Anticipative and non-anticipative controller design for network

18 Pages·2011·0.2 MB·English

Anticipative and non-anticipative controller
design for network control systems
Payam Naghshtabrizi1 and Joa~o P. Hespanha2
1 Departmentof Electrical andComputer Engineering, Univ.of California, Santa
Barbara, CA 93106-9560 [email protected]
2 Departmentof Electrical andComputer Engineering, Univ.of California, Santa
Barbara, CA 93106-9560 [email protected]
Summary. We propose a numerical procedure to design a linear output-feedback
controller for a remote linear plant in which the loop is closed through a network.
The controller stabilizes the plant in the presence of delays, sampling, and packet
dropouts in the (sensor) measurement and actuation channels. We consider two
types of control units: anticipative and non-anticipative. In both cases the closed-
loop system with delays, sampling, and packet dropouts can be modeled as delay
di(cid:11)erential equations. Our method of designing the controller parameters is based
ontheLyapunov-Krasovskiitheoremandalinearconecomplementarityalgorithm.
Numerical examples show that the proposed design method is signi(cid:12)cantly better
thanthe existing ones.
1 Introduction
Network Control Systems (NCSs) are spatially distributed systems in which
the communication between plants, sensors,actuators, and controllersoccurs
throughasharedband-limiteddigitalcommunicationnetwork.Usingnetwork
as a medium to connect spatially distributed elements of the system results
in (cid:13)exible architectures and generally reduces wiring and maintenance cost,
sincethereisnoneedforpointtopointwiring.Consequently,NCSshavebeen
(cid:12)nding application in a broad range of areassuch as mobile sensor networks,
remote surgery, haptics collaboration over the Internet and unmanned aerial
vehicles[7].However,the useofasharednetwork,incontrasttousingseveral
dedicated independent connections, introduces new challenges.
To transmit a continuous-time signal over a network, the signal must be
appropriately sampled to be carried over a network as atomic units called
packet. Hence there are some similarities between NCSs and sampled-data
systems due to the sampling e(cid:11)ect. However, NCSs are signi(cid:12)cantly di(cid:11)erent
from standard sampled-data systems since the delay in the feedback loop
can be highly variable due to both access delay (i.e., the time it takes for a
2 Payam Naghshtabrizi andJoa~o P.Hespanha
shared network to accept data) and transmission delay (i.e., the time during
which data are in transit inside the network). Both types of delays depend
onhighlyvariablenetworkconditionssuchascongestionandchannelquality.
Since access and transmissiondelays havethe same e(cid:11)ect with respect to the
stabilityofNCSs,throughoutthepaperweusetheterm(NCS) delay torefer
to access/transmissiondelay.
Another signi(cid:12)cant di(cid:11)erence between NCSs and standard digital control
is the possibility that data may be lost while in transit through the network.
Typically,packet dropout resultsfromtransmissionerrorsinphysicalnetwork
links (which is far more common in wireless than in wired networks) or from
bu(cid:11)erover(cid:13)owsduetocongestion.Longtransmissiondelayssometimesresult
inout of order delivery, whichcanamounttoapacketdropoutif thereceiver
discards \outdated" arrivals. Reliable transmission protocols, such as TCP,
guarantee the eventual delivery of packets. However, these protocols are not
appropriate for NCSs since the re-transmission of old data is generally not
very useful.
We consider two types of control units: non-anticipative and anticipative.
Anon-anticipative controlunitsendscontrolupdatesattimesta; ‘2N,equal
‘
to fu : 8‘ 2 Ng; where u is a single-value control command to be applied
‘ ‘
to the plant and held until the next control update arrives. An observer-
basedcontroller,constructsestimatex^of the plant’sstatex.Theseestimates
are used to compute the updates u := (cid:0)Kx^(ta) sent through the actuation
‘ ‘
channeltotheactuatorsattimesta,‘2Nsu(cid:11)eringa(possiblyvariable)delay
‘
of (cid:28)a (cid:21)0. In a loss-less network, the control signal u(t) is therefore updated
‘
according to
u(t)=u =(cid:0)Kx^(ta); 8t2[ta+(cid:28)a;ta +(cid:28)a ); ‘2N:
‘ ‘ ‘ ‘ ‘+1 ‘+1
An anticipative controller attempts to compensate the sampling and de-
lay introduced by the actuation channel. For simplicity, we assume that the
actuation channel is sampled with constant sampling interval ha =ta (cid:0)ta,
‘+1 ‘
8‘ 2 N and that its delay is constant and equal to (cid:28)a = (cid:28)a, 8‘ 2 N. At each
‘
sampling time ta = ‘ha, ‘ 2 N the controller sends a time-varying control
‘
signal u ((cid:1)) that should be used from the time ‘ha +(cid:28)a at which it arrives
‘
untilthetime(‘+1)ha+(cid:28)a atwhichthenextcontrolupdatewillarrive.This
leads to
u(t)=u (t); 8t2[‘ha+(cid:28)a;(‘+1)ha+(cid:28)a); ‘2N;
‘
where u (t) is equal to (cid:0)Kx^(t). However, the estimates x^((cid:1)) needed in the
‘
interval [‘ha+(cid:28)a;(‘+1)ha+(cid:28)a) must be available at the transmission time
‘ha,whichrequiresthecontrolunittoestimatetheplant’sstateuptoha+(cid:28)a
time units into the future.
Branicky et al. [1], [15] model NCSs as discrete-time systems. However,
this approach requires the assumption that the total delay in the control
Title SuppressedDueto Excessive Length 3
loop is smaller than the sampling interval, otherwise the analysis becomes
signi(cid:12)cantly more complex [10]. Alternatively NCSs can be modeled as delay
di(cid:11)erentialequations(DDEs)forwhichtheaftermentionedrestrictioncanbe
easilylifted[7].AnNCSwithLTIplantmodel,anticipativeornon-anticipative
controller, with delays, sampling, and packet dropouts can be modeled as a
DDE of the form
2
x(cid:22)_(t)=A0x(cid:22)(t)+ Aix(cid:22) t(cid:0)(cid:28)i(t) ; (cid:28)i(t)2[(cid:28)imin;(cid:28)imax ; (cid:28)_i(t)=1 a:e: (1)
Xi=1 (cid:0) (cid:1) (cid:1)
The delay’s bounds (cid:28)imin and (cid:28)imax for i 2 f1;2g, are positive and depend
on the sampling intervals, maximum number of consecutive packet dropouts,
and upper and lower bounds on the delay in the measurement and the ac-
tuation channels. We (cid:12)nd su(cid:14)cient conditions for the stability of (1), based
on a Lyapunov-Krasovskiifunctional, in the form of matrix inequalities. For
a given controller,the matrix inequalities turn out to be linear (LMIs). How-
ever, in design problems in which the controller parameters also appear as
unknowns, the matrix inequalities are bilinear (BMIs) and non-convex. We
propose a numerical method based on the linear cone complementarity al-
gorithm introduced in [6] to solve this problem. This method converts the
feasibility of the original non-convex matrix inequalities into a sequence of
the convex optimizations of a linear function subject to a set of LMIs, which
can be e(cid:11)ectively solved by numerical packages such as MATLAB.
Related work
ThestabilityofNCSshasreceivedsigni(cid:12)cantattentionintheliterature.Zhang
andBranicky[19]modelNCSswithtime-varyingsamplingintervalsashybrid
systemsand(cid:12)ndsu(cid:14)cientconditionforstability.MontestruqueandAntsaklis
[11] study the stability of model-based NCSs in which the network delays
are negligible and the measurement channel is loss-less and the controller is
directly connected to the plant. The authors use an explicit model of the
plant to produce an estimate of the plant’s state between transmission times
which allows them to reduce the network usage. Lin et al. [9] investigate
the stability of NCSs with uncertain time delays and packet dropouts in the
framework of switched systems. Signi(cid:12)cant work has been devoted to (cid:12)nding
upper bounds on transmission intervals, i.e., tk+1 (cid:0) tk, 8k 2 N for which
stability of the closed-loopis guaranteed.These upper bounds aresometimes
called the maximum allowable transfer interval (MATI). Walsh et al. [16, 17]
(cid:12)nd MATI for linear and nonlinear systems. Nesic and Teel [14] study the
input-outputstabilitypropertiesofgeneralnonlinearNCSsusinganargument
based on small gain theorem. Yu et al. [18] design an observer-type output
feedback controller to stabilize a plant through a network with admissible
bounds on data packet losses and delays, by modeling NCSs as DDEs and
using the Razumikhin theorem to prove the stability of DDEs. Our work
4 Payam Naghshtabrizi andJoa~o P.Hespanha
is inspired by [18], but instead of the Razumikhin theorem, which generally
leadstoconservativedesigns,ouranalysisisbasedonanewdescriptorsystem
approach and the Lyapunov-Krasovskiifunctional proposed by Fridman and
Shaked [4]. We refer the readers to [7] for a more through review of related
literature. The results in this paper generalized those in [13] by considering
variable intervals.
Notations
Thenotationisintroducedasandwhennecessary.Thesetofallnon-negative
integers is denoted by N. Superscripts ’s’ and ’a’ are used to refer to the
(sensor) measurement and the actuator channels and subscripts k and j are
discreteindicestakingvaluesinNusedforthemeasurementandtheactuator
channeleventsrespectively.Forexamplets andta refertothek-thandthej-
k j
thsamplingtimesinthemeasurementandtheactuationchannelsrespectively.
This paper is organized as follows: In section 2 we introduce anticipative
PSfrag replacements
andnon-anticipativecontrolunits.Weshowthattheresultedclosed-loopsys-
tems can be written as (1). In section 3, we provide a su(cid:14)cient condition for
asymptoticstabilityof (1)asasetofLMIs.Insection4anumericalprocedure
x^
is proposed to design a controller that stabilizes the plant for given admis-
sible bound on delays, number of consecutive packet dropouts and sampling
intervals. Then through examples we illustrate the use of our method.
2 Network control systems modeling
ActuationCh. Measurement Ch.
u(t) y(t) tsk
Hold Plant
u^j y(tsk)
Networkwith bounded delayand packetdropouts
u‘ ta‘ x^ y^(t) y^(tsk)
K Observer Hold
u^(t)
Networkmodel
Fig. 1. Two-channel feedbackNCS with observer-basedcontroller.
Fig.1showsanNCSconsistingofaplant,actuators,sensors,andacontrol
unit where the plant, actuators, and sensorsare compound. The plant is LTI
with state space model of the form
x_(t)=Ax(t)+Bu(t); y(t)=Cx(t); (2)
Title SuppressedDueto Excessive Length 5
where x(t) 2 Rn, u(t) 2 Rl, y(t) 2 Rm are the state, input, and output
of the plant respectively. The measurements are sampled and sent at times
ts;k 2N. Assuming, for now, that there are no dropouts, the measurements
k
fy(ts) : k 2 Ng are received by the control unit at times ts +(cid:28)s. These are
k k k
used to construct an estimate of the plant state using
x^_(t)=Ax^(t)+Bu^(t)+L y(ts)(cid:0)Cx^(ts) ; 8t2 ts +(cid:28)s;ts +(cid:28)s ; (3)
k k k k k+1 k+1
(cid:0) (cid:1) (cid:2) (cid:1)
whereu^(t)isanestimateof theplant’sinput attime t.Sinceuisconstructed
from data sent by the controller, in general we have u^ =u. We consider two
types of control units: non-anticipative and anticipative. The two units defer
on the construction of the control signal u.
2.1 Non-anticipative control unit
Control signal The control unit sends control updates at times ta, ‘ 2 N,
‘
equaltof(cid:0)Kx^(ta): ‘2Ng:Inthe absenceofdrops,thesearriveattheplant
‘
at times ta +(cid:28)a , ‘2N, leading to
‘+1 ‘+1
u(t)=(cid:0)Kx^(ta); 8t2 ta+(cid:28)a;ta +(cid:28)a : (4)
‘ ‘ ‘ ‘+1 ‘+1
(cid:2) (cid:1)
Delay di(cid:11)erential equation formulation De(cid:12)ning
(cid:28)s(t):=t(cid:0)ts; 8t2 ts +(cid:28)s;ts +(cid:28)s ;
k k k k+1 k+1
(cid:28)a(t):=t(cid:0)ta; 8t2(cid:2)ta+(cid:28)a;ta +(cid:28)a (cid:1);
‘ ‘ ‘ ‘+1 ‘+1
(cid:2) (cid:1)
which we call them (cid:12)ctitious delays(as opposed to the actual networkdelays
(cid:28)s and (cid:28)a), we can re-write (3) and (4) as 3
k j
x^_(t)=Ax^(t)+Bu^(t)+L y(t(cid:0)(cid:28)s)(cid:0)Cx(t(cid:0)(cid:28)s) ; (5)
u(t)=(cid:0)Kx^(t(cid:0)(cid:28)a): (cid:0) (cid:1)
Note that
(cid:28)s 2 minf(cid:28)sg;maxfts (cid:0)ts +(cid:28)s g ; 8k 2N ;(cid:28)_s =1 a.e.;
k k+1 k k+1
k k
(cid:2) (cid:1)
(cid:28)a 2 minf(cid:28)ag;maxfta (cid:0)ta+(cid:28)a g ; 8k 2N ;(cid:28)_a =1 a.e.;
‘ ‘+1 ‘ ‘+1
‘ ‘
(cid:2) (cid:1)
Fig.2.ashowsthe evolution of (cid:28)s with respecttotime where(cid:28)s =(cid:28)(cid:22)s;8k 2N
k
and constant sampling interval ts (cid:0)ts = hs;8k 2 N. The derivative of
k+1 k
(cid:28)s is almost always one, except at the times that new packets arrive to the
controller side at ts +(cid:28)(cid:22)s, where (cid:28)s (the (cid:12)ctitious delay) drops to (cid:28)(cid:22)s (the
k+1
actual network delay).
3 Forsimplicityof notationthroughthepaperweuse(cid:28)s and(cid:28)a as(cid:28)s(t)and(cid:28)a(t)
respectively.
6 Payam Naghshtabrizi andJoa~o P.Hespanha
(cid:28)s (cid:28)s
2hs+(cid:28)(cid:22)s
PSfragreplacements
hs+(cid:28)(cid:22)s hs+(cid:28)(cid:22)s
(cid:28)(cid:22)s (cid:28)(cid:22)s
tsk(cid:0)1+(cid:28)(cid:22)s tsk+(cid:28)(cid:22)s tsk+1+(cid:28)(cid:22)s t tsk(cid:0)1+(cid:28)(cid:22)s tsk+(cid:28)(cid:22)s tsk+1+(cid:28)(cid:22)s tsk+2+(cid:28)(cid:22)s t
(a) (b)
Fig. 2. Evolution of (cid:28)s with respect to time when (a) There is no packetdropout,
(b)The packetsentat ts is dropped.
k
Packet dropouts Packetdropoutsinthemeasurementandactuationchan-
nels can be viewed as the delay that grow beyond the de(cid:12)ned bounds. If ms
and ma consecutive packet dropouts happen in the measurement and actua-
tion channels then
(cid:28)s 2 minf(cid:28)sg;maxfts (cid:0)ts +(cid:28)s g ;
k k+ms+1 k k+ms+1
k k
(cid:2) (cid:1)
(cid:28)a 2 minf(cid:28)ag;maxfta (cid:0)ta+(cid:28)a g :
‘ ‘+ma+1 ‘ ‘+ma+1
‘ ‘
(cid:2) (cid:1)
Fig. 2.b shows the situation that the measurement packet sent at t = ts is
k
dropped and as a result (cid:28)s grows up to 2hs+(cid:28)s.
Closed-loop De(cid:12)ninge:=x(cid:0)x^,withregardto(2)and(5),theclosed-loop
can be written as
x_(t) A 0 x(t) 0 0 x(t(cid:0)(cid:28)s) (cid:0)BK BK x(t(cid:0)(cid:28)a)
= + +
(cid:20)e_(t)(cid:21) (cid:20)0 A(cid:21)(cid:20)e(t)(cid:21) (cid:20)0(cid:0)LC(cid:21)(cid:20)e(t(cid:0)(cid:28)s)(cid:21) (cid:20) 0 0 (cid:21)(cid:20)e(t(cid:0)(cid:28)a)(cid:21)
or alternatively, as
x^_(t) A 0 x^(t) 0 LC x^(t(cid:0)(cid:28)s) (cid:0)BK 0 x^(t(cid:0)(cid:28)a)
= + + :
(cid:20)e_(t)(cid:21) (cid:20)0 A(cid:21)(cid:20)e(t)(cid:21) (cid:20)0(cid:0)LC(cid:21)(cid:20)e(t(cid:0)(cid:28)s)(cid:21) (cid:20) 0 0(cid:21)(cid:20)e(t(cid:0)(cid:28)a)(cid:21)
2.2 Anticipative control unit
Control signal For simplicity, we assume that the actuation channel is
sampled with constant sampling interval ha = ta (cid:0)ta, 8‘ 2 N and that
‘+1 ‘
its delay is constant and equal to (cid:28)(cid:22)a = (cid:28)a, 8‘ 2 N. At each sampling time
‘
ta = ‘ha, ‘ 2 N the controller sends a time-varying control signal u ((cid:1)) that
‘ ‘
should be used from the time ‘ha + (cid:28)(cid:22)a at which it arrives until the time
(‘+1)ha+(cid:28)(cid:22)a at which the next control update will arrive. This leads to
u(t)=u (t); 8t2[‘ha+(cid:28)(cid:22)a;(‘+1)ha+(cid:28)(cid:22)a); ‘2N: (6)
‘
Tostabilize(2),u (t)shouldbeequalto(cid:0)Kx^(t),wherex^(t)isanestimateof
‘
x(t).However,theestimatesx^((cid:1))neededintheinterval[‘ha+(cid:28)(cid:22)a;(‘+1)ha+(cid:28)(cid:22)a)
mustbeavailableatthetransmissiontime‘ha,whichrequiresthecontrolunit
to estimate the plant’s state up to ha+(cid:28)(cid:22)a time units into the future.
Title SuppressedDueto Excessive Length 7
Remark 1. Anticipative controllers send actuation signals to be used during
time intervals of duration ha, therefore the sample and hold blocks in Fig. 1
should be understood in a broad sense. In practice, the sample block would
send over the network some parametric form of the control signal u ((cid:1)) (e.g.,
‘
the coe(cid:14)cients of a polynomial approximation to this signal).
State predictor Anestimatez(t)ofx(t+ha+(cid:28)(cid:22)a)isconstructedasfollows:
z_(t)=Az(t)+Bu^(t+ha+(cid:28)(cid:22)a)+L y^(ts)(cid:0)Cz(ts (cid:0)ha(cid:0)(cid:28)(cid:22)a) ; (7)
k k
(cid:0) (cid:1)
for 8t 2 [ts +(cid:28)s;ts +(cid:28)s ); k 2 N. To compensate for the time varying
k k k+1 k+1
delayandpacketdropoutsinthe actuationchannel,z wouldhavetoestimate
xfurtherintothefuture.Hencetheassumptionsofconstantdelayandloss-less
actuation channel can be relaxed by predicting x further into the future.
Controlsignalconstruction Withsuchestimateavailable,thesignalu (t)
‘
sent at times ta =‘ha, to be used in ‘ha+(cid:28)(cid:22)a;(‘+1)ha+(cid:28)(cid:22)a , is then given
‘
by (cid:2) (cid:1)
u (t)=(cid:0)Kz(t(cid:0)ha(cid:0)(cid:28)(cid:22)a); 8t2[‘ha+(cid:28)(cid:22)a;(‘+1)ha+(cid:28)(cid:22)a); ‘2N; (8)
‘
which only requires the knowledge of z(:) in the interval t2 (‘(cid:0)1)ha;‘ha ,
and therefore is available at the transmission time ‘ha. (cid:2) (cid:1)
Delay di(cid:11)erential equation formulation De(cid:12)ning
(cid:28)s(t):=t(cid:0)ts; 8t2 ts +(cid:28)s;ts +(cid:28)s ;
k k k k+1 k+1
(cid:2) (cid:1)
and assuming that u^=u, we conclude from (6),(7), and (8) that
z_(t)=(A(cid:0)BK)z(t)+L y(t(cid:0)(cid:28)s)(cid:0)Cz(t(cid:0)ha(cid:0)(cid:28)(cid:22)a(cid:0)(cid:28)s) ; (9)
(cid:28)s(t)2 minf(cid:28)sg;maxfts (cid:0) (cid:0)ts +(cid:28)s g ; 8k 2N; (cid:28)_s =(cid:1)1 a.e.
k k+1 k k+1
k k
(cid:2) (cid:1)
Closed-loop De(cid:12)ning e(t):=x(t+ha+(cid:28)(cid:22)a)(cid:0)z(t) and considering (2),(8),
and (9), the closed-loop can be written as
z_(t) A(cid:0)BK 0 z(t) 0 LC z(t(cid:0)ha(cid:0)(cid:28)(cid:22)a(cid:0)(cid:28)s)
= + : (10)
(cid:20)e_(t)(cid:21) (cid:20) 0 A(cid:21)(cid:20)e(t)(cid:21) (cid:20)0(cid:0)LC(cid:21)(cid:20)e(t(cid:0)ha(cid:0)(cid:28)(cid:22)a(cid:0)(cid:28)s)(cid:21)
3 Stability of delay di(cid:11)erential equations
Itwasshowninsection2thatbothtypesofcontrolunitsresultinthe closed-
loop models of the following form
2
x(cid:22)_(t)=A0x(cid:22)(t)+ Aix(cid:22) t(cid:0)(cid:28)i(t) ; (cid:28)i(t)2[(cid:28)imin;(cid:28)imax ; (cid:28)_i(t)=1 a:e: (11)
Xi=1 (cid:0) (cid:1) (cid:1)
8 Payam Naghshtabrizi andJoa~o P.Hespanha
Untilrecentlyitwasbelievedthattheonlyavailabletooltostudythestability
of delay equations of the form (11) was the Razumikhin theorem. Fridman
and Shaked [5] were able to use the Lyapunov-Krasovskii theorem to study
the stability of system (11). In [3] they studied the stability of sampled-data
systems with input delays as DDEs of the form (11) where (cid:28)1 2 [0;hs). In
sampled-data systems, at each sampling time delay drops to zero. However
in NCSs as new information arrives the (cid:12)ctitious delay drops to the actual
networkdelay.Thenexttheoremgivesasu(cid:14)cientconditionfortheasymptotic
stability of (11) where the matrices A is 2n(cid:2)2n for i2f0;1;2g.
i
Theorem 1.The system (11) is asymptotically stable, if there exist 2n(cid:2)2n
matricesP1 > 0,P2,P3,Si,Ri >0and4n(cid:2)4nmatricesZ1i,Z2i and2n(cid:2)4n
matrices T ;i=1;2, that satisfy the following set of matrix inequalities 4:
i
0 0
(cid:9) P0 (cid:0)T0 P0 (cid:0)T0
2 (cid:20)A1(cid:21) 1 (cid:20)A2(cid:21) 23
<0; (12a)
(cid:3) (cid:0)S1 0
6 7
6(cid:3) (cid:3) (cid:0)S2 7
4 5
R 0A0 P
i i >0; 8i2f1;2g; (12b)
(cid:20)(cid:3) (cid:2) Z2i(cid:3) (cid:21)
R T
i i >0; 8i2f1;2g; (12c)
(cid:20)(cid:3) Z1i(cid:21)
where
0
P := P1 0 ; (cid:9) :=P0 0 I + 0 I P +(cid:8) (13)
(cid:20)P2 P3(cid:21) (cid:20)A0 (cid:0)I(cid:21) (cid:20)A0 (cid:0)I(cid:21)
2 0
S 0 T T
(cid:8):=Xi=1(cid:16)(cid:20)0i (cid:28)imaxRi(cid:21)+(cid:28)iminZ1i+((cid:28)imax(cid:0)(cid:28)imin)Z2i+(cid:20)0i(cid:21)+(cid:20)0i(cid:21) (cid:17):
Proof of Theorem 1 See Appendix. (cid:4)
Remark 2. The\triangular"structure of (10) is unique tothe anticipatecon-
troller.Withthistypeofcontroller,ifwechooseK sothatA(cid:0)BK isHurwitz,
asymptotic stability of the NCS is equivalent to the asymptotic stability of
the (decoupled) dynamics of the error e(t) :=x(t+ha+(cid:28)(cid:22)a)(cid:0)z(t), which is
given by the following DDE
e_(t)=Ae(t)(cid:0)LCe t(cid:0)(cid:28)(t) ; t(cid:21)0;
(cid:0) (cid:1)
with (cid:28) 2[(cid:28)min;(cid:28)max), (cid:28)_ =1, a.e., where
(cid:28)min :=mk2iNnf(cid:28)ks+ha+(cid:28)(cid:22)ag; (cid:28)max :=mk2aNxftsk+1+ms (cid:0)tsk+(cid:28)ks+1+ms +ha+(cid:28)(cid:22)ag;
and Theorem 1 provides the stability test with A0 =A and A1 =(cid:0)LC.
4 Matrixentriesdenotedby’*’areimplicitlyde(cid:12)nedbythefactthatthematrixis
symmetric.
Title SuppressedDueto Excessive Length 9
Remark 3. Suppose that the matrices of the system (cid:10) := A0 A1 A2 are not
exactlyknownandthat insteadthey aregivenaspolytopi(cid:2)c uncertain(cid:3)tygiven
by
N N
(cid:10) 2 f (cid:10) ; 0(cid:20)f (cid:20)1; f =1 ;
j j j j
nXj=1 Xj=1 o
where the N vertices of the polytope are described by (cid:10) := Aj Aj Aj .
j 0 1 2
Stability of the system can be checked by solving the LMIs in Th(cid:2)eorem 1 fo(cid:3)r
each of the individual vertices and the same matrix variables.
4 Observer-based controller design for NCSs
When the controllerparametersL and K are known, the system matrices A
i
for i 2 f0;1;2g, are constant and known. Hence (12a)-(12c) are in the form
of LMIs and the stability of NCSs can be established easily.Howeverwhen L
andK arealsovariablestobede(cid:12)ned,sinceA containunknowns,thematrix
i
inequalities in Theorem 1 are in the form of BMIs. The next lemma taken
from [2], plays a central role in transforming BMIs to a suitable form for a
numerical algorithm we will develop.
Lemma 1.Assume that Q is a symmetric matrix. There exists a symmetric
matrix N >0 such that
J0NU +U0NJ +Q<0; (14)
if and only if there exist symmetric matrices X, Y and a scalar (cid:11) > 0 such
that X = (cid:11)2Y(cid:0)1 and
U0XU (cid:0)QJ0+(cid:11)U0
>0: (15)
(cid:20) (cid:3) Y (cid:21)
The proof is obtained by SchurLemma and the equation X =(cid:11)N. Lemma 1
changesthematrixinequality(14),with productsofunknownsJ andN,into
the inequality (15) in which suchproducts areabsent. However,(15)requires
the non-convex constraint X = (cid:11)2Y(cid:0)1. Our goal now is to rewrite the
matrix inequalities in Theorem 1 in the form of (14) and ultimately in the
formof(15).Itturnsoutthatthisformissuitablefordevelopinganumerical
procedure to compute K, L, and the other matrix variables in Theorem 1.
SupposethatP2 >0,P3 >0,aftermatrixmanipulations(12a)canbewritten
as J0(K;L)0NU0+U00NJ0(K;L)+Q0 <0; where
(cid:0) (cid:0)T0 (cid:0)T0
Q0 =2(cid:3)(cid:3) (cid:0)(cid:3)S11 (cid:0)0S223; J0(K;L)=(cid:20)AA00 (cid:0)(cid:0)II AA11 AA22(cid:21); U0 =(cid:20)I0 I0 0000(cid:21);
4 5
P2 0 0 P1
N = >0; (cid:0) = +(cid:8):
(cid:20) 0 P3(cid:21) (cid:20)P1 0 (cid:21)
10 Payam Naghshtabrizi andJoa~o P.Hespanha
Thematrixinequality(12b)canbewrittenasJ (K;L)0NU +U0NJ (K;L)+
i i i i
Q <0; where
i
R 0 A 00 0I 0
Q =(cid:0) i ; J (K;L)=(cid:0) i ; U = ;
i (cid:20)0 Z2i(cid:21) i (cid:20)Ai 00(cid:21) i (cid:20)00 I(cid:21)
for i2f1;2g. Theorem 1 can be read as the following theorem:
Theorem 2.Theanticipativeornon-anticipativecontrollergiveninsection2
with the matrix variables K and L asymptotically stabilizes the plant (2) for
given (cid:28)imin and(cid:28)imax,ifthereexist2n(cid:2)2nmatricesP1 >0,X1 >0,X2 >0,
Y1, Y2, Si, Ri > 0, 4n(cid:2)4n Z1i, Z2i, 2n(cid:2)4n matrices Ti, n(cid:2)1 matrix L,
1(cid:2)n matrix K and (cid:11) > 0 that satisfy the following matrix inequalities for
i2f1;2g:
U00XU0(cid:0)Q0 J00 +(cid:11)U00 >0; Ui0XUi(cid:0)Qi Ji0+(cid:11)Ui0 >0; Ri Ti >0;
(cid:20) (cid:3) Y (cid:21) (cid:20) (cid:3) Y (cid:21) (cid:20) (cid:3) Z1i(cid:21)
(16)
where
X =(cid:11)2Y(cid:0)1; X = X1 0 ; Y = Y1 0 :
(cid:20) 0 X2(cid:21) (cid:20)0 Y2(cid:21)
Since A0, A1, A2 are linear functions of K and L, the matrix inequalities in
(16) are in the form of LMIs also in the unknowns. However, the condition
X = (cid:11)2Y(cid:0)1 is not convex. In the next section we introduce a numerical
procedure to solve this non-convex problem.
Remark 4. A simple but conservative way to make the matrix inequalities in
Theorem 1 suitable for controller synthesis for anticipative controllers with
regard to Remark 2, consists of requiring that P2 > 0; P3 = (cid:26)P2; for
some positive constant (cid:26) > 0 and making the (bijective) change of variables
Y =P2L, which transforms (12) into
YC
(cid:9) (cid:0) (cid:0)T0 R (cid:0) C0Y0 (cid:26)C0Y0 R T
2 (cid:20)(cid:26)YC(cid:21) 3<0; >0; >0;
(cid:3) (cid:0)S (cid:20)(cid:3) (cid:2) Z2 (cid:3)(cid:21) (cid:20)(cid:3) Z1(cid:21)
4 5
with (cid:9) given by (13). This inequality is linear in the unknowns and can
therefore be solved using e(cid:14)cient numerical algorithms. The observer gain
is found using L = P(cid:0)1Y. This procedure introduces some conservativeness
2
because it restricts P3 to be a scalar multiple of P2.
Numerical procedure
We modify the complementarity linearization algorithm introduced in [6] to
construct a procedure to determine the feasibility of the matrix inequalities
in Theorem 2 by solving a sequence of LMI’s as follows: