A Strategy for Dynamic Controller Emulation in Packet-based

Networked Control

S. Falasca

, M. Gamba

, and A. Bicchi

1,2

Interdepartmental Research Center “E. Piaggio”, Via Diotisalvi 2, 56126 Pisa, Italy

Istituto Italiano di Tecnologia, Genova, Italy

Keywords:

Networked Control System, Packet-switching Networks, Model-based Control, Time-varying Delays.

Abstract:

The problem of the stabilisation of a nonlinear system via output-feedback dynamic control is addressed, under

the assumption that every communication between the plant and the controller is subject to network-induced

constraints. These constraints include variable transfer intervals; time varying, large communication delays;

non-simultaneous access to the network. A control technique that copes with these constraints is presented

which is based on recent results addressing the problem of static state-feedback. The stability of the resulting

nonlinear networked control system is assessed. Network-in-the-loop experiment results are presented, which

conﬁrm that the proposed method is effective.

1 INTRODUCTION

The past decade has witnessed a dramatic growth in

interest for control over distributed networked archi-

tectures, which have the strong potential to increase

ﬂexibility and scalability of a plant, while inducing

a remarkable reduction of costs for both installation

and maintenance. However, because of the use of

the network and because of the system being dis-

tributed, some problems arise: e.g. bandwidth limi-

tations, time-delays and packet losses, which cannot

be ignored in the control design. The state-of-the-art

is reported and discussed in (Heemels et al., 2010).

An essential aspect of many Networked Con-

trol Systems (NCS), such as those using Ethernet

as a communication layer, is that they organise data

transmission in packets. Such networks carry larger

amount of data with less predictable rates with respect

to circuit-switching communication channels. Pack-

etised transmissions substantially alters the band-

width/performance trade-off of traditional design.

The potentially large size of packet payload can be

exploited to reduce data transmissions without de-

grading the overall NCS performance. (Bemporad,

1998) pioneered the idea of sending feed-forward

control sequences computed in advance on the ba-

sis of a model-based predictive (MBP) scheme to the

aim of compensating for large delays in communica-

tion channels. The technique has been generalised to

address time-varying delays and transfer intervals in

(Polushin et al., 2008).

In (Greco et al., 2012) a control strategy (namely

Packet-Based Control) for stabilising an uncertain

nonlinear NCS affected by varying transmission in-

tervals, varying large delays, and constrained access

to the network is presented. A model of the plant is

used to build a prediction of the control law. Feedback

is provided by measuring the plant state. The state is

measured by distributed sensors. A network proto-

col is in charge of deciding which sensor node can

communicate at each instant. The control sequence

sent by the remote controller is stored in an embed-

ded memory on the plant side, a control command in

the sequence is chosen based on the time stamp con-

tained in the packet.

One major limitation of (Greco et al., 2012) is that

it only applies to static-feedback controllers. The sta-

bilisation of NCSs by means of dynamic controllers

has been considered in (Nesic and Liberzon, 2009)–

where it is addressed under the assumption of small

delays–and in (Polushin et al., 2008)–where the au-

thors solve the problem in the assumption that all

the plant state is sent simultaneously. In this paper

we aim at extending the Packet-Based Control to dy-

namic controllers, hence allowing for large delays and

non-simultaneous transmissions. Indeed, a direct ap-

plication of the aforementioned framework to the use

a dynamic controller would require updating the in-

ternal state of both the system model and the con-

troller by means of a protocol ensuring some nice

Falasca S., Gamba M. and Bicchi A..

A Strategy for Dynamic Controller Emulation in Packet-based Networked Control.

DOI: 10.5220/0004427100290038

In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 29-38

ISBN: 978-989-8565-71-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

error-decreasing properties.

We depart from the basic idea of updating the in-

ternal state of the dynamic controller in a way consis-

tent with the behavior it would have had if it were

directly connected to the plant. We then devise a

method which produces the same effects as having

a controller on the plant side which sends its inter-

nal state through the network towards the remote con-

troller, the same way the state of the plant is sent. We

consider the effects of our algorithm as virtual send-

ings. The only information we need to consistently

run the controller is the history of the inputs to the

controller. The virtual transmission of the controller

state is hence realised by sending the history of the

outputs of the plant and by using these outputs to feed

the remote controller. The output history can be sent–

again–by exploiting the large payload of packets. If

output sensors are distributed, outputs are partitioned

and the history of each sensor is sent according to a

static protocol similar to Round Robin. The draw-

back of the virtual sendings–especially in the case of

outputs partitioned over many nodes–is that a poten-

tially large delay on the arrival of the virtual packets

is introduced. Such a delay has to be directly taken

into account in the conditions ensuring the stability of

the overall system. We prove the exponential stabil-

ity of the NCS over a prescribed basin of attraction,

provided that some explicit bounds on the Maximum

Allowable Delay (MAD (Heemels et al., 2010)) and

on the Maximum Allowable Transfer Interval (MATI

(Walsh et al., 1999)) are satisﬁed. We ﬁnally apply

our technique to the control of a magnetic levitator

involving an output-feedback dynamic controller. It

will be shown that if the proposed technique is not

used, the network strongly affects the behavior of

the NCS. On the other hand, the presented algorithm

closely mimic the ideal closed-loop behavior; in ac-

cordance with the paradigm adopted i.e. the presence

of the network must be as transparent as possible to

the designer of the stabilising controller.

2 SYSTEM DESCRIPTION

In this section we provide the reader with all the char-

acteristics we assume the controller, the network and

the plant to have. Figure 1 shows the control archi-

tecture. The plant and the controller communicate

via a shared-bus communication network. An output-

feedback dynamic control law for the system is as-

sumed to be available. The plant is equipped with

network-enabled devices for actuation and sensing.

The sensors measure the internal state of the system

and its output. A protocol grants access to the net-

OSn

SSn

Plant Model

Dynamic control law

Controller

Shared Bus Network

Packet FIller

Local Dynamic

Dynamic

control law

Plant

Actuator Node

SSn

OSn

Compound Protocol

Figure 1: The proposed control architecture.

work to one node at a time. Output-measuring sensors

send the whole history of readings in tranches. State-

measuring sensors send the current reading each time

they are granted access to the network.

On the controller side, upon reception of output

data, the exact knowledge of the control law is ex-

ploited in order to infer a suitable value for the in-

ternal state of the controller at a given time. The

so computed internal state of the controller, together

with the received information about the internal state

of the plant, is used to initialise a model for the ideal

closed loop composed by a model for the plant dy-

namics and the control law. By means of simulating

the ideal closed loop behavior, a sequence of control

actions is computed, which is intended to be used in

the future. The control actions are then sent to the

plant; they will be received on the actuator side and

used appropriately.

2.1 The Plant and the Controller

We address the stabilisation of a nonlinear

continuous-time system of the form

˙x

= f

,u) (1)

y = g

), (2)

where x

≥0

→R

is the plant state, y:R

≥0

→R

is the output, u : R

≥0

→R

represents the control in-

put, and f

: R

×R

→R

and g

: R

→R

de-

note locally Lipschitz functions. For this system, we

assume that a nominal dynamic feedback controller

of the form

˙x

= f

,y) (3)

u = g

,y) (4)

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

is available. Here x

: R

≥0

→ R

is the controller

state, and f

: R

×R

→ R

and g

: R

×R

→R

denote locally Lipschitz functions. Letting

x(t) ,



(t),x

(t)



∈ R

= R

and

f(x,u) ,



,u)

))



g(x) , g

)),

the closed-loop system (1)-(4) in the absence of net-

work effects simply reads

˙x = f(x,u) (5)

u = g(x). (6)

We assume that the nominal controller (3)-(4)

globally exponentially stabilises the plant (1)-(2) in

the absence of network effects.

Assumption 1 (. Nominal GES) The origin of the

system (1)-(2) in closed-loop with (3)-(4) is glob-

ally exponentially stable (GES), and there exists a

differentiable function V : R

→ R

≥0

and constants

,α,α,d > 0 such that the following conditions hold

for all x ∈R

αkxk

≤V(x) ≤ αkxk

∂V

∂x

(x) f(x, g(x)) ≤ −αkxk



∂V

∂x

(x)



≤ d kxk.

Local Lipschitz constants are assumed to be available

to the designer.

Assumption 2 (. Local Lipschitz) Given some con-

stants R

> 0, there exist some constants λ

,λ

0 and all u

∈B

, the following inequalities hold

kf(x

) − f(x

)k ≤ λ

(kx

−x

k+ ku

−u

(7)

kg(x

) −g(x

)k ≤ λ

−x

k. (8)

The control strategy analysed in this paper aims at

compensating the network-induced effects by relying

on a prediction of the plant behavior. To that aim, we

assume that a model for (1)-(2) is known:

ˆx

( ˆx

, ˆu) (9)

ˆy = ˆg

( ˆx

). (10)

This model in closed-loop with the nominal controller

(3)-(4) reads

ˆx =

f( ˆx, ˆu) (11)

ˆu = ˆg( ˆx), (12)

where ˆx , ( ˆx

, ˆx

)

: R

≥0

→ R

and

f( ˆx, ˆu) ,



( ˆx

, ˆu)

( ˆx

, ˆg

( ˆx

))



ˆg( ˆx) , g

( ˆx

, ˆx

ˆg

( ˆx

)).

The plant-model inaccuracy is assumed to be sector-

bounded.

Assumption 3 (. Sector-bounded Model Inaccuracy)

Given R

> 0, there exists a constant λ

≥ 0 such

that, for all x ∈ B

and all u ∈ B



f(x, u) −

f(x, u)



≤ λ

(kxk+ kuk). (13)

2.2 The Network

Measurements are taken by distributed sensors and

sent to the controller into packets. Sensors are as-

sumed to be synchronised with each other. We as-

sume that the measurement part of the network is par-

titioned in ℓ nodes and only a unique node at a time

can send its information (i.e. only partial knowledge

of the plant state is available at each time instant).

The controller is seen as a unique node. The over-

all state of the system x(t) ∈ R

is thus decomposed

in ℓ + 1 nodes as x(t) = [x

p,1

(t) , . . . ,x

p,ℓ

(t) , x

(t)]

with x

p,i

(t) ∈R

and

∑

ℓ

i=1

= n

Control sequences are sent as packets. An em-

bedded control device receives, decodes, synchro-

nises these packets and applies control commands to

the plant. We consider that measurements are taken

and sent at instants {τ

}, and are received by the

remote controller at instants {τ

+ T

}. In other

words, {T

} denotes the sequence of (possibly time-

varying) measurement data delays. Delays cover both

processing time and transmission delays on the mea-

surement chain. Similarly, control commands are sent

over the network at time instants {τ

}. They reach the

plant at instants {τ

+ T

}, where {T

} denotes the

sequence of delays accounting for both the computa-

tion time and the transmission delay from the remote

controller to the plant.

Assumption 4 (. Network) The communication net-

work satisﬁes the following properties:

i) (MATI) There exist two constants τ

,τ

∈ R

≥0

such that τ

i+1

−τ

≤ τ

and τ

i+1

−τ

≤ τ

, ∀i ∈

ii) (mTI) There exist constants ε

,ε

∈R

≥0

such that

≤ τ

i+1

−τ

andε

≤ τ

i+1

−τ

∀i ∈N.

iii) (MAD) There exist two constants T

∈ R

≥0

such that T

≤ T

and T

≤ T

, ∀i ∈ N;

AStrategyforDynamicControllerEmulationinPacket-basedNetworkedControl

Table 1: The sequences {s

} and {o

} for ℓ

= 3.

...

0 1 2 3 4 5 6 7 8 ...

2.3 The Network Protocol

The use of a dynamic controller imposes a careful up-

date of the controller internal model in order to gener-

ate meaningfulcontrol sequences. We propose a strat-

egy that consists in transmitting the measurement his-

tory of each output nodes over a prescribed time hori-

zon, as well as the instantaneous value of the plant’s

state when access is granted to the network. The sys-

tem thus involves two different kinds of sensor nodes:

ℓ

output-sending nodes (OSn) and ℓ state-sending

nodes (SSn).

The access to the network is ruled by a protocol

choosing, at each instant τ

, which node communi-

cates its data. In order to limit the cumulated delays

induced by this approach, we assume that the nodes

are granted access to the network according to the

following rule: after each SSn access, all OSn are re-

quired to send their data according to a prescribed or-

dering (Round Robin). Then access is again granted

to a SSn, and so on. This rule can be formally stated

by extracting from the sequence of access times {τ

}

two subsequences





and





. More precisely,

we deﬁne two sequences {s

}, {o

} having values in

N. Such sequences have the following meaning: at

time τ

, s ∈ {s

} a SSn is granted access to the net-

work; at time τ

, o ∈ {o

} an OSn has the ability to

send. The policy is such that the two sequences ex-

hibit the following properties

a) {s

}∪{o

} = N, {s

}∩{o

} =

b) s

= (ℓ

+ 1)i;

c) o

is strictly increasing.

Consider, as an example, ℓ

= 3; the sequences {s

}

and {o

} are shown in table 1.

We keep track of which OSn is granted access to

the network at a given time by means of the deﬁni-

tion of the sequence {ν

} having values in [1, ℓ

] ⊂ N

deﬁned as

i+1



1+ ν

ifν

< ℓ

1 otherwise

(14)

and we consider ν

= 1 to express the fact that the

OSn number 1 sends ﬁrst. The ν

−th OSn is thus

granted the access to the network at time τ

The SSn are granted access to the network accord-

ing to a protocol ruled by the map involving the error

(t) ∈ R

deﬁned as e

(t) , ˆx

(t) −x

(t):

(τ

) = h



i,e

(τ

)



, ∀i ∈ N. (15)

where h

: N ×R

→ R

. This protocol is as-

sumed to induce an exponential decrease of the error

when the inter-sample dynamics are neglected; i.e.

we are interested in UGES protocols. We recall here

a slightly modiﬁed version of the deﬁnition in (Nesic

and Teel, 2004) as given in (Greco et al., 2012).

Deﬁnition 1. A function h : N ×R

→ R

is said to

be an UGES protocol having parameters a

,a,ρ, c if

there exists a functionW : N ×R

→R

≥0

locally Lip-

schitz in its second argument and there exist constants

,a > 0;c > a and ρ ∈ [0, 1) such that the following

conditions hold for the auxiliary discrete time system

ξ(i+ 1) = h(i, ξ(i)):

kξk ≤W (i,ξ) ≤ a kξk

W (i+ 1, h(i, ξ)) ≤ ρW (i,ξ)

(16)

for all ξ ∈ R

and all i ∈ N, and



∂W

∂ξ

(i,ξ)



≤ c (17)

for almost all ξ ∈R

and all i ∈ N.

Assumption 5. The protocol (15) is UGES with pa-

rameters a

,ρ

Remark 1. The UGES class is often used in the net-

work control literature. Although it might seem to

be very conservative, it has to be stressed out that

it is not. Indeed, no limits are being posed on the

rate of convergence of the error. The network pro-

tocol is used as a control-oriented description of the

effects of the sending order (which is not necessarily

pre-deﬁned). UGES network protocols which are of-

ten used in practice are the Round Robin and the so

called Try Once Discard.

3 ALGORITHM DESCRIPTION

The algorithm we propose can be decomposed in dif-

ferent modules. At the plant side of the network,

three kinds of devices are needed, namely the actuator

node, the state-sending node and the output-sending

node. The controller is divided into two modules: the

ﬁrst one–namely Local Dynamics–is in chargeof con-

verting the output data it receives from the plant into

information about the internal state the controller is

supposed to have; the second one has state-related in-

formation as an input (either received from the net-

work or produced by the Local Dynamics); based on

its input it computes the controls to be sent to the ac-

tuator node.

In this section we describe the behavior of each

module and provide a model for the overall closed-

loop system.

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

3.1 The Plant

3.1.1 Actuator Node

Such node is in charge of receiving, decoding and

re-synchronise packets sent by the controller. Each

received packet contains a timestamp and a certain

number of control values which are stored in a local

buffer. Upon choosing which one of the control val-

ues is the most appropriate, it actuates the plant. More

precisely, the actuator node compares the timestamp

of the last packet it received with its internal clock

and moves within the control sequence it received up

to the corresponding starting point.

3.1.2 State-sensor Node

When such a node is granted access to the network

(see section 2.3 for a description of the policy used to

take this decision), it encodes the sensed values into

a network packet, timestamps it and sends it to the

controller.

3.1.3 Output-sensor Node

An output-sensor node continuously monitors the

sensed outputs, storing the readings into a buffer. The

buffer content is timestamped and encoded into pack-

ets sent to the controller when the network is avail-

able. Upon sending, data is discarded. The sending of

tranches of data is consistent with how control data is

sent in (Greco et al., 2012). Figure 2 shows an exam-

ple of the sendings carried out by the output-sensor

nodes.

Figure 2: Example of OSn sendings for ℓ

= 3.

Remark 2. It is tacitly assumed that the effects of

sending discrete-time values instead of continuous

function to the controller and to the actuator node can

be neglected.

3.2 The Controller

3.2.1 Local Dynamics

By exploiting output readings it is possible to mathe-

matically consider the controller as being executed on

the plant side and sending its internal state over the

network. We will deﬁne the sequence of time-instants

at which such virtual-sendings happen and the delay

each of these packets incurs into. Finally, we will use

such virtual packets, together with the state packets,

in order to design an overall UGES protocol acting on

the complete system as expressed in (5).

First of all, we need give some deﬁnitions.

Deﬁnition 2. Given T ∈ R

≥0

, we will say that y

[0,τ]l

is known at time τ + T if there exist j ∈ {o

},k ∈ N

such that:

a) τ

≥ τ, τ

+ T

≤ τ+ T;

b) o

= j, ν

= l.

Moreover, we will say that y

[0,τ]

is known at time

τ+ T if ∀l ∈ [1, ℓ

] ⊂ N we have that y

[0,τ]l

is known

at time τ+ T.

This deﬁnition reﬂects the natural notion of the

controller having already received a packet related to

which was sent no later than time τ.

Deﬁnition 3. Given T ∈ R

≥0

, we will say that x

(τ) is

known at time τ + T if either τ = 0 or y

[0,τ]

is known

to the controller at time τ + T .

This deﬁnition formalises the fact that given σ such

that x

(σ) is known at time τ + T , it is possible to

use the solution for the differential equation φ

[σ,τ]

[σ,τ]×[σ,τ]×R

×R

[σ,τ]

→R

in order to compute

x(τ), since

(τ) = φ

[σ,τ]



τ,σ,x

(σ), y

[σ,τ]



(18)

With the previous deﬁnitions in mind we can state the

following.

Proposition 4 (. Virtual Packets) Given i ≥ℓ

−1,i∈

N the value x



(

i−(ℓ

−1)

)



is known at time

(

i−(ℓ

−1)

)

+ ℓ

+ T

This means that we can consider that at time

(

i−(ℓ

−1)

)

a packet containing x



(

i−(ℓ

−1)

)



is sent

by the plant; such a virtual packet incurs a delay

which is no longer than ℓ

+ T

We will consider those virtual packets in conjunc-

tion with the packets containing x

(τ

), which arrive

at the controller at time τ

+ T

. The sequence of

time instants at which such packets are sent is {τ

As for the delays, we deﬁne a new sequence

Deﬁnition 5 (. Sequence of State-sending Delays)

The sequence of state-sending delays

, T

∈ R

≥0

such that:

(

if∃k : i = s

min

otherwise

(19)

AStrategyforDynamicControllerEmulationinPacket-basedNetworkedControl

where

, {T : x

(τ) is known at timeτ+ T}. (20)

By virtue of Proposition 4, the following inequal-

ity holds:

≤ ℓ

+ T

= T

. (21)

From now on we will consider the packets con-

taining the state of the system and the virtual packets

containing the state of the controller. The information

they gather will be used in order to design a proto-

col which acts on the error e(t) = (e

(t), e

(t))

ˆx(t) −x(t), where e

(t) ∈ R

: e

(t) , ˆx

(t) −x

(t).

We will show that, provided that an UGES protocol

acting on e

is in use (see Assumption 5), the de-

signed protocol is UGES.

Proposition 6 (. Compound Protocol) The function

h : N ×R

→ R





,ξ





7→









(k, ξ

) if∃k : s

= i

otherwise



if∃k : s

= i

0 ∈ R

otherwise







(22)

deﬁnes an UGES protocol. Here ξ

∈R

,ξ

∈ R

3.2.2 Computing the Control Law

When a new measurement is received, the remote

controller uses the new data in order to update an es-

timate of the internal state of the plant. The controller

then computes a prediction of the control signal over

a ﬁxed time horizon

≥ T

+ T

+ τ

(23)

by numerically running the model (11)-(12). Such

computation generates values for the function ˆu(t)

(cf. equation (12)) which are then coded, marked with

the appropriate timestamp, and put in a single packet

which is sent at the next network access .

3.3 The Overall Model

The loop composed of the system (1)-(2) and the con-

troller node which executes the algorithms described

in Section 3.2 can be modelled by means of the fol-

lowing equations (see also (Greco et al., 2012)).

The NCS model has a state x(t) which models the

internal state of the plant as well as the state of the

controller as it would act if it were connected to the

outputs of the plant. Moreover, N vectors of addi-

tional state variables are used for modelling the esti-

mations of the vector ˆx. N represents the number of

packets, either real or virtual, that can be received by

the controller during the time T

. It is deﬁned as

N ,



−τ



+ 1. (24)

By means of the deﬁnition of ¯x(t), ˜x(t),e(t) ∈

deﬁned as ¯x(t) ,



(t), . . . ,x

(t)



and e(t) ,



(t), . . . ,e

(t)



, e

(t) ∈ R

, the closed-loop dy-

namics of the NCS can be compactly written as

˙x = F(t, ¯x,e) (25a)

˙e = G(t, ¯x, e) (25b)

e(τ

) = H(i,e(τ

)), (25c)

where

F(t, ¯x, e) = f(x, υ(t,e+ ¯x)) (26a)

G(t, ¯x, e) =





f(e

+ x, ˆg(e

+ x)) − f (x, υ(t, e+ ¯x))

f(e

+ x, ˆg(e

+ x)) − f (x, υ(t, e+ ¯x))





(26b)

H(i,e) =







+ (h(i,e

) −e

)η(i, 1)

+ (h(i,e

) −e

)η(i, 2)

+ (h(i,e

N−1

) −e

)η(i,N)







, (26c)

where η : N ×{1, . . . , N}→{0, 1}identiﬁes the index

of the relevant state estimate

η(i,r) ,



1 if µ(i) = r

0 otherwise

(27)

and µ : N → {1,...,N} is deﬁned as

µ(i) , ((i−1)modN) + 1 . (28)

The control signal υ in (26a) and (26b) is deﬁned as

υ : R

≥0

×R

→ R

(t,



,... , ξ



) 7→

∑

k=1

ˆg(ξ

)ζ(t, k) (29)

where ξ

∈ R

and ζ : R

≥0

×{1, . . . , N} → {0, 1} is

the map

(t, k) 7→







if ∃j ∈ N s.t. µ(γ( j)) = k and

t ∈ (τ

+ T

,τ

j+1

+ T

j+1

]

0 otherwise

(30)

and γ : N → N

j 7→ max

i ∈ N | τ

+ T

< τ

(31)

denotes the index of the latest measurement received

before τ

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

4 MAIN RESULT

Theorem 7. Assume that assumptions 1, 4, 5 hold.

Given some R > 0, ﬁx R

= R and R

= λ

R and sup-

pose that assumptions 2 and 3 hold with these con-

stants. Let a

,ρ

α, α,α,d,λ

,λ

and λ

generated by these assumptions. Pick

= a

a = a

, ρ = ρ

ℓ

, c = c

(32)

and deﬁne a







a ifN = 1

min









otherwhise

. (33)

Assume that the following conditions on

,τ

,ε

hold:

∈[ε

,τ

∗

), τ

∗

log



Hγ

ρL



N =

+τ

+ 1

(34)

where

L ,



(1+ λ

)

√

Nλ

√

Nλ



√

N −1+ N −1



M , (1+ λ

)cNλ

. (35)

Then the origin of the NCS (25) is exponentially stable

with radius of attraction

R ,

(36)

where K ,

√

1−γ

max{(1+ γ

,(1+ γ

}, γ

exp(Lτ

)−1

L(1−ρexp(Lτ

))

H, k

ρa

and k

Conditions expressed in (34) establish a re-

lation between the relevant parameters, namely

,τ

and τ

. Notice that (21) can be used to

express such a relation in terms of T

and ℓ

instead

of T

. Note that since Theorem 7 guarantees only

local properties, Assumption 1 could be relaxed to lo-

cal exponential stability of the nominal plant, over a

sufﬁciently large domain.

Remark 3. The presented formulation of the MATI

and the expression for the radius of convergence are

based on (Greco et al., 2012) where examples show-

ing that the MATI constitutes an improvement over the

previously existing state-of-the-art can be found. It is

easily seen that due to the high number of variables

involved in the presented expressions, it is impractical

to actually compute the MATI and the radius of con-

vergence for a real plant. In fact, the same can be said

for most of the similar results which can be found in

literature. In this case the presented theorem can be

seen as an existence result: it assesses that it is pos-

sible to stabilise the system by means of the presented

architecture.

5 NETWORK-IN-THE-LOOP

EXPERIMENTS

In this section, we provide results of the experiments

carried out for the networked control of a magnetic

levitator. The setup uses two computers, one for the

controller and the other for simulating the plant with

sensors and actuators. The computers are connected

through a real Ethernet link. The experimental net-

work setup is such that

1×10

−3

[s] ≤ τ

i+1

−τ

≤ 5×10

−3

[s]

1×10

−3

[s] ≤ τ

i+1

−τ

≤ 5×10

−3

[s]

. (37)

Figure 3 shows the measured round-trip time. Based

on the measurements, we can consider the maximum

delays

to be T

maxRTT

≈26×10

−3

[s].

The experiments have been carried out by means

of a software for networked control systems based on

the one presented in (Falasca et al., 2010) which al-

lows for network-in-the-loop tests to be carried. For

this purpose a module implementing the local dy-

namic algorithm has been designed.

0 5000 10000 15000 20000

RTT [ms]

Time 10

-3

[s]

Figure 3: Round Trip Time.

˙x =







sin(α)g−





−







y = x

Figure 4: Magnetic levitator and its model

The measurements include both the network-induced

delays and some additional delays which have been added

via software in order to simulate the effects of additional

trafﬁc. The program

has been used on both

Linux

hosts

to provide additional sending delays, which have a normal

probability distribution of N (15,5). Hence, the mean value

of the added round trip time is 30ms. Additional delays

account for the larger portion of the overall measured delay.

AStrategyforDynamicControllerEmulationinPacket-basedNetworkedControl

The plant parameters (equations are shown in Fig-

ure 4) are α =

, g = 9.8

, m = 0.05[Kg], c =

0.5[Hm], L = 1[H] , R = 10[Ω].

Following the philosophy of section 2.1, we as-

sume that a stabilising controller is given for the nom-

inal plant. In our case, the nominal controller is the

result of the straightforward application of numeri-

cal self-tuning routines in Matlab, and has the transfer

function:

C(s) = −

+ a

s+ a

(s+ b)

(38)

where a

= 1418,a

= 767, a

= 377 and b = 29.

The plant model used in the controlling computer,

is subject to parametric uncertainties. The parame-

ters for the model it uses are very different from the

real ones, i.e. α =

, g = 9.8

, m = 1[Kg] , c =

1[Hm], L = 5[H], R = 0.1[Ω]

Figure 5 shows the results of the experiments for

a reference signal x

= 0.05[m]. One of the trajec-

tories shows the behavior of the ideal closed-loop;

the second one shows the networked system. Finally,

the behavior of the networked system when the al-

gorithm taking into account local dynamics (cf. sec-

tion 3.2.1) is not used is shown, i.e. the protocol

h : N ×R

→ R





,ξ





7→







(k, ξ

) if∃k : s

= i

otherwise





is used. Experiments show that the proposed algo-

rithm manages to produce results resembling the ideal

closed loop behavior. If the proposed algorithm is

not used, the behavior is altered; for instance–for the

given example–the steady state error is not zero.

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0 5000 10000 15000 20000 25000 30000

Position [m]

Time 10

-3

[s]

Networked

Networked: no local dynamics

Ideal closed loop

0.047

0.05

0.053

5000 10000 15000 20000 25000 30000

Figure 5: Trajectory x

(t).

The PBC approach has been experimented with

on a real plant, namely a Furuta Pendulum in (Fabbri

et al., 2013). Those experiments assess the real-world

applicability of the approach and give a comparison

on the issues encountered in implementing the PBC

control scheme and the classical local control.

6 PROOFS

We start by giving a proof for Proposition 4.

Proof 1 (. Virtual Packets) Pick i ∈ N,i ≥ ℓ

−1 and

assume that x

(τ

i−(ℓ

−1)

) is not known at time T =

i−(ℓ

−1)

+ ℓ

+ T

. Then there exists l ∈ [1,ℓ

] :



0,τ

i−(ℓ

−1)



is not known at time T. It follows

that ν

i−(ℓ

−1)

6= l and that there exists j < ℓ

i−(ℓ

−1)+ j

= l. Hence y

0,ν

i−(ℓ

−1)+ j

is known at

time τ

i−(ℓ

−1)+ j

+ T

i−(ℓ

−1)+ j

. But since τ

i−(ℓ

−1)+ j

i−(ℓ

−1)+ j

≤ τ

i−(ℓ

−1)+ j

+ T

≤ τ

i−(ℓ

−1)

+ jτ

≤τ

i−(ℓ

−1)

+ℓ

= T we can conclude that

0,ν

i−(ℓ

−1)+ j

is known at time T; which is an absurd.

In order to prove Proposition 6 we need to state some

preliminary results.

Lemma 8 (. Sum of Protocols) Given two UGES pro-

tocols h

: N ×R

→ R

and h

: N ×R

→ R

with relative constants and functions as contained in

Deﬁnition 1, distinguished by the subscript s and p,

let deﬁne e ,



, p



, with s ∈ R

, p ∈ R

. The

protocol:

h(i,e) ,



(i,s)

(i, p)



(39)

is UGES with parameters a

= min





a =

max{a

}, ρ = max{ρ

,ρ

} and c = max{c

Proof 2. With the parameters deﬁned in the lemma

and the Deﬁnition 1 in mind, we deﬁne the function

W : N ×R

)

→ R

≥0

W(i, e) =

(i,s)

(i, p)

(40)

Hence

kek

= a



ksk

+ kpk



≤W(i, e)

≤



ksk

+ kpk



kek

W (i+ 1, h(i, e))

≤ ρ



(i,s)

(i, p)



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Furthermore



∂W

∂e





∂W

∂s

, W

∂W

∂p





∂W

∂s



∂W

∂p



≤

= c,

by means of which the theorem is proven.

Lemma 9 (. N-dilation of a Protocol) Let us con-

sider a UGES protocol h : N × R

→ R

and a

not-decreasing surjective function k : N → N, the

protocol deﬁned as:

(i, e) =



h(k(i), e) if k(i+ 1) 6= k(i)

e otherwise

(41)

is UGES, provided that ∃

ι : k(i) 6= k(i + ι)∀i. In par-

ticular, if h is UGES with parameters a

,a,ρ, c , h

UGES with parameters ρα

,α, ρ

and c.

Proof 3. Given the function W associated with the

UGES protocol h, we deﬁne a function W

(0,e) = W(0, e)

(i+ 1, e) =



W(k(i+ 1), e) if k(i+ 1) 6= k(i)

(i,e) otherwise

(42)

The following conditions hold:

ρa

kek ≤W

(i,e) ≤ akek.

Furthermore, when k(i+ 1) = k(i)

(i+ 1, h

(i,e)) ≤ ρ

(i,e).

Consider now the case k(i+1) 6= k(i). Take n ≤

ι : k(i) =

k(i−n+ 1) 6= k(i−n). We have:

(i−n + 1,e) = W (k(i−n + 1), e) (43)

and

(i, e) = ρ

n−1

W (k(i), e) ≥ ρ

ι−1

W (k(i) , e) . (44)

Therefore:

ρW (k(i), e) ≤ ρ

(i,e). (45)

We can then write W

(i+ 1, h

(i,e)) =

W (k(i+ 1),h(k(i) , e)) ≤ρW (k(i) , e) ≤ρ

(i,e).

Since ρ < 1, it is apparent from the very deﬁnition of

that



∂W(i, e)

∂e



≤ c ⇒



∂W

(i,e)

∂e



≤ c,

which concludes the proof.

We can now give a proof for Proposition 6.

Proof 4. Deﬁne

(i,e

) =



(k, e

) if ∃k : s

= i

otherwise

(46)

it is easily seen that h

can be obtained by applying

Lemma 9 to the protocol h

(i,e

) by deﬁning

(i+ 1) =



(i) + 1 if ∃k : s

= i

(i) otherwise

, (47)

(0) = 0. From the deﬁnition of s

, it follows that

(i + ℓ

+ 1) 6= k

(i)∀i ∈ N. Hence h

(i,e

) is

UGES with parameters a

= a

, ρ

ℓ

and c

= c

Deﬁne now h

(i,e

) = 0, which is an UGES pro-

tocol having parameters a

= c

= a

and ρ

. If we deﬁne

(i,e

) =



if ∃k : s

= i

0 otherwise

(48)

it is easily seen that h

can be obtained by applying

Lemma 9 to the protocol h

(i,e

) by deﬁning

(i+ 1) =



(i) if ∃k : s

= i

(i) + 1 otherwise

, (49)

(0) = 0. From the deﬁnition of o

and s

, it follows

that k

(i+2) 6= k

(i)∀i ∈N. Hence h

(i,e

) is UGES

with parameters a

= a

, ρ

= ρ

and

= a

Now, the protocol in Proposition 6 can be ob-

tained by applying Lemma 8 to the protocols h

and

, hence it is UGES with parameters a

= a

a = a

, ρ = ρ

ℓ

and c = c

In order to prove Theorem 7, we are now going to

invoke theorem 1 from (Greco et al., 2012). The as-

sumption on the network made there are satisﬁed by

means of (21) which follows from our Assumption 4

in conjunction with the deﬁnition of the sequence of

state-sending times and delays. The assumption re-

garding the protocol being UGES is here satisﬁed by

means of Proposition 6. The ideal closed-loop system

has the nominal GES property, as stated in Assump-

tion 1. Moreover, Assumptions 2 and 3 are the same

required in (Greco et al., 2012).

Theorem 7 is a rewriting of Theorem 1 in (Greco

et al., 2012) in terms of the quantities involved in the

writing of system (25).

The function W

(i, e

) = a

k can be used to show

this.

AStrategyforDynamicControllerEmulationinPacket-basedNetworkedControl

7 CONCLUSIONS

The networked stabilisation of a nonlinear plant via

an output-feedback dynamic controller has been con-

sidered. An algorithm is proposed which exploits

the packet-based nature of the considered network.

Sufﬁcient condition for the local exponential stabil-

ity of the resulting system are given. The stabilisation

of a magnetic levitator is presented as an example.

Network-in-the loop experiment results show that the

resulting network controlled system closely mimics

the behavior of the ideal closed-loop system. If–on

the contrary–the proposed algorithm is not used, the

network is shown to strongly affect the behavior of

the controlled system.

ACKNOWLEDGEMENTS

The research leading to these results has re-

ceived funding from the European Union Seventh

Framework Programme [FP7/2007-2013] under grant

agreement n257462 HYCON2 Network of excel-

lence”

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