Distributed Implementation of Minimax Adaptive Controller For Finite Set of Linear Systems Venkatraman Renganathan Anders Rantzer and Olle Kjellqvist

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Distributed Implementation of Minimax Adaptive Controller

For Finite Set of Linear Systems

Venkatraman Renganathan, Anders Rantzer, and Olle Kjellqvist

Abstract— This paper deals with a distributed implemen-

tation of minimax adaptive control algorithm for networked

dynamical systems modeled by a ﬁnite set of linear models. To

hedge against the uncertainty arising out of ﬁnite number of

possible dynamics in each node in the network, it collects only

the historical data of its neighboring nodes to decide its control

action along its edges. This makes our proposed distributed ap-

proach scalable. Numerical simulations demonstrate that once

each node has sufﬁciently estimated the uncertain parameters,

the distributed minimax adaptive controller behaves like the

optimal distributed H-inﬁnity controller in hindsight.

I. INTRODUCTION

Control of large-scale and complex systems is often

performed in a distributed manner [1] as it is practically

difﬁcult for every agent in the network to have access to the

global information about the overall networked system while

deciding its control actions. On the other hand, designing

optimal distributed control laws when the networked system

dynamics are uncertain still remains an open problem. This

naturally calls for a learning-based controller to be employed

in such uncertain settings and it was shown in [2] that an

adaptive controller can learn the true system dynamics online

through sufﬁcient parameter estimation and then control

the system. Multiple model-based adaptive control problems

have been extensively studied in the control literature [3]–[8].

The minimax problem of handling adversarial disturbance

and the uncertain parameters leading to an unknown linear

system was introduced in [9]. It was specialized to scalar

systems with unknown sign for input matrix in [10], to

ﬁnite sets of linear systems in [11]. However, designing

optimal distributed control laws that address the uncertainty

prevailing over true model of the networked system out of

the ﬁnite set of linear models still remains an open problem.

Minimax adaptive control problems are challenging

mainly due to the exploration and exploitation trade-off

that inevitably comes with the learning and controlling

procedure. Aiming for a distributed implementation on top

of it complicates things further. However, there are certain

classes of systems for which scalable implementation of

distributed minimax adaptive control is possible, such as the

the one with linear time-invariant discrete time systems with

symmetric and Schur state matrix. Such system models are

common in irrigation networks. For instance, the authors in

This project has received funding from the European Research Coun-

cil (ERC) under the European Union’s Horizon 2020 research and in-

novation program under grant agreement No 834142 (Scalable Con-

trol). The authors are with the Department of Automatic Control LTH,

Lund University, Sweden. (e-mail: (venkatraman.renganathan,anders.rantzer,

olle.kjellqvist)@control.lth.se).

[12] computed a closed-form expression for a decentralised

H∞optimal controller with diagonal gain matrix for network

systems with acyclic graphs. Similarly, the authors in [13]

computed a closed-form expression for the distributed H∞

optimal state feedback law for systems with symmetric

and Schur state matrix, where the total networked system

comprised of subsystems with local dynamics, that share

only control inputs and each control input affecting only two

subsystems.

Contributions: We extend the problem setting in [13]

by considering ﬁnite number of possible local dynamics in

each node and control action along each edge. Our main

contributions in this paper are as follows:

1) We develop scalable and distributed implementation of

minimax adaptive control for networked systems where

each node in the network is required to maintain the

history of just its own neighboring nodes to hedge

against the uncertainty in its local system dynamics.

2) We demonstrate the effectiveness of our proposed

approach using a buffer network based numerical ex-

ample to advocate the use of scalable and distributed

minimax adaptive control law for ﬁnite set of linear

systems.

Following a short summary of notations, this paper is orga-

nized as follows: In §II, the main problem formulation of

distributed minimax adaptive controller and its distributed

implementation is presented. The effectiveness of the pro-

posed algorithm is then demonstrated in §III. Finally, the

paper is closed in §IV along with a summary of results and

directions for future research.

NOTATIONS

The cardinality of the set Ais denoted by |A|. The set of

real numbers, integers and the natural numbers are denoted

by R,Z,Nrespectively. The subset of real numbers greater

than a given constant say a∈Rand between two constants

a, b ∈Nwith a<bare denoted by R>a and [a:b]

respectively. A vector of size nwith all values being one

is denoted by 1n. For a matrix A∈Rn×n, we denote its

transpose and its trace by A>and Tr(A)respectively. We

denote by Snthe set of symmetric matrices in Rn×nand

the set of positive (semi)deﬁnite matrices as Sn

++(Sn

+). A

symmetric matrix P∈Snis said to be positive deﬁnite

(positive semi-deﬁnite) if for every vector x∈Rn\{0},

we have x>P x > 0 (x>P x ≥0) and is denoted by

P0(P0) . An identity matrix in dimension nis

denoted by In. Given x∈Rn, A ∈Rn×n, B ∈Rn×n,

the notations |x|2

Aand kBk2

Amean x>Ax and Tr B>AB

arXiv:2210.00081v1 [eess.SY] 30 Sep 2022

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DistributedImplementationofMinimaxAdaptiveControllerForFiniteSetofLinearSystemsVenkatramanRenganathan,AndersRantzer,andOlleKjellqvistAbstractThispaperdealswithadistributedimplemen-tationofminimaxadaptivecontrolalgorithmfornetworkeddynamicalsystemsmodeledbyanitesetoflinearmodels.Tohedgeagainsttheun...

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Distributed Implementation of Minimax Adaptive Controller For Finite Set of Linear Systems Venkatraman Renganathan Anders Rantzer and Olle Kjellqvist

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