control problem. In this

way, consensus robustness will be ensured for a general feedback formulation of the control

action. Additionally, we consider the large–agent limit and show that the robustness is

guaranteed independently of the number of agents.

Furthermore, we will discuss the numerical realization of system (1.1) employing uncertainty

quantiﬁcation techniques. In general, at the numerical level, techniques for uncertainty

quantiﬁcation can be classiﬁed into non-intrusive and intrusive methods. In a non-intrusive

approach, the underlying model is solved for ﬁxed samples with deterministic schemes,

and statistics of interest are determined by numerical quadrature, typical examples are

Monte-Carlo and stochastic collocation methods [19, 53]. While in the intrusive case, the

dependency of the solution on the stochastic input is described as a truncated series ex-

pansion in terms of orthogonal functions. Then, a new system is deduced that describes

the unknown coeﬃcients in the expansion. One of the most popular techniques of this

type is based on stochastic Galerkin (SG) methods. In particular, generalized polynomial

chaos (gPC) gained increasing popularity in uncertainty quantiﬁcation (UQ), for which

spectral convergence on the random ﬁeld is observed under suitable regularity assumptions

[19, 35, 36, 53]. The methods, here developed, make use of the stochastic Galerkin (SG) for

the microscopic dynamics while in the mean-ﬁeld case we combine SG in the random space

with a Monte Carlo method in the physical variables.

The manuscript is organized as follows, in Section 2 we introduce the problem setting

and propose diﬀerent feedback control laws; in Section 3 we reformulate the problem in

the setting of H∞control and provide conditions for the robustness of the controls in

the microscopic and mean-ﬁeld case. Section 4 is devoted to the description of numerical

strategies for the simulation of the agent systems, and to diﬀerent numerical experiments,

which assess the performances and compare diﬀerent methods.

2 Control of interacting agent system with uncertainties

The following notation is introduced with the control of high-dimensional systems of in-

teracting agents with random inputs. We consider the evolution of Nagents with state

v(t, θ)∈RN×das follows

dtvi(t, θ) = 1

j=1

¯p(vj(t, θ)−vi(t, θ)) + ui(t, θ) +

k=1

θk(2.1)

with deterministic initial data vi(0) = v0

ifor i= 1, . . . , N, and where θk∈Ωk⊆Rdfor

k= 1, . . . , Z are random inputs, distributed according to a compactly supported probability

density ρ≡ρ1⊗ ··· ⊗ ρZ, i.e., ρk(θ)≥0a.e., supp(ρk)⊆Ωkand RΩkρk(θ)dθ = 1. For

simplicity, we also assume that the random inputs have zero average E[θk]=0. The control

signal u(t, θ)∈RN×dis designed minimizing the (parameterized) objective

u∗(·, θ) = arg min

u(·,θ)J(u;v0) := Z+∞

exp(−rτ)

1

j=1

(|vj(τ, θ)−¯v|2+ν|uj(τ, θ)|2)

dτ,

(2.2)

with ν > 0being a penalization parameter for the control energy, the norm |·|being

the usual Euclidean norm in Rd. The discount factor exp(−rτ)is introduced to have a

well-posed integral.

We assume that ¯vis a prescribed consensus point, namely, in the context of this work we

are interested in reaching a consensus velocity ¯v∈Rdsuch that v1=. . . =vN= ¯v, and

w.l.o.g. we can assume ¯v= 0. Note that ¯v= 0 is also the steady state of the dynamics

in absence of disturbances. Hence, we may view u(·, θ)as a stabilizing control of the zero

steady state of the system. Furthermore, we will be interested in feedback controls u.

Recall that the (deterministic) linear model (2.1), without uncertainties, allows a feedback

stabilization by solving the resulting optimal control problem through a Riccati equations

[2, 3, 33]. The functional Jin (2.2), in absence of disturbances, reads as follows

J(u;v0) = Z+∞

exp(−rτ)v>Qv +νu>Rudt

where Q≡R=1

NIdN. In this case the controlled dynamics (2.1) is reformulated in a

matrix-vector notation

dtv(t) = Av(t) + Bu(t), u(t) = −N

νKv(t),(2.3)

with B=IdNthe identity matrix of order N, and

(A)ij =(ad=¯p(1−N)

N, i =j,

ao=¯p

N, i 6=j. (2.4)

The matrix Kassociated to feedback form of the optimal control has to fulﬁlll the Riccati

matrix-equation of the following form

0 = −rK +KA +A>K−N

νKK +Q. (2.5)

For a general linear system, we need to solve the N×Nequations to ﬁnd K, which can

be costly for large-scale agent-based dynamics. However, we can use the same argument of

[3] and exploit the symmetric structure of the Laplacian matrix Ato reduce the algebraic

Riccati equation. Unlike in [3], where they investigate the case with ﬁnite terminal time,

here we state the following proposition for the inﬁnite horizon case with discount factor r

Proposition 2.1 (Properties of the Algebraic Riccati Equation (ARE)).For the linear

dynamics (2.3), the solution of the Riccati equation (2.5) reduces to the solution of

0 = −rkd−2¯p(N−1)

N(kd−ko)−N

νk2

d+ (N−1)k2

o+1

0 = −rko+2¯p

N(kd−ko)−N

ν2kdko+ (N−2)k2

o.

(2.6)

The entries (i, j)of the matrix Kof the algebraic Riccati equation (2.5) is given by

(K)ij =δij kd+ (1 −δij )ko.

In order to allow the limit of inﬁnitely many agents N→ ∞, we introduce the following

scalings

kd←Nkd, ko←N2ko, α(N) = N−1

and keeping the same notation also for the scaled variables kd, ko, the system (2.6) reads

0 = −rkd−2¯pα(N)kd−ko

N−1

νk2

d+α(N)

Nk2

o+ 1,

0 = −rko+ 2¯pkd−ko

N−1

ν2kdko+α(N)k2

o−1

Nk2

o.

(2.7)

The previous considerations motivate to extend formula (2.3) to the parametric case (2.1).

Hence, in the presence of parametric uncertainty the feedback control is written explicitly

as follows

ui(t, θ) = −1

ν(Kv(t, θ))i=−1

ν

kd−ko

Nvi(t, θ) + ko

j=1

vj(t, θ)

.(2.8)

The question arises if the given feedback is robust with respect to the uncertainties θ. In

the following, we will provide a measure for the robustness of control (2.8) in the framework

of H∞control. Some additional remarks allow generalizing formula (2.8).

Remark 2.1 (Non-zero average).In presence of general uncertainties with known expec-

tations, we modify the control (2.8) for model (2.1) including a correction factor given by

the expected values of the random inputs,

ui(t, θ) = −1

ν

kd−ko

Nvi(t, θ) + ko

j=1

vj(t, θ)

−

k=1

µk,(2.9)

for µk=E[θk]for k= 1, . . . , Z.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Robustfeedbackstabilizationofinteractingmulti-agentsystemsunderuncertaintyGiacomoAlbi*,MichaelHerty,andChiaraSegalaAbstractWeconsidercontrolstrategiesforlarge-scaleinteractingagentsystemsunderun-certainty.Theparticularfocusisonthedesignofrobustcontrolsthatallowtoboundthevarianceofthecontrolledsyst...

展开>> 收起<<

Robust feedback stabilization of interacting multi-agent systems under uncertainty.pdf

共27页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Robust feedback stabilization of interacting multi-agent systems under uncertainty

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: