Systems & Control Letters 62 (2013) 943–948

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Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

Output feedback networked control with persistentdisturbance attenuation✩

Eloy Garcia a,∗, Panos J. Antsaklis b

a Infoscitex Corporation, Dayton, OH 45431, USAb Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

h i g h l i g h t s

• Present a method for stabilization and rejection of persistent disturbances.• Consider output feedback using the state observer to estimate states and disturbances.• Apply in NCS to reduce the number of transmitted messages.• Improved performance is shown with respect to recent and similar work.

a r t i c l e i n f o

Article history:Received 11 May 2012Received in revised form21 April 2013Accepted 3 July 2013Available online 15 August 2013

Keywords:Networked control systemsModel-based controlDisturbance attenuationState observers

a b s t r a c t

This paper presents a model-based control approach for output feedback stabilization and disturbanceattenuation of continuous time systems that transmit measurements over a limited bandwidthcommunication network. Necessary and sufficient conditions for asymptotic stability of the networkedsystem in the presence of persistent external disturbances are given. The results in this paper provide asignificant improvement in the performance of the system and provide a considerably reduction of thenecessary network bandwidth with respect to similar approaches in the literature.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In Networked Control Systems (NCS) a digital communicationnetwork is used to transfer information among the components ofa control system including actuators, controllers, and sensors. Thistype of implementation differs significantly from classical controlsystems where all system components are attached directly to thecontrol plant exchanging information using dedicated wiring [1].The use of a shared communication channel for control systemsmakes feedback measurements inaccessible to the controller forlong intervals of time. An approach followed by different authorsis to keep the latest received feedbackmeasurement constant untilnew information arrives from the controller, i.e. using a Zero-Order-Hold (ZOH), [2–4].

✩ Partial support of the National Science Foundation under Grant No. CNS-1035655 is gratefully acknowledged.∗ Corresponding author. Tel.: +1 5742173592.

E-mail address: elgarcia@infoscitex.com (E. Garcia).

0167-6911/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.sysconle.2013.07.005

The use of amodel of the system in the controller node providesbetter performance in general since an estimate of the state ofthe system can be used for control during the time intervals thatfeedback measurements are not available.

Model-based frameworks have been used by different au-thors for control of networked systems [5–14] or for control ofsystems with limited feedback communication [15]. The workin [5–7,13,14] considers model uncertainties and no external dis-turbances while the authors of [8,9] focus on attenuation of con-stant input disturbances by considering no plant-modelmismatch.Similar model-based frameworks have been studied by differentauthors, for instance, the work in [11] considers discrete time sys-tems subject to packet dropouts and the model dynamics are usedto generate the control inputs when no feedback measurementsare successfully received. The authors of [10,12] used the model-based framework in [6] for, respectively, stabilization of singularlyperturbed systems and stabilization of coupled networked uncer-tain systems.

In the present paper we address the case of disturbance attenu-ation and consider the situation of output feedback using state ob-servers. The main contributions of this paper are as follows: first,

944 E. Garcia, P.J. Antsaklis / Systems & Control Letters 62 (2013) 943–948

Fig. 1. Model-based networked system with external disturbance.

we introduce an augmented state observer that provides estimatesnot only of the states of the disturbed system but also estimates ofthe unknown external disturbance. We provide observability con-ditions for the augmented system in terms of the original plant pa-rameters. Second, the augmented state observer is implementedin the networked system and we provide stability conditions as afunction of the update interval parameter which dictates how of-ten the sensor transmits the observed variables to the controllernode. Constant update intervals are used throughout this paper.It is shown that this approach considerable improves the perfor-mance of the networked system compared to similar work in [8,9].

The paper is organized as follows. Section 2 introduces themodel-based architecture for control over networks. Section 3describes the augmented state observer and provides observabilityconditions on the augmented system. Section 4 provides stabilityresults for continuous time systems that transmit measurementsusing a limited bandwidth communication network. Illustrativeexamples are given in Section 5, and Section 6 concludes the paper.

2. Model-based networked architecture

The model-based networked setup illustrated in Fig. 1 makesuse of an explicit model of the plant which is added to theactuator/controller node to compute the control input based on thestate of the model rather than on the plant state. The state of themodel is updated when the controller receives the measured stateof the plant that is sent from the sensor node every h seconds.

Consider a Model-Based Networked Control System (MB-NCS)as shown in Fig. 1. We concern ourselves with continuous timesystems that are disturbed by an unknown external disturbanceas shown in that figure. Assume that the disturbance is a persis-tent disturbance; in particular, we assume that it is a constantsignal. This approach can be used in practice for piece-wise con-stant disturbances. In this case the output of the system is notasymptotically stable since there could be an infinite number ofdiscrete changes on the disturbance as time goes to infinity. How-ever, bounded stability can be obtained as it is shown in this paper.

The dynamics of the system are given by

x(t) = Ax(t) + B(u(t) + w(t))y(t) = Cx(t)

(1)

where x(t) ∈ Rn, u(t) ∈ Rm, and y(t) ∈ Rp represent the state,control input, and measurable output of the system, respectively.w(t) ∈ Rm represents the unknown constant external disturbance.It is assumed that the disturbance is bounded as follows ∥w(t)∥ ≤

W . The model dynamics are given by the nominal system dynam-ics:

˙x(t) = Ax(t) + Bu(t)

y(t) = Cx(t).(2)

We do not assume that the entire state vector is available for mea-surement but only the output y(t). We implement a state observer

in the controller node in order to estimate the state of the systemand use the observer state to update the state of the model. Sincethe input disturbance w(t) cannot be measured we augment thesystem in order to estimate both the states and the disturbance.This approach will be used in Section 4 in order to attenuate theundesired effects of the disturbance when the system is controlledusing a networked architecture as shown in Fig. 1.

3. Augmented state observer

Consider a linear systemgiven by (1). In this sectionwe consideran augmented state observer under the assumption of continuousaccess to the input and output signals, u(t) and y(t) as in atraditional closed loop control system. This state observer will beused later in the networked implementation represented in Fig. 1.The purpose of this augmented observer is to provide estimatesnot only of the state of the system x(t) but also to obtain estimatesof the unknown disturbance w(t). Define the augmented statevector:

xe(t) =x(t)T w(t)T

T. (3)

The augmented system is given by

xe(t) =

A B0 0

xe(t) +

B0

u(t) = Aexe(t) + Beu(t)

y(t) =C 0

xe(t) = Cexe(t).

(4)

A state observer is designed for the augmented system (4) and it isgiven by

˙xe(t) = (Ae − LCe)xe(t) + Beu(t) + Ly(t). (5)

Theorem 1. Assume that the continuous time pair (A, C) is observ-able, then (Ae, Ce) is observable if N (M) = {0}, the nullspace of Mcontains only the zero vector, where

M =

A BC 0

. (6)

Proof. Let λi represent the eigenvalues of A for i = 1, . . . , n, withassociated eigenvectors vi, and let λe

i represent the eigenvalues ofAe for i = 1, . . . , n+m, with associated eigenvectors ve

i . Note that

λei = λi for i = 1, . . . , n.

λei = 0 for i = n + 1, . . . , n + m.

(7)

For any λei = 0 we have that

(λei I − Ae)v

ei =

λiI − A −B

0 λiI

vnvm

= 0 (8)

since λi = 0 then vm = 0 and

(λiI − A)vn = 0 ⇒ vn = vi

⇒ vei =

vi0

.

(9)

Additionally, assume that λei = 0 is an unobservable eigenvalue of

(Ae, Ce), then by the PBH unobservable eigenvalue condition [16]we have that

λei I − AeCe

v = 0 (10)

for some v = 0. Eq. (10) can be expressed asλiI − A −B

0 λiIC 0

vnvm

= 0 (11)

E. Garcia, P.J. Antsaklis / Systems & Control Letters 62 (2013) 943–948 945

which results in vm = 0 and(λiI − A)vn

C vn

= 0 (12)

but since (A, C) is observable (12) is only satisfied by vn = 0then we have a contradiction and consequently any λe

i = 0 is anobservable eigenvalue of (Ae, Ce).

Now consider λei = 0 and, similarly, assume that λe

i = 0 isan unobservable eigenvalue of (Ae, Ce), then we have that (10) issatisfied for some v = 0. Eq. (10) now results in the following:

−Avn − BvmC vn

= 0 (13)

but from the condition on M expressed in the theorem, (13) canonly be satisfied by v =

vTn vT

m

T= 0 and we have a contradic-

tion; therefore any λei = 0 is an observable eigenvalue of (Ae, Ce)

and the augmented system (4) is observable. �

4. Continuous time systems with limited output feedback anddisturbance estimation

The networked system in Fig. 1 contains a model (2) of thesystem in order to generate an estimate of the state of the disturbedsystem (1) and to compute the control input between updateintervals. The use of the extended observer described in theprevious section provides an estimate of the unknown externaldisturbance that can be transmitted to the controller node in orderto improve the control action by compensating for the effects ofthe disturbance w(t), that is, the control input is given by

u(t) = Kx(t) − w(t) (14)

where w(t) represents the latest update of the disturbance esti-mate received at the controller node, i.e.

w(t) = w(tµ) for t ∈ [tµ, tµ+1) (15)

where tµ represents the update instants and tµ+1 − tµ = h is theupdate period and it is assumed to be constant, i.e. we use periodiccommunication. At the update instants the augmented observerstate xe(tµ) is received at the controller node. The observer statecontains estimates of both, the states of the plant and the distur-bance. The observed disturbance is kept constant in the controllernode during the next update interval as described in (15) and themodel state is updated using the part of the observer state corre-sponding to the estimated plant states, i.e.

x(tµ) = x(tµ). (16)

After a model update takes place the model keeps its normal exe-cution as in (2).

The sensor node contains the augmented observer and it hascontinuous access to the output of the system. In order to providethe observer with the control input u(t) a copy of the modelis implemented in the sensor node which is updated using theobserver state at the same time instants that the model in thecontroller. The purpose is to stabilize the system by maintainingminimum communication between the sensor and controllernodes.

Define the state vector

z(t) =xe(t)T e(t)T xe(t)T

T (17)

that contains the state of the augmented system (3), the state ofthe observer, and the state error:

e(t) = xe(t) −

x(t)w(t)

. (18)

Between model update instants the networked system operates inopen loopmode since feedbackmeasurements are not transmittedto the controller node. The evolution of (17) during the intervalst ∈ [tµ, tµ+1) is given by

z(t) = Λz(t) (19)

where

Λ =

Ae + BKB −BKB 00 Ae 0

BKB + LCe −BKB Ae − LCe

(20)

BKB =

BK −B0 0

. (21)

At the update instants tµ we have that

xe(tµ) = xe(t−µ )

e(tµ) = xe(t−µ ) − xe(t−µ )

xe(tµ) = xe(t−µ ).

(22)

Proposition 2. The system described by (19) with updates (22) andwith initial conditions z0 = z(t0) has the following response:

z(t) = eΛ(t−tµ)

I 0 0I 0 −I0 0 I

eΛh

µ

z0 (23)

for t ∈ [tµ, tµ+1) where h = tµ+1 − tµ.

Proof. Without loss of generality consider t0 = 0. On the intervalt ∈ [0, t1) the response of the system is given by

z(t) = eΛtz0. (24)

At t = t1 the state of the model is updated using the state of theobserver, this can be represented as follows:

z(t1) =

I 0 0I 0 −I0 0 I

eΛhz0. (25)

Continuing with the interval t ∈ [t1, t2) we have

z(t) = eΛ(t−t1)

I 0 0I 0 −I0 0 I

eΛhz0. (26)

At t = t2 the state of the model is updated once again using thestate of the observer, and we have the following:

z(t2) =

I 0 0I 0 −I0 0 I

eΛh

I 0 0I 0 −I0 0 I

eΛhz0

=

I 0 0I 0 −I0 0 I

eΛh

2

z0. (27)

By following the same analysis at every cycle t ∈ [tµ, tµ+1) weobtain the response of the system given by (23). �

Theorem 3. The system described by (19) with updates (22) isglobally exponentially stable around the solution

z =0n w 0n+m 0n w

T (28)

if and only if the eigenvalues of I 0 0I 0 −I0 0 I

eΛh (29)

946 E. Garcia, P.J. Antsaklis / Systems & Control Letters 62 (2013) 943–948

are within the unit circle of the complex plane, wherew represents thevalue of the constant disturbancew(t) and 0n represents a row vectorof n zeros.

Proof. Consider the following change of variables:

wc(t) = w(t) − w

wc(t) = w(t) − w

zc(t) = z(t) −0n w 0n+m 0n w

T⇒ zc(t) = z(t).

(30)

It is also easy to check that

zc(t) = eΛ(t−tµ)

I 0 0I 0 −I0 0 I

eΛh

µ

zc0. (31)

Sufficiency. Taking the norm of (31) we have

zc(t) =

eΛ(t−tµ)

I 0 0I 0 −I0 0 I

eΛh

µ

zc0

≤eΛ(t−tµ)

I 0 0

I 0 −I0 0 I

eΛh

µ zc0 . (32)

Let us analyze the first term on the right hand side of (32):eΛ(t−tµ) ≤ 1 + (t − tµ)σ (Λ) +

(t − tµ)2

2!σ (Λ)2 · · ·

= eσ (Λ)(t−tµ)

≤ eσ (Λ)h= K1 (33)

where σ (Λ) is the largest singular value of Λ.The second term in (32) can be bounded if and only if the

eigenvalues of (29) are within the unit circle of the complex plane: I 0 0

I 0 −I0 0 I

eΛh

µ ≤ K2 e−α1µ (34)

for some K2, α1 > 0.Sinceµ is a function of timewe can bound the right term of (34)

in terms of t:

K2 e−α1µ < K2 e−α1t−1h = K2 e

α1h e−

α1h t

= K3 e−αt (35)

for some K3, α > 0.From (32), (33) and (35) we can conclude thatzc(t) ≤ K1K3 e−αt

zc0 (36)

that is, zc(t) converges exponentially to zero and z(t) to (28).Necessity. We will now proof the necessity part of the theorem

by contradiction. Assume that the system is stable and also assumethat (29) has at least one eigenvalue outside the unit circle. Sincethe system is stable, a periodic sample of the response should bestable as well. In other words the sequence product of a periodicsample of the response should converge to zero with time.Wewilltake the sample at times t−µ+1.We can express the solution zc(t−µ+1)as

zc(t−µ+1) = ξ(µ) = eΛ(t−µ+1−tµ)

I 0 0I 0 −I0 0 I

eΛh

µ

zc0 (37)

we also know that (29) has at least one eigenvalue outside theunit circle. This means that (37) will in general grow with µ. In

other words, we cannot ensure that ξ(µ) will converge to zero forgeneral initial condition zc0:zc(t−µ+1)

= ∥ξ(µ)∥ → ∞ as µ → ∞; (38)

this clearly means that the system cannot be stable, and thus wehave a contradiction. �

Remark 1. Note that the states of the disturbed system (1) areasymptotically stable around the equilibriumpoint x = 0. Thenon-zero components of the solution (28) correspond to the externaldisturbance, its value cannot be controlled, and to the elementsof the observer state corresponding to the observation of thedisturbance.

The model-based approach with disturbance estimation pre-sented in this paper can also be applied when the disturbance isa piece-wise constant signal. This is possible since the observeris able to estimate the new setpoint of the disturbance. In thiscase the output of the system is not asymptotically stable sincethere could be an infinite number of discrete changes on the dis-turbance as time goes to infinity. A bound on the state can beobtained assuming that the time between changes of the distur-bance is large compared to the time response of the compensatedmodel and the observer. In addition, the update period hb shouldbe selected smaller than the minimum time between disturbancechanges in order to transmit the new estimated disturbance to thecontroller node, hb < min(h, τw) where the period h is a stabiliz-ing period obtained from Theorem 3 and τw is the minimum timebetween changes on the disturbance. Under the previous condi-tions we can estimate an upper bound on the state of system (1)with model-based control input (14), observer (5), and with peri-odic updates hb.

System (1) can be written as

x(t) = Ax(t) + B(Kx(t) − w(t−ν ) + w(t+ν )) (39)

where w(t−ν ) represents the estimated value of the disturbancebefore a step change in the disturbance and w(t+ν ) representsthe value of the disturbance after a step change has occurred.Since sufficient time has elapsed since the previous step changeon the disturbance and the conditions in Theorem 3 are satisfied,there exists positive constants c, c , and cw such that

x(t−v ) ≤

c,x(t−v )

≤ c , andw(t−v )

=w(t−v ) − w(t−v )

≤ cw . Letus consider the worst case scenario where the disturbance stepchange occurs just after an update instant tµ whichmeans that thenew estimated states and disturbance will be received until timetµ+1 = tµ + h and the step change is the maximum step changegiven by

w(t+v ) − w(t−v ) = 2W . The response of (39) is given by

x(tµ+1) = eAhbx(t−ν ) +

hb

0eA(hb−s)BK x(s) ds

+

hb

0eA(hb−s)Bwν ds, t ∈ [tµ, tµ+1) (40)

where wν = w(t+ν ) − w(t−ν )and we have set x(tµ) = x(t−ν ), thenan upper bound for the system response (40) is given by

∥x(t)∥ ≤ c · ahb,0 + c ∥BK∥

hb

0ahb,sa · ds

+ (2W + wc) ∥B∥ hb

0ahb,s · ds (41)

where ax,y =eA(x−y)

and a =e(A+BK)s

.

E. Garcia, P.J. Antsaklis / Systems & Control Letters 62 (2013) 943–948 947

Fig. 2. Example in [9]. Solid lines represent plant states x, dotted lines representobserver states x, and dashed lines represent model states x. The dark arrows at thebottom represent the update instants.

5. Examples

Example 1. Consider the same example given in [9] where asimilar technique to the present paper was applied except that adisturbance observer is not used and the communication is basedon events in that example. The main idea of the approach in [9]is the same as in this paper, that is, to use the nominal modelto generate estimates of the current state of the system, whilethe system is disturbed by an unknown constant disturbance. Theresults in this example show the significant improvements that canbe obtained using the approach described in the present paper interms of both system performance and network communication.

Consider the continuous time system given by

x(t) =

−0.1 0.1112

−18

x(t) +

0.10

u(t) +

0.10

w(t)

y(t) =0 0.5

x(t).

The simulation in [9] has been repeated in Fig. 2. It can be seenthat the states of the system are bounded but disturbance rejectionis poor. For the framework studied in the present paper, weconsider the same initial conditions x(0) =

3 1

T and constantdisturbance w(t) = 0.5 as in [9]. The initial conditions for themodel, the augmented observer, and the disturbance estimate inthe controller node are all equal to zero.

Fig. 3 shows the simulation results following the approachdescribed in the present paper. The update period is h = 100 s.The first update takes place at t1 = 20 s, at this instant the modeland the estimated disturbance are first updated using the currentobserver quantities and the response of the system considerablyimproves in comparison to the first 20 swhen themodel states andthe estimated disturbance in the controller w(t) were all equal tozero. The performance of the system ismuch better than the resultsin [9], which were shown in Fig. 2, as measured by the state normduring the interval shown and we also use fewer communicationinstants during the same simulation interval as shown by the darkarrows at the bottomof both figures.We are able to achieve amuchbetter disturbance rejection only at the cost of degraded initialresponse, that is, the initial seconds of the execution of the systemwhich are shown in detail in Fig. 4.

For completeness, we show in Fig. 5 the simulation of the sameexample using the same disturbance for the case of a traditional

Fig. 3. Response of Example 1 using the model-based approach with disturbanceobserver described in this paper. Solid lines represent plant states x, dotted linesrepresent observer states x, and dashed lines represent model states x. The darkarrows at the bottom represent the periodic update instants.

Fig. 4. Response corresponding to the first 40 s of Fig. 3. Solid lines represent plantstates x, dotted lines represent observer states x, and dashed lines represent modelstates x. The dark arrows at the bottom represent the periodic update instants.

networked systemwhere a model-based approach is not used nei-ther a disturbance observer. In this case the received measure-ments are held constant in the controller, that is, a ZOH model isimplemented. The top of Fig. 5 shows the case when the transmis-sion period is equal to 30 s, the system becomes unstable for pe-riods larger than about 40 s. Good disturbance rejection is not ob-tained even if we decrease the transmission period as shown at thebottom of Fig. 5, where the transmission period is equal to 5 s.

Example 2. Consider the 2 input–2 output unstable system givenby

x(t) =

0.8 −1.5 1.60 0.3 −0.5

−0.5 1 −0.7

x(t) +

1 20 31 0

(u(t) + w(t))

y(t) =

1 0 0.51 −0.3 0

x(t).

Fig. 6 shows the response of the system subject to piece-wiseconstant disturbances for h = 10 s. Our estimation-control scheme

948 E. Garcia, P.J. Antsaklis / Systems & Control Letters 62 (2013) 943–948

Fig. 5. Example 1 using a ZOHmodel. Normof the state for periodic communicationh = 30 s (top), and h = 5 s (bottom).

Fig. 6. Response of the system in Example 2 showing the states of the plant, thenorm of the plant states, and the piece-wise constant disturbance.

is able to asymptotically stabilize the unstable system and to rejectthe effects of the external disturbance. The same figure also showsthe disturbances thatwere used in this example. The system is ableto react to changes in the value of one or both of the disturbancesand to bring the plant states to its equilibrium point after thesedisturbance variations.

Remark 2. Although in general very large stabilizing update peri-ods can be obtained as given by the conditions in previous sections,it is not recommended using very large update intervals in the caseof piece-wise disturbances since the transient response could notbe satisfactory. By using very large periodic communication inter-vals the system may react to the changes in the disturbance set-point values after long time since the disturbance changed. The

system can be stabilized again but the system response during thattime interval may not be desired in general.

6. Conclusion

Output feedback control of networked systems in the presenceof persistent disturbances has been addressed in this work. Anestimate of the unknown external disturbance is used in thecontroller node along with the model state in order to stabilizethe system and compensate for the disturbance effects on thesystem output. An augmented state observer was implementedin the sensor node to provide estimates of both the externaldisturbance and the states of the disturbed system. Observabilityconditions for the augmented system were also provided in termsof the original system parameters. The overall networked systemis shown to be asymptotically stable even in the presence ofpersistent disturbances in comparison to similar methods thatonly offer bounded output results. This shows that our approachis able to reject the effects of unknown piece-wise constantdisturbances on the networked system. In addition, this approachrequires very infrequent communication between sensor andcontroller which reduces network traffic and releases the networkso it can be used by other systems and applications to transmitinformation. Future researchwill address dynamic communicationschemes in order to improve the transient response of the systemwhen the disturbances present sudden changes since the periodiccommunication approach does not react instantaneously underthat situation.

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