On a Canonical Distributed Controller in the Behavioral Framework

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arXiv:2210.06268v2 [math.OC] 10 Apr 2023

On a Canonical Distributed Controller in the Behavioral

Framework⋆

Tom R.V. Steentjesa,∗, Mircea Lazara, Paul M.J. Van den Hofa

aDepartment of Electrical Engineering, Eindhoven University of Technology, P.O. Box

513, Eindhoven, 5600 MB, The Netherlands

Abstract

Control in a classical transfer function or state-space setting typically views

a controller as a signal processor: sensor outputs are mapped to actuator

inputs. In behavioral system theory, control is simply viewed as intercon-

nection; the interconnection of a plant with a controller. In this paper we

consider the problem of control of interconnected systems in a behavioral

setting. The behavioral setting is especially ﬁt for modelling interconnected

systems, because it allows for the interconnection of subsystems without im-

posing inputs and outputs. We introduce a so-called canonical distributed

controller that implements a given interconnected behavior that is desired,

provided that necessary and suﬃcient conditions hold true. The controller

design can be performed in a decentralized manner, in the sense that a local

controller only depends on the local system behavior. Regularity of intercon-

nections is an important property in behavioral control that yields feedback

interconnections. We provide conditions under which the interconnection of

this distributed controller with the plant is regular. Furthermore, we show

that the interconnections of subsystems of the canonical distributed con-

troller are regular if and only if the interconnections of the plant and desired

behavior are regular.

Keywords: behavioral control, distributed control, canonical controller,

⋆This work has received funding from the European Research Council (ERC), Advanced

Research Grant SYSDYNET, under the European Union’s Horizon 2020 research and

innovation programme (Grant Agreement No. 694504).

∗Corresponding author

Email addresses: t.r.v.steentjes@tue.nl (Tom R.V. Steentjes), m.lazar@tue.nl

(Mircea Lazar), p.m.j.vandenhof@tue.nl (Paul M.J. Van den Hof)

Preprint submitted to Systems & Control Letters April 12, 2023

interconnected systems

1. Introduction

When physical systems are interconnected, no distinction between inputs

and outputs is made. Think for example of the interconnection of two RLC-

circuits through their terminals or the interconnection of two mass-spring-

damper systems. Typical transfer-function and input-state-output represen-

tations inherently impose an input-output partition of system variables. One

of the main features of the behavioral approach to system theory, is that it

does not take an input-output structure as a starting point to describe sys-

tems: a mathematical model is simply the relation between system variables.

In the case of dynamical systems, the set of all time trajectories that are

compatible with the model is called the behavior. The behavioral approach

has been advocated as a convenient starting point in several applications,

among which in the context of interconnected systems [1] and the context of

control [2].

In the context of interconnected systems, modelling can be performed

through tearing (viewing the interconnected system as an interconnection of

subsystems), zooming (modelling the subsystems), and linking (modelling

the interconnections) [1]. Interconnection of systems in a behavioral setting

means variable sharing. When two masses are physically interconnected, the

laws of motion for the ﬁrst mass involve the position of the second mass and

vice versa; the laws of motion of both masses together dictate the behavior

of the interconnected system. Thinking of system interconnections makes

the modelling of interconnected systems remarkably simple. Partitioning

variables into input and output variables is appropriate in signal processing,

feedback control based on sensor outputs and other unilateral systems, but

often unnecessary for physical system variables [1].

Feedback control based on sensor outputs to generate actuator inputs,

where the controller is viewed as a signal processor [3], holds an impor-

tant place in control theory. It has been argued that many practical con-

trol devices cannot be interpreted as feedback controllers, however, such as

passive-vibration control systems, passive suspension systems or operational

ampliﬁers [2]. Indeed, such control systems do not inherit a signal ﬂow, but

can be interpreted as an interconnection in a behavioral setting. Control by

interconnection allows the control design to take place without distinguish-

ing between control inputs and measured outputs, a priori [2], and can be

performed for, e.g., stabilization [4], H∞control [5] and robust control [6].

Control by interconnection in a behavioral setting means restricting the

behavior of the system that is to be controlled, by interconnecting it with a

controller. By specifying a behavior that is desired for the controlled system,

an important control problem is to determine the existence of a controller

such that the controlled system’s behavior is equal to the desired behavior.

This is called the implementability problem [3]. The canonical controller

plays a major role in the implementability problem: the canonical controller

implements the desired behavior if and only if the desired behavior is imple-

mentable [7], [8].

In this paper, we will consider distributed control in a behavioral setting.

In particular, we will consider distributed control of interconnected linear

time-invariant systems. As a natural consequence of behavioral interconnec-

tions, we consider a distributed controller to be an interconnected system

itself, i.e., we consider it to consist of subsystems that are interconnected

without imposing signal ﬂows between subsystems. Several types of inter-

connections become of interest in this problem: interconnections between

subsystems of the to-be-controlled interconnected system (plant), intercon-

nections between subsystems of the plant and subsystems of the distributed

controller, and interconnections between subsystems of the distributed con-

troller. Given a desired behavior for the controlled interconnected system

that has the same interconnection structure as the plant, the considered

distributed control problem is to determine the existence of a distributed

controller that implements the desired behavior. We introduce a canonical

distributed controller which implements the desired interconnected behavior

under necessary and suﬃcient implementability conditions on the manifest

plant and desired behavior. The distributed canonical controller has an at-

tractive interconnection structure, in the sense that two of its subsystems are

interconnected only if two subsystems of the plant or desired behavior are

interconnected.

Distributed control with input-output partitioning and communication

between subsystems of the distributed controller follows as an important

special case of distributed control in a behavioral setting. An important

question is: when can the canonical distributed controller be implemented

with feedback interconnections? Following up on this question: When can

the interconnections between controller subsystems be implemented as com-

munication channels? The main concept in the solution to these problems is

regularity of the corresponding interconnections. We will analyze regularity

of the canonical distributed controller. In particular, we show that the con-

nections between subsystems of this distributed controller are regular if and

only if connections between subsystems of the plant and desired behavior are

regular.

2. Preliminaries

Behavioral notions

For the notions of systems in the behavioral setting, we will follow the

notation in [3]. A dynamical system is deﬁned as a triple Σ = (T, W, B),

where T⊆Ris the time axis, Wis the signal space and B⊆WTis the

behavior. Consider two dynamical systems Σ1= (T, W1×W3,B1) and Σ2=

(T, W2×W3,B2) with the same time axis, and trajectories (w1, w3)∈B1

and (w2, w3)∈B2, respectively. The interconnection of Σ1and Σ2through

w3yields the dynamical system

Σ1∧w3Σ2:= (T, W1×W2×W3,B),

with B:= {(w1, w2, w3)|(w1, w3)∈B1and (w2, w3)∈B2}. The manifest

behavior of Σ1with respect to w1is

(B1)w1:= {w1:T→W1| ∃w3so that (w1, w3)∈B}.

The set Lwdenotes the set of all linear diﬀerential systems Σ = (R,Rw,B),

with w∈Nvariables, where the behavior is

B:= {w∈C∞(R,Rw)|R(d

dt)w= 0},

with a polynomial matrix R∈Rg×w[ξ], g∈N>0, and C∞(R,Rw) denotes the

set of inﬁnitely often diﬀerentiable functions from Rto Rw.

Consider a behavior B∈Lw. The components of w∈Ballow for a

component-wise partition1such that w= (u, y), with uinput and youtput.

The partition w= (u, y) is called an input-output partition if uis free, i.e.,

for all uthere exists a yso that (u, y)∈B, and ydoes not contain any

further free components, i.e., uis maximally free [9, Deﬁnition 3.3.1], cf.

1Up to re-ordering of the components in w.

[10, Deﬁnition 2.9.2]. The number of components in the input and output,

called the input and output cardinality, is invariant, i.e., independent of the

input-output partition. Henceforth, m(B) denotes the input cardinality and

p(B) denotes the output cardinality, which implies that p(B) + m(B) = w.

For a kernel representation Rd

dtw= 0 of B, the output cardinality is

p(B) = rank R.

Control by interconnection

A controlled interconnection is the interconnection of a plant Σp= (T, W ×

C, P) and a controller Σc= (T, C, C), with the same time axis, and trajecto-

ries (w, c)∈ P and c∈ C, respectively. The plant has two types of variables:

wis the to-be-controlled variable and cis the control variable. The controlled

interconnection is thus P ∧cC. A general control problem can now be formu-

lated as: Given the plant behavior Pand a desired behavior K ⊆ WT, does

there exist a controller Cso that K= (P ∧cC)w, i.e., is Kimplementable? The

implementability problem has been extensively studied in [3, 11]. Necessary

and suﬃcient conditions for implementability are recalled in the following

theorem.

Theorem 1 ([11]).Let P ∈ Lw+cbe a plant with (P)w∈Lwits manifest

behavior and N:= {w∈ P | (w, 0) ∈ P} its hidden behavior. Then K ∈ Lw

is implementable by a controller C ∈ Lcif and only if

N ⊆ K ⊆ (P)w.

3. Control of interconnected systems

3.1. Plant interconnections

For the design of a distributed controller, we consider Lsystems (plants)

Σpi= (T, Wi×Si×Ci,Pi), i∈Z[1:L]:= Z∩[1, L], having trajectories

(wi, si, ci)∈ Pi, with withe to-be-controlled variable, sithe inter-plant con-

nection variable and cithe control variable. Partition the inter-plant con-

nection variable siinto sij , the variable that behavior Pishares with Pj.

The variable sharing is symmetric in the sense that if Pishares variable

sij with Pj, then Pjshares variable sji with Piand hence sij =sji. The

interconnection of Piand Pjis given by

Pi∧sij Pj={(wi, wj, sij, ci, cj)|(wi, sij , ci)∈ Piand

(wj, sij, cj)∈ Pj}.

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摘要：

arXiv:2210.06268v2[math.OC]10Apr2023OnaCanonicalDistributedControllerintheBehavioralFramework⋆TomR.V.Steentjesa,∗,MirceaLazara,PaulM.J.VandenHofaaDepartmentofElectricalEngineering,EindhovenUniversityofTechnology,P.O.Box513,Eindhoven,5600MB,TheNetherlandsAbstractControlinaclassicaltransferfunctionors...

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