A ROBUST LIMITED-INFORMATION FEEDBACK FOR A CLASS

OF UNCERTAIN NONLINEAR SYSTEMS

Alessio Franci

LSS - Universit

e Paris Sud - Sup

elec, 3 rue Joliot-Curie, 91192 Gif sur Yvette, France

Antoine Chaillet

EECI - LSS - Universit

e Paris Sud - Sup

elec, 3 rue Joliot-Curie, 91192 Gif sur Yvette, France

Keywords:

Limited-information feedback, Robustness, Nonlinear systems.

Abstract:

We propose a variant of the recently introduced strategy for stabilization with limited information recently

introduced in (Liberzon and Hespanha, 2005) and analyze its robustness properties. We show that, if the nom-

inal plant can be made Input-to-State Stable (ISS) with respect to measurement errors, parameter uncertainty

and exogenous disturbances, then this robustness is preserved with this quantized feedback. More precisely, if

a sufﬁcient bandwidth is available on the communication network, then the resulting closed-loop is shown to

be semiglobally Input-to-State practically Stable (ISpS).

1 INTRODUCTION

The always greater use of digital communication de-

vices for control applications makes quantization a

crucial issue. The limitations on the communica-

tion rate between the plant sensors and the con-

troller imposes to develop new approaches that are

able to guarantee good performance even when only

limited information on the plant’s state is available.

Despite strong technological improvements, the bit

rate available for a given control application may in-

deed be strongly limited due to scalability or energy-

saving concerns, or due to harsh environment con-

straints. Stabilization in this context becomes partic-

ularly challenging in presence of model uncertainties,

measurement errors or exogenous disturbances.

These observations explain why limited-

information control feedback has been widely

studied recently: (Nair et al., 2007; Hespanha et al.,

2007; Liberzon, 2009) and references therein for

representative examples. An important literature

already exists for linear systems, (Montestruque

and Antsaklis, 2004; Liberzon, 2003; Petersen

and Savkin, 2001; Nair and Evans, 2004; Jaglin

et al., 2008; Jaglin et al., 2009). In particular, the

results of (Liberzon and Ne

c , 2007) provide a

coding/decoding strategy that achieves Input-to-State

Stabilization of quantized linear control systems. The

proposed control strategy relies on a discrete time

zoom-in/zoom-out procedure. This construction is

based on the exact sampled dynamics of the system,

or at most on its discrete time approximation. This

is why the closed-loop system may lack robustness

with respect to parameter uncertainties. These results

were subsequently generalized to nonlinear systems

in (Kameneva and Ne

c , 2008).

In (Sharon and Liberzon, 2007), Input-to-State

Stabilization of quantized linear and nonlinear sys-

tems is achieved in the framework of continuous time

quantized control systems, that is exploiting hybrid

dynamics. It is based on a generalization of the dy-

namic quantization approach developed in (Liberzon

and Hespanha, 2005) and (Persis and Isidori, 2004)

for ISS and global asymptotically stable systems re-

spectively. For nonlinear systems, this control strat-

egy leads to local ISS. However, model uncertainties

can seriously compromise the efﬁciency of the pro-

posed algorithm and no estimates of the domain of

attraction can be obtained in general. We detail these

limitation in Section 5.

The purpose of this paper is to propose an al-

ternative dynamic quantization strategy, able to cope

with (time-varying) model uncertainties. It is based

on a simple and natural modiﬁcation of the one

proposed in (Liberzon and Hespanha, 2005). We

show that the quantized control strategy ensures

Franci A. and Chaillet A. (2010).

A ROBUST LIMITED-INFORMATION FEEDBACK FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS.

In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 45-53

DOI: 10.5220/0002950000450053

 SciTePress

semiglobal Input-to-State practical Stability. More

precisely, any compact set of initial conditions and for

any bounded time-varying measurement error, distur-

bance and model uncertainty, it is possible to achieve

the desired robustness properties by properly tuning

the controller parameters. On the other hand, practical

stability here does not guarantee convergence to the

origin for vanishing perturbations, although the size

of the stable subset depends on the tuning parameters

and can be somewhat reduced, provided a sufﬁcient

knowledge on the intensity of the perturbations. The

main contributions of our work are the robustness to

model uncertainties and its semiglobal characteriza-

tion for nonlinear systems.

The rest of the paper is organized as follows. In

Section 2 we introduce the needed notation. In Sec-

tion 3 we formally state the problem. In Section 4

we introduce our dynamic quantization strategy. We

then present the main results of the paper and com-

ment them in Section 5. In Section 6 we check their

application on the illustrative example of a DC motor

with nonlinear load. Proofs are given in Section 8.

2 NOTATION

For a set A ⊂ R, and a ∈ A, A

≥a

denotes the set

{x ∈ A : x ≥ a}. |x| denotes the inﬁnity norm of

the vector x, that is, if x ∈ R

, |x| := max

i=1,...,n

B(x,R) refers to the closed ball of radius R centered at

x in this norm, i.e. B(x,R) := {z ∈ R

: |x − z| ≤ R}.

kxk is the inﬁnity norm of the signal x(·), that is, if

x : R

≥0

→ R

, kxk = esssup

t≥0

|x(t)|. A continuous

function α : R

≥0

→ R

≥0

is said to be of class K if it

is increasing and α(0) = 0. It is said to be of class K

∞

if it is of class K and α(s) → ∞ as s → ∞. A func-

tion β : R

≥0

× R

≥0

→ R

≥0

is said to be of class K L

if β(·,t) ∈ K for any ﬁxed t ≥ 0 and β(s, ·) is contin-

uous, decreasing and tends to zero at inﬁnity for any

ﬁxed s ≥ 0.

3 PROBLEM STATEMENT

We are interested in the robustness properties of non-

linear plants of the form

˙x = f (x,µ,u,d), (1)

where x ∈ R

is the state, f : R

×R

→ R

is a locally Lipschitz function, µ : R

≥0

→ P ⊂ R

is a vector of (possibly time-varying) parameters, u :

≥0

→ R

is a control input and d : R

≥0

→ D ⊂

is a vector of measurable and locally essentially

bounded exogenous perturbations. We assume that

f (0,µ,0,0) = 0 for all µ ∈ P .

Limited-information feedback imposes that only

an estimate of the state is available to the controller.

This estimate is elaborated based on an encoded mea-

surement of the actual state. This encoded symbol is

then sent over the communication channel. The com-

munication channel is deﬁned by its constant sam-

pling period τ and by the number of symbols N

N ∈ N

, that can be transmitted at each sampling

time kτ, k ∈ N. We will assume, in this paper, that the

communication channel is noiseless and delay-free.

The overall structure of the controlled systems can be

summarized by Figure 1.

˙x = f (x,µ, ˆu,d)

PLANT

ENCODER

CHAN.

COM.

DECODER

ˆu

CONTROLLER

ˆx

κ( ˆx, ν)

Figure 1: Limited information feedback with exogenous

perturbations, measurement errors and uncertainties.

At each reception of a symbol, that is, at each time

instant kτ, k ∈ N, the decoder computes the state es-

timate that will be used in the applied feedback law.

The decoding is necessarily imprecise due to the lim-

ited bandwidth of the channel. This imprecision is re-

inforced by the uncertainty on the plant parameter µ,

by the presence of exogenous disturbances d and by

the possible measurement errors d

. We assume that

only a constant

approximation ν ∈ P of the (possi-

bly time-varying) parameter vector µ is available and

deﬁne µ − ν =: d

∈ R

as the parameter uncertainty.

Our ﬁrst assumption imposes that, without communi-

cation constraints, the plant (1) can be stabilized by a

state-feedback law that makes it ISS with respect to

exogenous disturbances, parameter uncertainties and

measurement errors.

Assumption 1 (ISS of the nominal plant). There ex-

ists a continuous feedback law κ : R

× R

→ R

, a

continuously differentiable function V : R

→ R and

class K

∞

functions α, α, α,χ,Γ,γ such that, for all

x ∈ R

, d ∈ D, d

∈ P and d

∈ R

α(|x|) ≤ V (x) ≤ α(|x|),

|x| ≥ χ(|d|) + Γ(|d

|) + γ(|d

|) ⇒ (2a)

In a second stage one may think of implementing an

adaptive control strategy.

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

∂V

∂x

f (x,µ,κ(x + d

,µ + d

),d) ≤ −α(|x|) . (2b)

Based on the Lyapunov characterization of ISS

systems (Sontag and Wang, 1995), condition (2) is

equivalent to ISS of (1) with respect to d, d

, d

(at

least locally as far as d and d

are concerned). As-

sumption 1 therefore constitutes a strong requirement,

but the following remarks may help establishing it in

some particular contexts.

Remark 1 (Systems in strict feedback form). For all

systems in strict feedback form it is possible to achieve

conditions of Assumption 1. Indeed, back-stepping al-

lows to iteratively make each subsystem ISS with re-

spect to (d,d

), using part of the state as a “vir-

tual” control input. See (Freeman and Kokotovich,

1993) for details.

Remark 2 (Globally Lipschitz systems). The condi-

tions of Assumption 1 can be achieved for all systems

which can be stabilized by a globally Lipschitz state

feedback that makes it ISS with respect to actuation

errors. Indeed if L denotes the global Lipschitz con-

stant of the nominal control law κ, then the effects

due to parameter uncertainties d

and measurement

errors d

can be described explicitly as an input dis-

turbance

d satisfying |

d| ≤ L|d

| + L|d

|. Hence,

if γ ∈ K

∞

is the ISS gain, the presence of measure-

ment errors and parameter uncertainties simply adds

γ(|

d|) ≤ γ(2L|d

|) + γ(2L|d

|) to the solution esti-

mate of the closed-loop solutions, hence proving ISS

with respect to (d

). Note that all systems which

can be made ISS by differentiable bounded control

trivially satisfy this global Lipschitz condition. Cf.

e.g. (A.Isidori, 1999, Chapters 12,13,14)

4 QUANTIZED CONTROLLER

In this section, we extend the encoding-decoding pro-

cedure presented in (Liberzon and Hespanha, 2005)

and (Persis and Isidori, 2004) to take into account

exogenous disturbances, measurement errors and pa-

rameter uncertainties. We assume that measurement

errors are bounded by some constant E > 0 such that

k ≤ E. (3)

4.1 Quantization Region

Given an estimate ˆx of the actual state x, the quantiza-

tion region Q is deﬁned by its centroid ˆx and its radius

L > 0 as

Q := B( ˆx, L).

Due to measurement errors, the information available

to the encoder about the system state, i.e. x + d

, be-

longs to the quantization region Q if and only if the

estimation error e := x − ˆx + d

is small enough, that

is |e| ≤ |x − ˆx| + |E| ≤ L. Note that the presence of the

estimation error e results from the combined effects

of quantization (x − ˆx) and measurement errors (d

Given the number N

of symbols that can be trans-

mitted through the communication channel, we parti-

tion the quantization region into N

identical hyper-

cubes. Q is then updated according to the encoding-

decoding procedure described below.

4.2 Dynamics of the Encoder

At each step k ∈ N, the centroid update law is given

by the following hybrid dynamics

ˆx = f ( ˆx,ν,κ( ˆx, ν), 0), (4a)

∀t ∈ [kτ,(k + 1)τ),

ˆx(kτ) = ˆc(kτ), k 6= 0, (4b)

ˆx(0

−

) = 0, (4c)

where ˆc(kτ) is the centroid of the sub-region of Q (kτ)

in which x(kτ) + d

lies. This sub-region is identiﬁed

by the variable q

, which constitutes the output of the

encoder. In other words, q

∈ N

≤N

denotes the in-

dex of the sub-region of Q (kτ) to which x(kτ) + d

belongs. Then, given some Λ > 1 (we will make it

precise in the sequel) and any ball of initial conditions

B(0,∆) with ∆ > 0, the radius update law is given, at

each step k ∈ N, by the following dynamics

L((k + 1)τ) = Λ



L(kτ)

+ E



+ E , (5a)

L(0) = ∆ + E . (5b)

This radius update law is a natural extension of the al-

gorithms proposed in (Liberzon and Hespanha, 2005;

Persis and Isidori, 2004). It takes into account pos-

sible measurement errors. We will show in the se-

quel (cf. Claim 2) that such a dynamics leads to a

sequence {L(kτ)}

k∈N

that decreases up to a constant

depending on E, Λ and N. This in turn imposes a

decrease of the estimation error, modulo the measure-

ment errors, as long as dynamics (5) applies. The idea

behind this dynamics can be roughly summarized as

follows. The parameter Λ > 1 accounts for the ex-

pansiveness between sampling times. The constant E

appearing inside the brackets of (5a) accounts for the

case in which the encoder individuates a wrong sub-

region due to measurements errors. In such a situation

the error between the real and the measured state is

indeed less than the size of the sub-region, L(kτ)/N,

plus the measurement error. The second E appearing

in (5a) prevents the measured state from falling out of

the quantization region while the real one is inside.

A ROBUST LIMITED-INFORMATION FEEDBACK FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

Note that, as long as x(kτ) + d

lies in Q (kτ), the

estimation error satisﬁes |e(kτ)| ≤ ¯e(kτ), where

¯e(kτ) :=

L(kτ)

+ E, ∀k ∈ N, (6)

is the maximum quantization error. Hence, at each

sampling time, the quantization procedure individu-

ates a hypercube B( ˆx(kτ), ¯e(kτ)) to which x(kτ) be-

longs, provided that x(kτ) + d

∈ Q (kτ).

However, due to uncertainties and disturbances, it

may happen that x(kτ) + d

falls out of the quantiza-

tion region anyway. Indeed, the expansion factor Λ

in (5a) ensures that the updated quantization region

is large enough to contain the measured state only if

the quantization error is large compared to the distur-

bances (see the proof of Theorem 1 for details). This

situation is deﬁned as an overﬂow. It is represented

by the symbol q

= 0. In particular, as detailed in the

proof of Theorem 1, an overﬂow can happen at time

(k + 1)τ only if the maximum quantization error (6)

at time kτ is strictly smaller than the size of pertur-

bations and uncertainties. If an overﬂow occurs at the

th sampling time, k

∈ N

, the encoder updates the

quantization region as follows:

ˆx(k

τ) = ˆx(k

−

), (7a)

L((k

+ 1)τ) = Λ(E +E) + E, (7b)

where E ∈ R

will be deﬁned later on. This means

that the hypercube individuated by the quantization

procedure (to which x(kτ)) belongs, is no longer

B( ˆx(kτ),

L(kτ)

+ E), but rather B( ˆx(kτ),E + E), while

the rest of the update law remains as in (4),(5).

4.3 Dynamics of the Decoder

By implementing the same evolution laws as

(4),(5),(7), the decoder is able to reconstruct the evo-

lution of the state estimate ˆx from the knowledge of

}

k∈N

4.4 Controller

Inspired by the principle of certainty equivalence, and

in view of Assumption 1, the applied control input is

given by

ˆu(t) = κ( ˆx(t),ν), (8)

where ˆx(·) is given by (4),(5),(7).

5 MAIN RESULTS

Our ﬁrst result establishes robustness properties of

the closed-loop system with the proposed limited-

information feedback in the case the number of trans-

mittable bits is ﬁxed and the sampling period can be

adjusted arbitrarily.

Theorem 1 (Fixed N). Let Assumption 1 hold for the

system (1). Then, there exist class K

∞

functions

χ,Γ,γ

and, given any compact sets P ⊂ R

and D ⊂ R

any constant ∆ ∈ R

and any N ∈ N

, there exist

positive constants τ, Λ, E,E and a class K L function

β such that the trajectories of the closed-loop system

˙x = f (x,µ, ˆu,d), (9)

where ˆu(t) is the output of the digital controller de-

ﬁned by (4),(5), (7),(8), satisfy, for all x(0) ∈ B(0,∆),

all ν ∈ P , all µ : R

≥0

→ P , all d : R

≥0

→ D and all

: R

≥0

→ B(0,E),

|x(t)| ≤ β(∆,t) + χ(kdk) + Γ(kµ − νk) + δ, (10)

where δ := γ



Λ(E + E) +



2 +

Λ+1

1−



Theorem 1 states that (9) is semiglobally ISpS

(Input-to-State practically Stable) in the sense of

(Jiang et al., 1994) with the proposed quantized con-

trol strategy. Our proof, provided in Section 8.1, is

constructive. The utilized bit-rate, given by

log

)

is ﬁxed by the condition that quantization resolution,

given by N

−1

, is small enough to compensate for the

expansiveness of the system between sampling times

Λ. An upper bound on this expansiveness expressed

in terms of the (local) Lipschitz constant of the system

L, is given by

Λ := e

Lτ

Based on the size of the initial conditions ∆, our

control strategy permits to build a forward invariant

region where the constant L can be computed (cf.

Claim 1 below). The explicit condition on the data

rate used in the proof is then given by

ΛN

−1

< 1.

It is interesting to note that the required data-rate is the

same as in (Liberzon and Hespanha, 2005), modulo

the size of the constructed forward invariant region.

The comparison functions involved in (10) can be

explicitly given

β(·,t) = α

−1

(α(2γ(·)γ(e

−λt

))),

χ(·) = α

−1

(α(4χ(·))),

Γ(·) = α

−1

(α(8Γ(·))),

γ(·) = α

−1

(α(8γ(·))).

where λ := −





, and the K

∞

functions α,α, γ,χ

and Γ are deﬁned as in Assumption 1. We note that,

since the function χ does not depend on the parame-

ters of the of the controller, but only on the nominal

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

comparison functions introduced in Assumption 1, it

is possible for a class of control and disturbance afﬁne

systems to ﬁnd a continuous feedback law for any de-

sired ISS attenuation gain χ (Praly and Wang, 1996;

Teel and Praly, 1998).

Due to the particular design of the encoding-

decoding procedure, measurement errors no longer

appear as an input. Indeed, their effects are embed-

ded in the last term of (10), which depends only on the

parameters of the digital controller. As already antic-

ipated, the constant Λ is an estimate of the expansion

of the system between two successive sampling times.

On the other hand, the constants E and E are propor-

tional to the upper bound on the size of disturbances-

uncertainties and measurement errors, respectively. In

particular E is deﬁned in (3) and

E = Λmax

(

sup

µ,ν∈P

|µ − ν|, sup

d∈D

|d|

)

Hence, the last term in (10), which constitutes an up-

per bound to the steady-state error, is a continuous

function of the known upper bound on the size of

exogenous disturbances, that vanishes at zero. This

guarantees that the steady-state error is small if distur-

bances are small, provided a sufﬁcient knowledge of

the plant. Moreover, when no perturbations apply, we

recover the exact same result as (Liberzon and Hes-

panha, 2005).

Robustness to model uncertainties is the main

contribution of this work if compared to the existing

representative examples in the literature ((Kameneva

and Ne

c , 2008) and (Sharon and Liberzon, 2007)).

In (Kameneva and Ne

c , 2008) this lack of robust-

ness is due to the digital nature of the controller,

which is based on the exact dynamics or at most on

its discrete time approximation (cf. Equation (2) in

that reference). In (Sharon and Liberzon, 2007) this

possible lack of robustness comes from the fact that

ISS of the quantized closed-loop system is achieved

through a cascade reasoning from the quantization er-

ror (which is ISS with respect to external disturbances

thanks to the particular encoding/decoding strategy)

to the system’s state (which is ISS by hypothesis).

This is possible because the evolution of the quanti-

zation error is shown to be independent from both the

controller’s and the system’s state (cf. Equation (12)

in that reference). This is no longer achievable if one

introduces parametric uncertainties, as the state of the

controller is fed back in the evolution equation of the

quantization error. However, it would be interesting

to study if this lack of robustness persists if under As-

sumption 1. Then, it would be worth comparing the

“gains” given by the different methods. These studies

are not presented here.

Another contribution compared to (Sharon and

Liberzon, 2007) is the non-local characterization of

robustness, which turns out to be semiglobal. In

the statement of Theorem 2 in that reference, which

gives an extension of the proposed algorithm to non-

linear systems, the admissible set of initial condi-

tions and external disturbances are built starting from

the K

∞

functions β

and γ

, whose explicit expres-

sion depends on the Lipschitz constant of the system

(cf. proof of Theorem 1 in that reference). Indeed,

given a region where to deﬁne the Lipschitz constant

(|x| < l

and |w| < l

), it is possible to ﬁnd the size

of the ball of admissible initial conditions ∆ and al-

lowed disturbances ε by satisfying the two relations

(δ) + γ(ε) < l

and ε < l

. It follows that the value

of ∆ and ε cannot be chosen a priori, and may re-

sult impossible to be arbitrarily enlarged, depending

on the explicit expression of the two K

∞

functions β

and γ

. On the other hand, given a compact set of ini-

tial conditions and a bound on the size of exogenous

disturbances, it is not possible either to build the K

∞

functions used in the statement of the theorem, as it is

not possible to build an “overshoot” region in which

the Lipschitz constant would be deﬁned. These obser-

vations show that in (Sharon and Liberzon, 2007) the

extension to nonlinear systems is only local. In this

paper we give a constructive way to build the over-

shoot region starting from an arbitrary ball of initial

conditions and an arbitrary size for the exogenous dis-

turbances.

However, considering the superior performances

in the steady-state error of the algorithm proposed in

(Sharon and Liberzon, 2007) (ISS instead of ISpS),

one may think of implementing some switching strat-

egy between the two methods to beneﬁt from the ad-

vantages of each procedure. In a ﬁrst step the state

would be estimated with the algorithm proposed here

even in the case of parametric uncertainties. In a sec-

ond time, once the parameters of the systems have

been identiﬁed and the state has entered a sufﬁciently

small region around the origin, one would switch

to the algorithm proposed in (Sharon and Liberzon,

2007).

In case of overﬂow, the size of quantization region

is set to Λ(E + E)+E (cf. (7)). It may happen, in par-

ticular for large sampling periods or highly nonlinear

systems, that Λ gets big. In this case, the quantization

error may become very large as E depends linearly

on Λ (cf. (21)), leading to a drop in performances.

This can be easily avoided by using a suitable E in

the encoding-decoding procedure. Indeed it follows

from Claim 2 (see below) that, as long as no overﬂow

occurs, the size of the quantization region converges

A ROBUST LIMITED-INFORMATION FEEDBACK FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

∞

:= σ

∞

where σ

∞

denotes a positive constant (see (23) be-

low). It follows that the maximum quantization error

(6) converges from above to

∞



∞

+ 1



E. (12)

As we show in the sequel (cf. (19)) an overﬂow can

occur only if the maximum quantization error gets

smaller than some constant η (deﬁned in (16)), which

denotes an upper bound on the size of perturbations

and uncertainties. In other words, it sufﬁces to set E

such that

∞

≥ η (13)

to avoid overﬂows. We then have the following theo-

rem, whose proof follows directly from that of Theo-

rem 1, together with Equations (12) and (13).

Theorem 2 (Fixed N - no overﬂows). Under the

assumptions of Theorem 1, the design parameters

τ,Λ,E,E can be picked in such a way that (10) holds

with δ = γ



1 +

Λ+1

1−



We point out that the size of the steady state error

δ deﬁned in Theorem 2 can be either larger or smaller

than the one obtained in Theorem 1, depending on the

parameters involved.

We ﬁnally state a similar result for the case when

the sampling period is imposed by technological con-

straints and we can only adjust the number of trans-

mittable bits. In this context, it appears that, due to

the presence of exogenous perturbations, τ cannot be

chosen arbitrarily large, as it happens in the ideal case

(cf. (Liberzon and Hespanha, 2005)). This fact is de-

tailed in the proof, given in Section 8.4.

Theorem 3 (Fixed sampling period). Let Assump-

tion 1 hold for the system (1). Then, there exist

class K

∞

functions χ, Γ, γ and, given any compact sets

P ⊂ R

, D ⊂ R

and any ∆ ∈ R

, there exists a time

max

∈ R

such that, for all τ ∈ (0,τ

max

), there exist

positive constants N,Λ, E,E and a class K L func-

tion β such that trajectories of the closed-loop system

(9), where ˆu(t) is the output of the digital controller

deﬁned by equations (4),(5), (7),(8), satisfy (10) for

all x(0) ∈ B(0, ∆), all ν ∈ P , all µ : R

≥0

→ P , all

d : R

≥0

→ D and all d

: R

≥0

→ B(0,E). That is, (9)

is semiglobally ISpS.

We stress that the functions involved in the trajec-

tories estimate of this result are the same as for Theo-

rem 1.

Remark 3. Theorems 1, 2 and 3 can be easily gen-

eralized to non-constant sampling periods, provided

that the time between two samples does not exceed

the value τ deﬁned in the above statements. 

6 ILLUSTRATIVE EXAMPLE

We check the application of our strategy on the con-

trol of a model of a DC motor with a load modeled

as a nonlinear torque. The uncertainty on the load

is modeled by unknown time-varying variables µ and

. Actuator errors are represented by an exogenous

disturbance d

˙x

= x

+ µx

+ d

˙x

= u + d

For the needs of the numerical simulations, we

have chosen µ(t) = 1 + P sin(t), d

(t) = D sin(t) and

(t) = D cos(t). At each sampling time the measure-

ment available to the encoder is perturbed by the mea-

surement error d

(t) = E(sin(t),cos(t))

. The sys-

tem being in strict feedback form, we follow (Free-

man and Kokotovich, 1993) to construct a continu-

ous ISS feedback law. We assume that only 2 bytes

can be transmitted at each sampling time. Our aim

is to stabilize every solution starting in B(0, 10) (i.e

∆ = 10) assuming the following values for the per-

turbation amplitudes, P = 0.5, D = 1.0, E = 0.1.

Note that this correspond to a 50% uncertainty on the

load parameter. Our control scheme with parameters

τ = 0.1s, Λ = 64, E = 64, E and ν = 1 successfully

stabilizes the system (cf. Figure 2).

0 1 2 3 4 5 6 7 8 9 10

time (s)

| x |

Figure 2: Evolution of the norm of the state from x(0) =

[−10,−10].

Since E > 0, the size of the quantization region

remains sufﬁciently large (i.e e

∞

≥ η, cf. (13)) and

no overﬂow occurs, as stated by Theorem 2. If an

overﬂow had occurred then the size of the quantiza-

tion region would have jumped to Λ(E + E) + E (cf.

(7)), leading to a big drop in performances. This il-

lustrates the fact that, even when no measurement er-

ror applies, setting an appropriate E > 0 may be very

proﬁtable in practice.

For ∆ = 5, and the same parameters, the sampling

period can be taken as large as 0.3s. We point out that

the good performance of our strategy are also due to

the ISS characteristics of the feedback strategy pro-

vided in (Freeman and Kokotovich, 1993).

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

In conclusion, with only two bytes, with a sam-

pling period of the order of the plant’s time scale,

and with disturbances magnitude of the order of the

nominal controlled dynamics values, our proposed

approach succeeds in stabilizing the system, with a

steady-state error of the same magnitude as the per-

turbations.

7 CONCLUSIONS

The proposed strategy for limited-information feed-

back control of nonlinear plants is shown to be robust

to exogenous disturbances and measurement errors,

even in presence of parametric uncertainty. Its appli-

cation is illustrated by the numerical simulation of a

DC motor control. Possible future extensions concern

output feedback (see (Sharon and Liberzon, 2008) for

a representative example) and robustness to delays.

8 PROOF OF THE MAIN

RESULTS

8.1 Proof of Theorem 1

Contraction of the quantization region: Suppose there

are no measurement errors, i.e E = 0. Then we want

the estimation error to decrease as long as no overﬂow

occurs, that is, as long as x(kτ) ∈ Q (kτ), we impose

L(kτ) < L((k − 1)τ). This means, in view of (5), that

< 1. (15)

Divergence between sampling times: During the

time intervals separating two consecutive sampling

times, the estimation error may increase. To eval-

uate this expansion, let us assume that a number

W > 0 is known

such that x(t) + d

∈ B(0,

W ) and

ˆx(t) ∈ B(0,

W ) for all t ≥ 0. In this case, it re-

sults from the continuity of κ that, for all t ≥ 0,

| ˆu(t)| ≤ max

ˆx∈B(0,

W ),ν∈P

|κ( ˆx, ν)| =: U < ∞. Let

W ) be the Lipschitz constant of f over the re-

gion {(x,µ,u,d) ∈ R

n+p+m+h

: |x| ≤

W ,µ ∈ P ,|u| ≤

U, d ∈ D}, then, in view of (4) and exploiting

the Bellman-Gronwell Lemma, it holds that, for

all x, ˆx ∈ B(0,

W ),

|e(t)| ≤ | f (x,µ(t), ˆu,d(t)) −

f ( ˆx,ν, ˆu,0)| ≤ L(

W )max

{

|e(t)|,η(t)

}

, where η(t) =

max{|µ(t) − ν|, |d(t)|}. Note that η(t) ≤ η, where

η = max

(

sup

,ν

∈P

|µ

− ν

|, sup

∈D

)

. (16)

We will demonstrate the existence of such

W , by con-

structing it, in the sequel (cf. Claim 1).

Hence, it holds that, for all t 6= kτ, k ∈ N,

|e(t)| ≥ η(t) ⇒

|e(t)| = L(

W )|e(t)|, and

|e(t)| < η(t) ⇒

|e(t)| = L(

W )η(t).

By the fact that |e(0)|e

W )t

< |e(0)| + ηL(

W )t only

if |e(0)| < η

W )t

−1

≤ η, it follows that, for all t ∈

[0,τ],

|e(0)| ≥ η ⇒ |e(t)| ≤ |e(0)|e

W )t

. (17)

Deﬁning Λ = e

W )τ

, we can give a natural interpre-

tation to (5). The constant N

−1

describes the effect of

measuring the state, which individuates a smaller hy-

percube to which the state belongs, while Λ describes

the increase in the size of this hypercube during the

time before the next sampling to make sure the state

belongs to the new quantization region. Recalling that

|e(kτ)| ≤ ¯e(kτ) for all k ∈ N, we claim that

¯e(kτ) ≥ η ⇒ |e((k + 1)τ

−

| ≤ Λ ¯e(kτ), (18)

that is x((k + 1)τ) ∈ Q ((k + 1)τ) and, consequently,

k+1

> 0 (i.e. no overﬂow at step k + 1). Indeed,

note that if η ≤ |e(kτ)| ≤ ¯e(kτ), then (18) follows

from (17), while, if |e(kτ)| < η, there exists

t :=

min{t ≥ 0 : |e(kτ)| + ηL(

W )t ≥ η}. If

t ≥ τ then

|e((k + 1)τ

−

)| < η < Λ ¯e(kτ), while, if

t < τ, then,

by (17), |e((k +1)τ

−

)| ≤ ηe

(τ−

t)L(

W )

< Λη ≤ Λ ¯e(kτ),

which shows (18).

Furthermore, by reversing (18), we obtain that

|e((k + 1)τ

−

| > Λ ¯e(kτ) only if ¯e(kτ) < η, that is,

deﬁning E = sup

∈E

|, |e((k+ 1)τ

−

|+E > L((k +

1)τ) only if ¯e(kτ) < η, which implies that an overﬂow

may occur only when the maximum quantization er-

ror (6) is below the bound η:

k+1

= 0 ⇒ ¯e(kτ) < η. (19)

Moreover, we establish the following upper bound on

the size of the estimation error right before an over-

ﬂow:

k+1

= 0 ⇒ |e((k + 1)τ

−

)| < Λη. (20)

Note that, from (19), there exists a time

:= min{t >

0 : |e(kτ)| + ηL(

W )t ≥ η}. If

≥ τ then |e((k +

1)τ

−

)| ≤ η < Λη. On the other hand, if

< τ then,

by (17), |e((k + 1)τ

−

)| ≤ ηe

(τ−

)L(

W )

< Λη, which

establishes (20).

Trajectory boundness: Fix any Λ > 1 and let

E = Λη. (21)

By equation (20) and as long as x(t) + d

∈ B(0,

W ),

this implies that, in the eventuality of an overﬂow at

time k

τ (i.e. q

= 0),

x(k

τ),x(k

τ) + d

∈ B( ˆx(k

τ),E +E), (22)

A ROBUST LIMITED-INFORMATION FEEDBACK FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

In view of (19) and (7), this implies that x((k

+1)τ) ∈

Q ((k

+ 1)τ), that is q

> 0. This means that it is

not possible to have two successive overﬂows.

Let Ω

:= {x ∈ R

: V (x) ≤ c}, where

c = α



χ(sup

∈D

|) + Γ(sup

,ν

∈P

|µ

− ν

|) +



max



∆ + E + σ

∞

E, Λ(E +E) + E





where σ

∞

is deﬁned as

∞

:= (Λ + 1)N/(N − Λ). (23)

Let

W := W + max



∆+E +σ

∞

E, Λ(

E + E)+ E



, (24)

W := max



∆,sup

x,z∈Ω

|x − z|



, (25)

and pick the sampling period τ as τ =

ln(Λ)

W )

. We prove

the following in Sections 8.2 and 8.3 respectively.

Claim 1. The solutions of the closed-loop system sat-

isfy |x(t)| ≤ W , | ˆx(t)| ≤

W , ∀t ≥ 0.

Claim 2. As long as no overﬂow occurs, it holds that

|e(t)| ≤ e

−λt

L(0) + σ

∞

E, where λ = −





Conclusion: From the proof Claim 1, it results

that, for all t ≥ 0, |e(t)| ≤ max



(∆ + E)e

−λt

∞

E, Λ(E + E) + E



≤ ∆e

−λt

+ Λ(E + E) + (2 +

∞

)E, ∀t ≥ 0. From Assumption 1, this implies that

the trajectories of the closed loop system (9), with pa-

rameters {N,τ,Λ,E,ν, E}, satisfy

|x(t)| ≤ α

−1





γ(∆e

−λt

) + χ(kdk) + Γ(kµ − νk) +

γ(Λ(E + E) + (2 + σ

∞

)E)





for all x(0) ∈ B(0,∆), all ν ∈ P , all µ : R

≥0

→ P , all

d : R

≥0

→ D and all d

: R

≥0

→ B(0,E). From this

and from the fact that σ(a + b) ≤ σ(2a) + σ(2b) for

all nondecreasing function σ and all a, b ≥ 0, the the-

orem is proved with

β(·,t) = α

−1

(α(2γ(·)γ(e

−λt

))) (26a)

χ(·) = α

−1

(α(4χ(·))) (26b)

Γ(·) = α

−1

(α(8Γ(·))) (26c)

γ(·) = α

−1

(α(8γ(·))). (26d)

8.2 Proof of Claim 1

Let Θ := inf{t ∈ R

: |x(t)| > W or | ˆx(t)| >

W }.

This time is well deﬁned as |x(0)| ≤ ∆ ≤ W and,

from the fact that q

> 0 by construction, ˆx(0) ∈

B(0,∆ + E) ⊂ B(0,

W ). For all t ∈ [0,Θ), L(

W ) can

be correctly interprated as an upper bound on the ex-

pansion of the system. In particular, Claim 2, (19) and

(20) hold for all t ∈ [0, Θ).

Deﬁne k

τ as the time of the ﬁrst overﬂow. Then,

by Claim 2, it holds that, for all t ∈ [0, min(Θ, (k

−

1)τ)), |e(t)| < L(0)e

−λt

+σ

∞

E ≤ ∆+E +σ

∞

E, where

λ =

|ln(

If Θ < (k

− 1)τ, then, from Assumption 1 and

(A.Isidori, 1999, Section 10.4), it results that

the set Ω

˜c

= {x ∈ R

: V (x) ≤ ˜c}, where ˜c :=

α(χ(sup

∈D

|)+Γ(sup

,ν

∈P

|µ

−ν

|)+γ(∆+E +

∞

E)), is an invariant attractive set, and, noting that

˜c ≤ c, it follows that Ω

˜c

⊆ Ω

, which, by the deﬁ-

nition of W in (25), implies |x(t)| ≤ W for all t ∈

[0,Θ]. This in turn ensures that sup

t∈[0,Θ]

| ˆx(t)| ≤

W + sup

t∈[0,Θ]

|e(t)| ≤ W + ∆ + E + σ

∞

E ≤

W (cf.

(24)). This contradicts the deﬁnition of Θ and hence

we conclude that Θ ≥ (k

− 1)τ.

If Θ ∈ [(k

− 1)τ,k

τ), then, by (19) and (20) and

the deﬁnition of E in (21), it results that |e(t)| < E

for all t ∈ [t

− τ,Θ]. With the same arguments as

before, this contradicts the deﬁnition of Θ and hence

we conclude Θ ≥ k

τ.

If Θ ∈ [k

τ,(k

+ 1)τ), by E > η and (22), we get

that |e(t)| ≤ Λ(E + E) for all t ∈ [k

τ,Θ], again con-

tradicting the deﬁnition of Θ. Hence we can conclude

Θ ≥ (k

+ 1)τ.

If Θ = (k

+ 1)τ, then by construction q

> 0

and |e((k

+ 1)τ)| ≤ L((k

+ 1)τ) = Λ(E + E) + E,

again contradicting the deﬁnition of Θ. Hence Θ >

+ 1)τ.

The system properties established along the whole

proof being uniform in time, we can set t

= t − (k

1)τ and apply the same arguments with new “initial”

condition L(0) = Λ(E + E) + E until the next over-

ﬂow. By reiterating for successive overﬂows, we con-

clude Θ = ∞, which is enough to prove the claim.

8.3 Proof of Claim 2

The ﬁrst line in (5), can be rewritten as

L(kτ) = R

L(0) + E(Λ + 1)

∑

i=0

L(kτ), (27)

where

L : R

≥0

→ R

is deﬁned as

L(t) = R

t/τ

L(0) + E(Λ + 1)

∞

∑

i=0

, (28)

and R := Λ/N < 1 (cf. (15)).

If Λ is chosen appropriately to compensate for er-

ror divergences between sampling times, if no over-

ﬂow occurs, then, recalling the deﬁnition of the maxi-

mum quantization error (6), forall t ∈ [kτ, (k + 1)τ),

|e(t)| ≤ Λ

t−kτ



L(kτ)

+ E



. Substituting (27) in the

this equation, we obtain that, for all t ∈ [kτ,(k + 1)τ),

ICINCO 2010 - 7th International Conference on Informatics in Control, Automation and Robotics

|e(t)| ≤





t/τ

L(0) +

t−kτ

E(Λ + 1)

∑

i=0





ΛE <

L(t). Recalling ﬁnally that the geometric se-

ries in the deﬁnition of

L (28) converges,

∑

∞

i=0





N−Λ

, this ﬁnishes to establish Claim 2 by recalling

the deﬁnition of σ

∞

in (23) and by noticing that the

error can only decrease at the sampling times.

8.4 Proof of Theorem 3

The proof follows along the same lines as that of The-

orem 1. The difference stands in the fact that Λ can no

longer be chosen arbitrarily. Instead, given any τ > 0,

we deﬁne Λ = e

W )τ

, where

W is deﬁned as before

(cf. (24)). Note that Λ enters the deﬁnition of

W , in

particular

W and L(

W ) are both continuous increas-

ing functions of Λ. Hence, Λ is required to satisfy the

following two equations Λ = e

W )τ

and

W =

W (Λ).

This set of equations admits other solutions than the

trivial one (Λ = 1, τ = 0) provided that

τe

W (Λ))τ

dΛ

W (Λ))



Λ=1

< 1.

By continuity of the equations in τ and Λ, we con-

clude that there exists τ

max

> 0 such that a solution

exists for all τ ∈ (0,τ

max

). The rest of the proof fol-

lows that of Theorem 1.

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A ROBUST LIMITED-INFORMATION FEEDBACK FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS