Automatica 42 (2006) 137–144www.elsevier.com/locate/automatica

Brief paper

Packet-based control: The H2-optimal solution�

Daniel Georgiev∗, Dawn M. TilburyDepartment of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109-2125, USA

Received 23 November 2004; received in revised form 28 April 2005; accepted 21 August 2005Available online 7 November 2005

Abstract

This paper presents a new control method for networked control systems. Motivated by a more efficient use of the packet structure, multipointpackets are used to reduce network traffic and computation time. The control problem for the multipoint-packet system is shown to be equivalentto a multirate control problem, which in turn is reduced to a synthesis problem with a constraint on the feedthrough matrix. This problem issolved in general using H2 optimization techniques and the solution to the general problem is applied to the multipoint-packet control problem.The paper concludes with an example that compares the performance of the proposed method to more traditional ones.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Networked control systems; H2 optimal control; Multirate control; Causality constraint

1. Introduction and background

The use of networks is becoming ubiquitous in control sys-tems. Controlling a system over a network is conducive to theimplementation of large systems as well as modularity and re-configurability. There are several challenges, however, that arisewhen a control system is networked. These challenges can beattributed to either the sharing of the communication mediumor the extra complexity associated with data transmission. Aspointed out by Lian, Moyne, and Tilbury (2002), networkedcontrol systems (NCS) are fundamentally different from digitalcontrol systems. Performance of digital control systems asymp-totically approaches the continuous time performance level asthe sampling period goes to zero. This is not the case withnetworked control systems. In networked control systems, dis-crete signals are encoded into a packet, sent across the network,

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor TongwenChen under the direction of Editor Ian Petersen. This material is based uponwork supported under a National Science Foundation Graduate ResearchFellowship and in part by the University of Michigan ERC/RMS under grantNSF 95-29125 and NSF 98-76039.

∗ Corresponding author. Tel.: +1 734 904 1341.E-mail addresses: dgeorgie@umich.edu (D. Georgiev),

tilbury@umich.edu (D.M. Tilbury).

0005-1098/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2005.08.011

and then decoded at the destination node. As the sampling fre-quency increases so does the network congestion and hence thenetwork induced delays and possibly packet losses. As a result,a tradeoff arises between the performance gain associated withincrease in transmission period (as in digital control systems)and the performance degradation associated with increase innetwork delays. The size and distributions of the delays aris-ing in NCS were studied by Lian, Moyne, and Tilbury (2003).It was found that for network protocols such as DeviceNet,ControlNet, and Ethernet, both computation and traffic relateddelays are significant. Therefore, NCS design methods need toconsider all components of the system as well as the controlobjectives.

The unique characteristics of NCS have opened up a wholenew area of control systems design, see Tipsuwan and Chow(2003), Antsaklis and Baillieul (2004), Chow (2004) for recentsurveys. Design methods for NCS can be classified into twobroad categories. First, one can design the controller withoutconsidering the network and manipulate the size and trans-mission instances of the data to optimize closed loop systemperformance (Lian, Moyne, & Tilbury, 2001). One way to dothis is to send fewer but more informative packets (Rehbinder& Sanfridson, 2000; Walsh & Hong, 2001). All packets carrysome overhead; therefore, reducing the number of transmis-sions while increasing the information per transmission canincrease the rate of information transfer per byte sent. For

138 D. Georgiev, D.M. Tilbury / Automatica 42 (2006) 137–144

Pad ChecksumDataLengthPreamble Start of

DelimiterDestination

AddressSource

Address Data

Overhead = 26 Bytes Data = 46 – 1500 Bytes

Fig. 1. Typical Ethernet packet (Lian et al., 2001).

example the minimum size of an Ethernet packet is 72 bytes,see Fig. 1, while a typical data point will only consume 2 bytes.The second approach to NCS design is to treat the networkprotocol and traffic as given parameters and design a controllerrobust to the communication disturbances such as delays andpacket loss. Many control methods designed to directly dealwith communication delays have been derived for this purpose,for example see Nilsson and Bernhardsson (1997).

The objective of this paper is neither to derive an optimalcontroller in the presence of time delays nor to derive opti-mal scheduling of packet transmission, but rather to exploit thepacket structure of network communication for improved NCSperformance without introducing additional network traffic. Anovel communication protocol is defined and the associatedoptimal control problem for this system is then solved in theH2 framework which facilitates the combination of this workwith some of the preexisting H2-optimal control techniques.The outline of the paper is as follows: Section 2 explicitly de-fines all critical components of network communication thatwill be used throughout the paper, Section 3 briefly reviewssome of the results used in the subsequent derivations, Section4 includes the problem statement and the packet-based controlsolution, Section 5 contains numeric performance analysis, andfinally Section 6 summarizes the results and proposes some fu-ture research directions.

2. Network structure

The contribution of this paper is the derivation of an optimalcontroller for a novel communication protocol. The novelty ofthe protocol lies in the way the signals are transmitted. Beforeformally defining the packet-based protocol, recall some net-work communication basics. All data travels across a networkin packets. A packet is first encoded at the source node, it thenawaits its turn to be transmitted to the destination mode, whereit is finally decoded. The network can become saturated in twoways. First, the network itself has a certain capacity. Second,the encoders and decoders required by all networked deviceshave a finite capacity. As mentioned above, all packets carryoverhead. Therefore, the rate of information transfer of a con-trol network depends on the number of data points per packet,assuming that packets are appropriately coordinated. One wayto minimize overhead is to group multiple data points into apacket by collocating sensors and/or actuators. However, thismay not be a desirable solution since it constraints the system.In this paper we propose a different solution whereby multipledata points from the same sensor or actuator are put togetherinto a packet.

transmission

instance

packet

Time0TS 2TS 3TS 4TS 5TS

6TS 7TS 8TS 9TS 10TS

kth control packet& u (k)

y (k)

Fig. 2. Packet-based communication protocol.

In NCS it typically is the case that the capacity of the NCS isnot limited by the A/D or D/A hardware but by the network andits associated components. Therefore, in the most general NCSwhere the actuators, sensors, and controller are all located atdifferent points on the network, samples are collected and thecontrol inputs can be updated at a rate faster then the networkcan service them. Different methods of handling the rate dis-parity exist. For some examples see the Maximum-Error-First-Try-Once-Discard (MEF-TOD) solution in Walsh and Hong(2001), or the model based control approach of Montestruqueand Antsaklis (2003). These methods provide a way of deter-mining which of the sample points should be transmitted whenthere is insufficient network capacity for all the samples. Inthis paper, the problem of choosing one representative datapoint is circumvented through the use of a network protocol inwhich many data points are transmitted for approximately thesame “cost” (network capacity) as a single point. The networkprotocol is as follows. Multiple data points—consisting of allsample points acquired since last transmission instance—aregrouped with the most recent point and transmitted together inone packet, called the output packet. Furthermore, the controlinput over one transmission period is partitioned to match theA/D period and an array of control input values correspond-ing to each subinterval is transmitted in each packet sent fromthe controller to the actuator. A packet sent from the controllerto the actuator is referred to as the control packet. Fig. 2 il-lustrates an example where the packet transmission period isfive times the A/D and D/A periods. It is not obvious howmuch performance can be gained by using this communica-tion protocol. First, the increase in packet size with respectto additional data points has to be determined. This will bestrongly dependent on the particular protocol. For example, anEthernet packet can hold up to 23 samples, of 2 bytes each,at no extra cost to the encoder/decoder hardware or the net-work. The second factor to consider is how much performancecan be gained by transmitting the old samples and the multi-ple control input values. The second question is addressed inSection 5.

D. Georgiev, D.M. Tilbury / Automatica 42 (2006) 137–144 139

The network and the associated hardware are assumedto satisfy the following assumptions. First, the networkchannel is idealized as a single-link error-free transmis-sion medium with no transmission overlap errors. Second,it is assumed that the absolute network delay (transmissiondelay + propagation delay + encoding/decoding delay . . .), isnegligible with respect to the sampling time. Third, the A/Dand D/A operation periods (hA/D and hD/A) are assumed to beconstant and equal to one sampling period hs. Fourth, all out-put measurements and control inputs are assumed to arrive atthe controller and actuators simultaneously. Finally, the packettransmission period hp is also assumed to be constant and equalto an integer multiple of hs, i.e. hp = Nhs. These assumptionsare highly idealized and would not be warranted for most NCS.However, the concern here is the derivation of a controller forthe proposed communication protocol. All other NCS issuescan be considered once this foundation has been established.

The network model can be formalized as follows. Let I ={0, 1, . . . , N − 1} represent the set of data points in a packetand designate the position of a given data point by a packetindex, i ∈ I, where the first data point (oldest output sampleor first control input) will be given a value of 0. Due to theassumptions made, the only manifestation of the network is thepacket transmission period, assuming N > 1. The transmissionperiod introduces an additional causality constraint defined asfollows. Let {u} and {y} be the control input and measuredoutput sequences, respectively, and denote the discrete-timeindex by k. Let i ∈ I be the packet index, and N be thenumber of data points per packet. Define � as the discrete-timetruncation operator where �Nr+i is a truncation of the sequenceat the Nr + i value, for r ∈ N. Finally, note the transmissionperiod introduces a periodic delay that can be represented usinga periodic delay operator, � : I × Rp2 → Rp2 defined as�(i, y(k))=y(k − (N −1)+ i). In other words, the first (i =0)entry in a packet is received at the controller with a delay ofN − 1 sample periods and the last (i =N − 1) entry is receivedwithout delay.

As a result of the above network model, a packet-based con-troller Kpac is defined as a controller that is consistent withthe delay characteristic of �. The constraint on the controllerimposed by � is formalized in the following definition.

Definition 1. Packet signal transmission introduces a packetcausality constraint on the controller which can be written as

�Nr+i{u} = �Nr+iKpac{y} = Kpac�Nr{y}. (1)

In other words, the (Nr + i)th control input can onlydepend on output samples up to (Nr), because the samplesNr + 1, Nr + 2, . . . , Nr + i would not be received until the(r + 1)th packet arrives.

Thus, the problem addressed in this paper is the derivation ofan H2-optimal controller that satisfies the packet causality con-straint. The following approach is taken. First, using standardtechniques for periodic systems, a time-invariant plant modelwith vector valued control input u and vector valued output y

is obtained. Fig. 2 illustrates the composition of one element of

these signals. Then, a corresponding time-invariant representa-tion of the packet-based controller is derived from this model,which turns out to be equivalent to designing a controller thatis strictly causal with respect to the last N − 1 components ofeach element of y.

3. Related work

The solution to the packet-based control problem combinesresults from periodically time varying control and H2-optimalcontrol. In this section a brief review of the work central to thedevelopment is provided. For details the reader is referred tothe cited work.

The communication protocol introduced above makes thesystem periodic, where the period is the transmission period ofthe network. Lifting allows a compact representation of a peri-odic system (Chen & Francis, 1995). The lifting operator hasthe following properties that make it suitable for controller syn-thesis: lifting preserves the system norm up to a multiplicativeconstant; lifting also preserves stability, controllability, observ-ability, invariance-zero structure, and invertibility; and solutionsto Riccati equations are invariant under lifting (Wimmer, 1992).

One downfall with using lifting methods is that the dimen-sion of the signals is increased. Optimal control techniquesare a common tool used to synthesize controllers for plantshaving multiple inputs and outputs. In this work, we use H2-optimization to synthesize the controller. The H2-optimizationproblem has been solved in many different ways. Methods fromSaberi, Sannuti, and Chen (1995) are used herein. In Saberiet al. (1995), the discrete-time H2-optimization problem is de-rived by first obtaining a lower bound on the H2 norm of theclosed-loop system that is a function of the feedthrough matrixonly. This lower bound is then minimized and an expressionfor the feedthrough matrix is found. It is then shown that thereexists a controller that attains this lower bound and the realiza-tion of this controller is given as a function of the feedthroughmatrix.

4. Packet-based control

In this section the packet-based control problem is definedand solved. The development proceeds as follows. First, an H2-optimal controller with a general constraint on its feedthroughmatrix is derived. Then, the solution to the general problem isused to derive the time-invariant representation of the packet-based controller.

4.1. H2-optimal control with a generalized causality constraint

In order to solve the H2-optimal packet-based control prob-lem, a more general H2-optimal controller, constrained by gen-eralized causality constraint (GCC), is first derived. A controlleris defined to be causal if the current control input is dependenton past measured outputs including the current output, i.e. u(k)

is allowed to depend on {y(l) : l�k}. A controller is definedas being strictly causal if it is only dependent on past measuredoutputs but not on the current output, i.e. u(k) is dependent on

140 D. Georgiev, D.M. Tilbury / Automatica 42 (2006) 137–144

G

K

u

wz

y

Fig. 3. Generalized discrete-time control system.

{y(l) : l < k}. This leads to the following definition of general-ized causality constraint and a generally causal controller.

Definition 2. Let y(k) ∈ Rp2 be the measured output (inputto the controller) and u(k) ∈ Rm2 be the corresponding con-trol input (output of the controller). A constraint on the depen-dence of ui(k) (ith component of u(k)) on yj (k) (jth compo-nent of y(k)) for all i and j is said to be a generalized causal-ity constraint (GCC) on the controller. A controller satisfyingthe generalized causality constraint will be referred to as beinggenerally causal.

To the best of the authors’ knowledge, the problem of find-ing an H2-optimal controller satisfying an arbitrary generalizedcausality constraint has not been solved. Different versions ofthis problem have been addressed in literature, for example seeQiu and Chen (1994), Shu and Chen (1995), Voulgaris, Dahleh,and Valavani (1991), Fujioka and Ito (2003), and Lu, Xie, andFu (2003). The most common version arises in the design ofmultirate control systems, which, if solved via lifting, inher-ently possess a causality constraint. This constraint requires thefeedthrough matrix to be lower-block triangular and is, there-fore, not general. There is a whole class of problems possessinga constraint on the feedthrough matrix that cannot be reducedto a lower block triangular constraint. The packet-based con-trol problem is one such case. Another example is a distributedsystem where the actuators are collocated with the sensors.For such a system, each actuator may have immediate accessto the local measurement but only delayed access to the othermeasurements. This imposes a block diagonal constraint on thefeedthrough matrix, which cannot be reduced to only a lowerblock triangular constraint. Therefore, the analytic results ob-tained below have a much wider applicability than packet-basedcontrol.

The H2-optimal control problem with a GCC is now solved.Consider the discrete-time system in Fig. 3. Let G be a linear,discrete-time, time-invariant plant represented by

(2)

and K be a linear, discrete-time, time-invariant controller withthe following representation:

(3)

Denote the closed loop system by T = F(G, K), where F(·, ·)is the lower linear fractional transformation. The discrete-timesignals {w}, {z}, {u}, {y}, where w(k) ∈ Rm1 , z(k) ∈ Rp1 ,u(k) ∈ Rm2 , and y(k) ∈ Rp2 , represent the plant disturbance,controlled output, control input, and measured output, respec-tively. It is assumed that the feedback loop is not ill-posed withrespect to the matrix D22 and that G satisfies the followingstandard assumptions, see (Chen & Francis, 1995):

(i) (A, B2) is stabilizable and (C2, A) is detectable.(ii) Matrices D12 and D21 are injective and surjective, respec-

tively.(iii) The following matrices have, respectively, full column and

row rank for all � on the unit circle:[A − �I B2

C1 D12

],

[A − �I B1

C2 D21

].

A GCC on a controller is equivalent to a null constraint onindividual elements of the controller feedthrough matrix DK .Because an expression for the GCC in Rm2×p2 would requirea column by column projection, instead we work with the vec-tor representation of DK . This can be done with the introduc-tion of two operators, the stack operator and the matrix Kro-necker product. The stack operator S associates DK ∈ Rm2×p2

with DSK ∈ Rm2p2 in a natural way, DKij —the element in

the ith row and j th column—translates to DKSi+(j−1)m2

, i.e.the columns stacked. The inverse stack operation will onlybe used when its meaning is unambiguous from the context.For definition and properties of the matrix Kronecker product⊗ : (Rl×k, Rr×s) → Rlr×ks the reader is referred to Graham(1981). Using the stack operator, a compact representation ofa GCC can be formed; ui(k) is independent of yj (k) if andonly if DK

Si+(j−1)m2

is equal to zero. This simple relation is re-stated using an orthogonal projection operator. Let D ⊂ Rm2p2

represent the allowable subspace for DSK based on the general-

ized causality constraint and PD be the associated orthogonalprojection operator. Then DS

K satisfies the GCC if and only if

DSK = PDDS

K . (4)

The H2-optimal control problem with a GCC can now be de-fined as the problem of finding a proper controller Kgc where

Kgc = arg minK

{‖F(G, K)‖2} (5)

and such that (4) is satisfied.Next, define the following matrices:

DTx Dx = DT

12D12 + BT2 XB2,

DTx Cx = DT

12C1 + BT2 XA,

DyDTy = D21D

T21 + C2YCT

2 ,

ByDTy = B1D

T21 + AYCT

2 ,

Q = −(DTx Dx)

−1DTx Cx ,

P = −ByDTy (DyD

Ty )−1, (6)

D. Georgiev, D.M. Tilbury / Automatica 42 (2006) 137–144 141

where X and Y are the positive-semidefinite stabilizing solut-ions to

X = ATXA − CTx Dx(D

Tx Dx)

−1DTx Cx + CT

1 C1,

Y = AYAT − ByDTy (DyD

Ty )−1DyB

Ty + B1B

T1 , (7)

which exist by the assumptions made about G.The following Lemma is essential to the solution to the prob-

lem.

Lemma 3. Consider the plant in (2), and the associated as-sumptions. Then a lower bound on the discrete-time H2 normof the closed loop system � = ‖T ‖2 can be written as

�2 � trace[BT1 XB1] + trace[DT

11D11]+ trace[(ATXA − X + CT

1 C1)Y ] + �(DK), (8)

where

�(DK) = 2(RS)TDSK + [(DT

y ⊗ Dx)DSK ]T

× [(DTy ⊗ Dx)D

SK ],

R = DTx CxYCT

2 + BT2 XB1D

T21 + DT

12D11DT21 (9)

and the unique DK that minimizes (8) under the constraint of(4) is given by

DgcK = −[(PD(DyD

Ty ⊗ DT

x Dx)PD)†RS]−S, (10)

where (·)† is the generalized inverse.

Proof. From Saberi et al. (1995, Lemma 6.5.4, Theorem 6.5.2)it is known that for a plant with D11 = 0 and D22 = 0 a lowerbound on the H2 norm can be written as

�20 � trace[BT

1 XB1] + trace[(ATXA − X + CT1 C1)Y ]

+ 2 trace[DT21D

TKBT

2 XB1] + 2 trace[DxDKC2YCTx ]

+ trace[(DxDKDy)(DxDKDy)T].

Eq. (8) follows immediately from the definition of the discrete-time H2 norm when D11 �= 0 and D22 �= 0. Next, using the

properties of the generalized inverse ()†

and (4), it can be shownthat by completing the square in (9), �(DK) can be rewritten as

�(DK) = [RS∗ + (DTy ⊗ Dx)D

SK ]T

× [RS∗ + (DTy ⊗ Dx)D

SK ] − (RS∗ )TRS∗ ,

where

RS∗ = [(DTy ⊗ Dx)PD][(DT

y ⊗ Dx)PD]†

× [D†y ⊗ (DT

x )†]RS,

which implies

infDS

K ∈D[�(DK)] = −(RS∗ )TRS∗

and DgcK = arg minDS

K ∈D[�(DK)] is as in (10).

Notice that thus far the approach has been completelygeneral—no assumptions besides the standard assumptionswere made. In fact, the only assumption that will need to bemade concerns the structure of D22. A nonzero feedthroughmatrix from the control input to the measured output intro-duces an algebraic loop in the feedback system. Therefore, ifthe feedback system is well posed this algebraic loop can beeliminated. The same idea applies here. The only exceptionis that the feedthrough matrix also has to be well posed withrespect to the causality constraint. In other words, if the ithcontrol input is to be independent of the j th measured outputbut the kth control input depends on the j th measured output,then all the output measurements that the ith control inputdepends on must also be independent of the kth control input.This condition can be easily checked explicitly by forming thealgebraic equation of the feedback loop.

The expression for the H2-optimal controller with ageneralized causality constraint is stated in the followingtheorem.

Theorem 4. Assume that the feedback loop is well posed withrespect to the generalized causality constraint. Then the uniqueH2-optimal controller with a GCC is

Kgc = F

([0 I

I −D22

], K

), (11)

where

Proof. Let G = F(G,

[0I

I−D22

]). This eliminates D22 from

the loop. Next, using the results of Saberi et al. (1995, Theorem6.7.3) it follows that the norm of the closed loop system is equalto (8) with DK = D

gcK . Therefore, by Lemma 3 Kgc is an opti-

mal controller satisfying (4). Uniqueness is guaranteed by the

assumptions made about G. Note that F([

0I

I−D22

], K

), sim-

ply solves the algebraic equation of the feedback loop. There-fore, if D22 is consistent with the generalized causality con-straint then the feedthrough matrix of Kgc which is equal toD

gcK (I −D22D

gcK )−1 will satisfy the generalized causality con-

straint. �

4.2. H2-optimal packet-based controller

In this section we construct the lifted representation of (2),show that the packet causality constraint is equivalent to a spe-cial case of the generalized causality constraint, and then useTheorem 4 to solve the packet-based control problem.

Let LN : �p → �pN (�p : Z+ → Rp is the space ofsequences taking values in Rp, not to be confused with the spaceof all p-power-summable sequences) and G=LNGL−1

N be thelifting operator and the representation of the lifted plant, defined

142 D. Georgiev, D.M. Tilbury / Automatica 42 (2006) 137–144

in Chen and Francis (1995), respectively. Let the representationof G be written as

(12)

Corollary 5 (H2-optimal packet-based controller). The uniqueH2-optimal controller that satisfies the packet-causality con-straint Kpac = L−1

N KpacLN where

Kpac = F

([0 I

I −D22

], Kp

)(13)

and

with

DKpac = [DK 0]

DK = (DxTDx)

−1DxTCxYC2

T(D21DT21 + C2YCT

2 )−1,

where Q, P , DxTDx , and Dx

TCx are defined in the same man-ner as in (6), i.e. the matrices representing G are replaced withthe matrices representing G; X and Y remain the same by theproperties of the lifting operation.

Proof. First note that by the properties of the lifting operatordiscussed in Section 3, Kpac is stabilizing if and only if Kpac isstabilizing and Kpac minimizes ‖T ‖2 if and only if Kpac min-imizes ‖T ‖2 where T := (G, K). Next, define the projectionoperator P

pacD as

PpacD

=[

INp2m2×Np2m2 0Np2m2×[Np2m2(N−1)]0[Np2m2(N−1)]×Np2m2 0[Np2m2(N−1)]×[Np2m2(N−1)]

],

where the subscript next to I and 0 denotes the dimension. Notethat Kpac satisfies the packet causality constraint if and onlyif DK

S = PpacD DK

S where DK is the feedthrough matrix ofKpac. The structure of D22 is well posed with respect to thepacket-causality constraint represented by P

pacD since all inputs

are allowed to depend on the same current measured outputs.The remainder follows from Theorem 4 and the structure of thelifted matrices described in Chen and Francis (1995). �

The inverse lifting transformation of Kpac can be done in astraightforward manner since at the time of packet transmis-sion, all control inputs are calculated from the same output mea-surement set and transmitted in one packet. In other words, thecontrol packet equals u(k). Although the correspondence be-tween the output packet and y(k) is not as simple due to a shift(see Fig. 2), all values necessary for the calculation of the nextcontrol input set are available at the same time. Therefore, thepacket-based controller would be programmed as it is written

in Corollary 5. Also note that there may be other approachesthat could solve the problem (Shu & Chen, 1995). Finallynote that the feedthrough term is only needed when the ab-solute network delay is small relative to the transmissionperiod. However, when the absolute network delay is small,the feedthrough term can provide significant performanceimprovements.

5. Performance analysis

In this section we compare the H2-optimal packet-based con-troller with two different H2-optimal single-rate controllers viasimulation. We take a continuous-time plant and use a gen-eralized H2 measure defined in Khargonekar and Sivashankar(1991) to derive an equivalent discrete-time plant, which is thenused for the controller synthesis. The discretization method ofthe continuous-time plant is not important to the above resultsbut by using a performance measure based in continuous timewe can provide a fair comparison of controllers operating atdifferent frequencies.

Consider the double integrator (1/s2). The state vector[x1 x2]T is defined as the plant output (position) and its deriva-tive (velocity), respectively. The continuous-time plant GC forthis example is defined as follows:

(14)

Fig. 4 contains the impulse response and the control input.Here the packet size, N, is equal to 10 and hs = 0.01 s. In thisfigure, the packet-based controller is compared against a singlerate controller that purely operates at sampling period hs anda single rate controller that purely operates at sampling periodhp = 10hs. It appears that the packet-based controller recoversabout 50% of the performance lost due to down-sampling bythe slow rate controller. The packet-based controller also resultsin a more continuous velocity response when compared to thesingle rate controller operating at the sampling period hp. Thecontrol input is larger in the first half and smaller in the secondhalf of the transmission period. Depending on the actuator, thisapparently inherent quality of the packet-based controller maynot be desirable. Fig. 5 contains the plot of generalized H2measure vs. transmission period for different fixed packet sizes.All cases are compared with the continuous time case. Thisfigure is a further indication that the system performance canbe significantly improved using packet-based control. In mostcases it appears that the packet-based controller can performas well as a single-rate controller with a transmission periodequal to one third of the packet-based controller’s transmission

D. Georgiev, D.M. Tilbury / Automatica 42 (2006) 137–144 143

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10-3

time (sec)

x 1

Single-Rate (hs)

Packet-Based

Single-Rate (hp)

0 0.2 0.4 0.6 0.8 1-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

time (sec)

u

Single-Rate (hs)

Packet-Based

Single-Rate (hp)

(a)

(b)

Fig. 4. (a) Impulse response—position vs. time: hs = 0.01 s, N = 10.(b) Impulse response—control input vs. time: hs = 0.01 s, N = 10.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.4

0.6

0.8

1

1.2

1.4

1.6

Transmission Period (hp)

Gen

eral

ized

H2

Mea

sure

(||

Tpa

c || 2

)

N = 10

N = 2

N = 4

N = 1

Continuous Time

Fig. 5. Generalized measure vs. transmission period for different packet sizes,here hs is adjusted to equal hp/N .

period. In other words, a control engineer could reduce thetransmission period by a factor of three and, by using packet-based methods introduced in this paper, retain the same levelof performance.

6. Conclusion

In this paper, a new approach to networked control systemdesign was presented. The synthesis problem was motivatedby traditional design criteria of most control networks togetherwith an efficient use of the packet structure. A specific proto-col, consisting of a multipoint-packet structure, was proposed.This was shown to be a means of transmitting more informationfor a fixed network capacity. H2-optimization techniques wereused to synthesize a controller with a generalized causality con-straint and then this solution was extended to the packet-basedsynthesis problem. Finally, the performance of the packet-basedcontroller was compared numerically against the performanceof two standard H2-optimal controllers. The results showed thatthe performance gained using the packet-based approach ap-pears to be significant enough to be considered for some appli-cations.

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Daniel Georgiev was born in the Czech Re-public on February 7th, 1980. He received theB.S. degree in Mechanical Engineering, summacum laude, from the University of New Mexicoin 2002 and he is currently working towardshis Ph.D. degree at the University of Michi-gan. In 2002, he received a National ScienceFoundation Graduate Research Fellowship andat the University of New Mexico he won theGeorge Breece prize for graduating first in hisclass. He is a member of ASME and IEEE. His

research interests include stochastic teams, networked control systems, anddecentralized control.

Dawn M. Tilbury received the B.S. degree inElectrical Engineering, summa cum laude, fromthe University of Minnesota in 1989, and theM.S. and Ph.D. degrees in Electrical Engineer-ing and Computer Sciences from the Universityof California, Berkeley, in 1992 and 1994, re-spectively. In 1995, she joined the faculty ofthe Mechanical Engineering Department at theUniversity of Michigan, Ann Arbor, where sheis currently an Associate Professor. She wonthe EDUCOM Medal (jointly with Professor

William Messner of Carnegie Mellon University) in 1997 for her work onthe web-based Control Tutorials for Matlab. An expanded version, ControlTutorials for Matlab and Simulink, was published by Addison-Wesley in1999. She is co-author (with Joseph Hellerstein, Yixin Diao, and SujayParekh) of the book Feedback Control of Computing Systems. She receivedan NSF CAREER award in 1999, and is the 2001 recipient of the DonaldP. Eckman Award of the American Automatic Control Council. She belongsto ASME, IEEE, and SWE, and is a member of the 2004–2005 class ofthe Defense Science Study Group (DSSG). Her research interests includedistributed control of mechanical systems with network communication, logiccontrol of manufacturing systems, and performance management and controlof computing systems.