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528IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 2, FEBRUARY 2012

[12] K.S.NarendraandJ.Balakrishnan,“AcommonLyapunovfunctionfor

stable LTI systems with commuting ?-Matrices,” IEEE Trans. Autom.

Control, vol. 39, no. 12, pp. 2469–2471, Dec. 1994.

[13] P. Peleties and R. DeCarlo, “Asymptotic stability of ?-switched sys-

tems using Lyapunov-like functions,” in Proc. Amer. Control Conf.,

Boston, MA, 1991, pp. 1679–1684.

[14] S.SastryandM.Bosdon,AdaptiveControl:Stability,Convergenceand

Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989.

[15] G. Tao, Adaptive Control Design and Analysis.

2003.

[16] K. S. Tsakalis, “Model reference adaptive control of linear

time-varying plants: The case of ‘jump’ parameter variations,”

Int. J. Control, vol. 56, no. 6, pp. 1299–1345, 1992.

New York: Wiley,

Iterative Distributed Model Predictive Control of Nonlinear

Systems: Handling Asynchronous, Delayed Measurements

Jinfeng Liu, Xianzhong Chen, David Muñoz de la Peña, and

Panagiotis D. Christofides, Fellow, IEEE

Abstract—Inthiswork,wefocusoniterativedistributedmodelpredictive

control (DMPC) of large-scale nonlinear systems subject to asynchronous,

delayed state feedback. The motivation for studying this control problem is

the presence of asynchronous, delayed measurement samplings in chem-

ical processes and the potential use of networked sensors and actuators

in industrial process control applications to improve closed-loop perfor-

mance. Under the assumption that there exist upper bounds on the time

interval between two successive state measurements and on the maximum

measurementdelay,wedesignaniterativeDMPCschemefornonlinearsys-

tems via Lyapunov-based control techniques. Sufficient conditions under

which the proposed distributed MPC design guarantees that the state of

the closed-loop system is ultimately bounded in a region that contains the

origin are provided. The theoretical results are illustrated through a cat-

alytic alkylation of benzene process example.

Index Terms—Asynchronous measurements, distributed model predic-

tive control (DMPC), measurement delays, nonlinear systems, process con-

trol.

I. INTRODUCTION

Model predictive control (MPC) is a popular control strategy for the

design of high performance process control systems and is typically

studied within the centralized control paradigm in which all the manip-

ulated inputsare optimized in a single optimization problem [1]. While

the centralized paradigm to MPC has been successful, in recent years,

there is a trend for the development of decentralized and distributed

MPC due to the significantly increased computational complexity, or-

ganization and maintenance difficulties as well as reduced fault toler-

ance of centralized MPC (e.g.,[2], [3]).

Manuscript received January 28, 2010; accepted August 04, 2011. Date of

publication August 15, 2011; date of current version January 27, 2012. Recom-

mended by Associate Editor E. Fabre.

J. Liu and X. Chen are with the Department of Chemical and Biomolec-

ular Engineering, University of California, Los Angeles, CA 90095-1592 USA

(e-mail: jinfeng@ucla.edu, xianzhongchen@gmail.com).

D. Muñoz de la Peña is with the Departamento de Ingeniería de Sistemas y

Automática Universidad de Sevilla, Sevilla 41092, Spain (e-mail: dmunoz@us.

es).

P. D. Christofides is with the Department of Chemical and Biomolecular En-

gineering,UniversityofCalifornia,LosAngeles,CA90095-1592USAandalso

with the Department of Electrical Engineering, University of California, Los

Angeles, CA 90095-1592 USA (e-mail: pdc@seas.ucla.edu).

Colorversionsofoneormoreofthefiguresinthistechnicalnoteareavailable

online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2011.2164729

In the literature, several approaches for the design of decentralized

and distributed MPC have been reported; please see [4], [2], [3] for

reviews of results in this area. Specifically, in [5], a distributed MPC

(DMPC) scheme for coupled nonlinear systems subject to decoupled

constraints was designed. In [6], a robust DMPC design was devel-

oped for linear systems with coupling between subsystems modeled

as bounded disturbances. In [7], a decentralized MPC was proposed

for nonlinear systems with no information exchange between the local

controllersand the stabilityof the decentralized controlsystemwasen-

sured by a set of contractive constraints. In [8], a cooperative DMPC

scheme was developed for linear systems with guaranteed stability of

theclosed-loopsystemandconvergenceofthecosttoitsoptimalvalue,

and in [9], a game theory based DMPC scheme for constrained linear

systems was proposed. In our previous work [10], [11], a sequential

DMPC architecture and an iterative DMPC architecture were designed

for nonlinear systems via Lyapunov-based control techniques. Specif-

ically, in the sequential DMPC architecture, the distributed controllers

communicate via one-directional communication, are evaluated in se-

quence and once in each sampling time; and in the iterative DMPC

architecture, the distributed controllers communicate via bi-directional

communication, are evaluated in parallel and iterate to achieve con-

vergence in each sampling time. However, all of the above results are

basedontheassumptionofcontinuoussamplingoftheentireplantstate

vectorandassumingnodelaysandperfectcommunicationbetweenthe

sensors/actuators and the controllers.

In many chemical process applications, the assumption of contin-

uous,undelayedprocessstatesamplingandperfectcommunicationbe-

tween the sensors/actuators and the controllers may not hold because

of measuring difficulties of some process states (e.g., species concen-

trations) and communication network malfunctions introducing data

losses and time-varying delays [12]. Previous work on MPC design

for systems subject to delayed feedback has primarily focused on cen-

tralized MPC designs [13]–[15] and little attention has been given to

the design of DMPC for systems subject to delayed measurements. In

[16],theissueofdelaysinthecommunicationbetweendistributedcon-

trollers was addressed. In our previous work [17], we developed se-

quentialDMPCschemesfornonlinearsystemssubjecttoasynchronous

and delayed state feedback. The approach used in [17] can be extended

to handle asynchronous measurements in an iterative DMPC, however,

it can not be used to handle measurement delays in iterative DMPC.

Motivatedbytheaboveconsiderations,inthiswork,wefocusoniter-

ative DMPC of large-scale nonlinear systems subject to asynchronous,

delayed state feedback. Under the assumption that there exist upper

boundsonthetimeintervalbetweentwosuccessivestatemeasurements

andonthemaximummeasurementdelay,wedesignaniterativeDMPC

scheme for nonlinear systems via Lyapunov-based control techniques.

Sufficient conditions under which the proposed distributed MPC de-

sign guarantees that the state of the closed-loop system is ultimately

bounded in a region that contains the origin are provided. The theo-

retical results are illustrated through a catalytic alkylation of benzene

process example.

II. PRELIMINARIES

The operator ??? is used to denote Euclidean norm of a vector while

????refers to the weighted Euclidean norm, defined by ???? ? ????.

A continuous function ? ? ????? ? ?????is said tobelong to class? if

it is strictly increasing and satisfies ???? ? ?. The symbol ?? is used

to denote the set ?? ?? ?? ? ??

positivedefinite,continuousdifferentiablefunction and? ??? ? ?,and

the operator ’?’ denotes set subtraction, that is, ??? ?? ?? ? ??

? ? ??? ? ? ??. The symbol ??denotes an estimate of ?. The symbol

? ? ??? ? ?? where ? is a scalar

?

0018-9286/$26.00 © 2011 IEEE

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 2, FEBRUARY 2012529

???????denotes asquare diagonal matrixwhose diagonal elementsare

the elements of vector ?.

We consider nonlinear systems of the form

? ???? ? ??????? ?

?

???

????????????? ? ???????????

(1)

where???? ? ??denotesthevectorofstatevariables,????? ? ??

? ? ???????, are ? sets of control (manipulated) inputs and ???? ?

??

denotes the vector of disturbance variables. The ? sets of in-

puts are restricted to be in ? nonempty convex sets ?? ? ??

? ? ???????, which are defined as ?? ?? ??? ? ??

????

?

?? ? ? ??????? where ????

?

tudes of the input constraints. The disturbance vector is bounded, i.e.,

???? ? ? where ? ?? ?? ? ??

We assume that ????, ?????, ? ? ???????, and ???? are locally

Lipschitz vector functions and that the origin is an equilibrium point

of the unforced nominal system (i.e., system of (1) with ????? ? ?,

? ? ???????, ???? ? ? for all ?) which implies that ???? ? ?.

We further assume that there exists an explicit control law ???? ?

?????????? ????????with ?? ? ?????, ? ? ???????, which ren-

ders(undercontinuousstatefeedback)theoriginofthenominalclosed-

loopsystemasymptoticallystablewhilesatisfyingtheinputconstraints

forall? insideagiven stabilityregion;pleasesee[19]forresultsonthe

explicit construction of ????. This assumption implies that there exist

functions ?????, ? ? ?, 2, 3, 4 of class ? and a continuously differ-

entiable Lyapunov function ? ??? for the nominal closed-loop system,

that satisfy the following inequalities [18], [19]:

,

,

? ???? ?

, ? ? ???????, are the magni-

? ??? ? ??? ? ??.

??????? ? ? ??? ? ????????

?? ???

??

????????

?? ???

??

???? ?

?

???

??????????? ? ???????

????? ? ??? ? ????????

(2)

with ? ? ??????? for all ? ? ?? ? ??

taken to be a level set of ? ???) denotes the stability region of the

closed-loop system under ????. We denote the region ?? ? ? as the

stability region of the closed-loop system under ????. The construc-

tion of ? ??? can be carried out in a number of ways using system-

atic techniques like, for example, sum-of-squares methods. Because of

the local Lipschitz property assumed for the vector fields ????, ?????,

? ? ???????, and ???? and of the boundedness of the manipulated

inputs ??, ? ? ???????, and the disturbance ?, there exists a positive

constant ? such that:

where ?? (typically

???? ?

?

???

???????? ????? ? ?

(3)

forall? ? ??.Moreover,ifwetakeintoaccountthecontinuous differ-

entiable property of the Lyapunov function ? ???, there exist positive

constants ??, ?? and ?? , ? ? ??????? such that:

??

?????? ???

??????? ??? ? ? ???

??

?????? ? ??

??

??????? ???

???????? ??? ?? ? ???? ? ? ???????

(4)

for all ????? ??, ?? ? ??, ? ? ???????, and ? ? ?.

III. ITERATIVE DMPC WITH ASYNCHRONOUS,

DELAYED MEASUREMENTS

In this section, we design an iterative DMPC scheme which takes

into account asynchronous, delayed measurements explicitly and pro-

Fig. 1. Iterative DMPC with asynchronous, delayed measurements.

videsdeterministicclosed-loopstabilityproperties.Intheproposedde-

sign, we will design ? Lyapunov-based MPC (LMPC) controllers to

compute ??,? ? ???????, and refer to the LMPC computing the input

trajectories of ?? as LMPC ?. A schematic of the proposed iterative

DMPC for systems subject to asynchronous, delayed measurements is

shown in Fig. 1.

We assume that the full state of the system (1) is received by the

controllers at asynchronous time instants ??where ?????? is a random

increasing sequence of times and that there are delays in the mea-

surements received by the controllers. In order to model the delays in

measurements, an auxiliary variable ?? is introduced to indicate the

delay corresponding to the measurement received at time ??, that is,

at time ??, the measurement ????? ??? is received. In order to study

the stability properties in a deterministic framework, we assume that

there exists an upper bound ?? on the interval between two succes-

sive measurements and the delays associated with the measurements

are smaller than an upper bound ?, which is, in general, related to

measurement sensors and data transmission networks. We note that for

chemical processes, the delay in the measurements received by a con-

troller are mainly caused in the measurement sampling process. We

also assume that the time instant in which a measurement is sampled

is recorded and transmitted together with the measurement. This as-

sumptionispracticalformanyprocesscontrolapplicationsandimplies

that the delay in a measurement received by the controllers can be as-

sumed to be known. Note that because the delays are time-varying, it

is possible that at a time instant ??, the controllers may receive a mea-

surement ???? ? ??? which does not provide new information (i.e.,

??? ?? ? ????? ????); that is, the controller has already received

a measurement of the state after time ?? ? ??. In this case, the con-

trollers only use measurements that provide new information. Based

on the above modeling of the measurements, we can calculate that the

maximum amount of time the system might operate in open-loop fol-

lowing ??is ? ? ??? ??[17]. This upper bound will be used in the

formulation of the iterative DMPC design below.

We propose to take advantage of the system model both to estimate

the currentsystem state from adelayed measurement and to control the

systeminopen-loopwhennewinformationisnotavailable.Tothisend,

whenadelayedmeasurementisreceived,thedistributedcontrollersuse

the system model and the input trajectories that have been applied to

the system to get an estimate of the current state and then based on

the estimate, MPC optimization problems are solved to compute the

optimal future input trajectory that will be applied until new measure-

ments are received. The proposed implementation strategy for the iter-

ative DMPC design is as follows:

1) When ????? ??? is available at ??, all the distributed controllers

receive it and check whether it provides new information. If it

does, go to step 2. Else, go to step 5.

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530IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 2, FEBRUARY 2012

2) The controllers estimate the current system state ?????? and then

evaluate their future input trajectories in an iterative fashion with

initial input guesses generated by ????.

3) At iteration ? (? ? ?):

3.1. Each controller evaluates its future input trajectory based

on ?????? and the latest received input trajectories of all the

other controllers (when ? ? ?, initial input guesses generated

by ???? are used).

3.2. The controllers exchange their future input trajectories.

Based on all the input trajectories, each controller calculates

and stores the value of the cost function.

4) If a termination condition is satisfied, each controller sends its

entire future input trajectory corresponding to the smallest value

of the cost function to its actuators; if the termination condition is

not satisfied, go to step 3 (? ? ? ? ?).

5) When a new measurement is received, go to step 1 (? ? ? ? ?).

In order to estimate the current system state ?????? based on ?????

???, the distributed controllers take advantage of the input trajecto-

ries that have been applied to the system from ??? ?? to ?? and the

system model of (1) with ???? ? ?. Note that since the controllers ex-

change their input trajectories at the end of each iteration, they are able

to determine the inputs the other controllers implement which corre-

spond to the smallest cost value in each sampling time. Let us denote

the input trajectories that have been applied to the system as ??

? ? ???????. Therefore, ?????? is evaluated by integrating the fol-

lowing differential equation:

????,

? ????? ? ???????? ?

?

???

???????????

????? ?? ? ???? ??????

(5)

with ?????? ??? ? ????? ???.

Inordertoproceed,wedefine????????for? ? ?????? asthenom-

inal sampled trajectory of the system of (1) associated with the feed-

back control law ????and sampling time ? starting from ??????.Note

that ? is the prediction horizon of the DMPC. This nominal sampled

trajectory is obtained by integrating the following differential equation

recursively:

? ???????? ???????????? ?

?

???

??????????????????????????

?? ??????? ? ????

(6)

where ? ? ??????? ? ?. Note that in (6), the control laws ??, ? ?

???????, are implemented in a sample-and-hold fashion. Based on

????????, we define:

?????????? ? ?????????????? ?? ? ?????? ? ????

(7)

where ? ? ??????? and ? ? ??????? ? ?. The sampled trajectory

???????? and the input trajectory ?????????? will be used in the design

of the LMPC to construct the stability constraint and used as the initial

input guess for iteration 1 (i.e., ????

?

ically, the design of LMPC ?, ? ? ???????, at iteration ? is based on

the following optimization problem:

? ????for ? ? ???????). Specif-

???

? ?????

??

?

?? ???????

?

???

????????

??

(8a)

?????? ???? ???? ??????

?

???

???? ??????????? ? ???? ? ??????

(8b)

????? ???????

?

??????? ?? ?? ?

(8c)

????? ? ??????

?

?????? ? ???? ?? ? ????????

(8d)

????? ???

?? ?? ?????

?? ?

(8e)

?

???? ????? ? ???? ??????????

??? ??????????

???

?

???????????? ? ?????????????????????? ?

?? ? ????????

?

(8f)

where????isthefamilyofpiece-wiseconstantfunctions,??and???,

? ? ???????, are positive definite weighting matrices, ? ? is the pre-

dicted state trajectory of the nominal system, and ???is the smallest

integer satisfying ???? ? ???????. The optimal solution to the

optimization problem of (8) is denoted ????

Accordingly, we define the final optimal input trajectory of LMPC ?

as ??

have different values at different time instants and has to be updated

before solving the optimization problems. The constraint of (8d) im-

poses a limit on the input change in two consecutive iterations, i.e.,

for LMPC ?, the magnitude of input change in two consecutive itera-

tionsisrestrictedtobesmallerthanapositiveconstant???.Giventhat

????? provides a feasible, stabilizing initial solution to the optimiza-

tion problem of LMPC ? (8), the constraint of (8d) allows LMPC ? to

gradually(dependingonthevalue of???)optimizeitsinputtrajectory

and ensures thatthe iterations can be terminated at any number without

loss of closed-loop stability. The constraint of (8f) is used to guarantee

the closed-loop stability.

Inthedesignof(8),thenumberofiterations?mayberestrictedtobe

smaller than a maximum iteration number ????(i.e., ? ? ????) and/or

the iterations may be terminated when a maximum computational time

is reached.

The manipulated inputs of the closed-loop system under the above

iterative DMPC with delayed measurements are defined as follows:

??????? for ? ? ??????.

???????. Note that the value of ??? depends on ??, so it may

????? ? ??

??? ? ??????? ? ? ?????????? ? ?????????

(9)

for all ??such that ??? ?? ? ???

the smallest integer that satisfies ????? ???? ? ??? ??.

Remark 1: For general nonlinear systems, there is no guaranteed

convergence of the optimal cost of the distributed optimization of (8)

to any value. Note also that the implementation strategy of the DMPC

guarantees that the optimal cost of the distributed optimization of (8)

is upper bounded by the cost of the controller ???? at each sampling

time. We further note that in the case of linear systems, the constraint

of (8f) can be written in a quadratic form with respect to ?? and it

can be verified that the optimization problem of (8) is convex. If the

input given by LMPC ? of (8) at each iteration is defined as a convex

combination of the current optimal input solution and the previous

one (e.g., ??

??????? ?

???

??????

?

????? ? ? with ? ? ?? ? ?, ????

the optimization problem of (8) and ????

the solutions obtained at iteration ? ? ?), then it can be proved that

the optimal cost of the distributed LMPC of (8) converges to the one

of the corresponding centralized control system [20], [8]. These con-

siderations imply that there is a balance between controller evaluation

time and closed-loop performance that should be struck in the control

system architecture (i.e., iterative or centralized) and/or the determi-

nation of the maximum iteration number, ????.

?????? ??and the variable ? denotes

??????

?

?????? ? ??????

is the current solution given by

is the convex combination of

? ?????? where

?

?

A. Stability Analysis

The stability properties of the iterative DMPC of (8)–(9) are stated

inTheorem 1below. To state Theorem 1, weneed thefollowing propo-

sitions.

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 2, FEBRUARY 2012531

Proposition 1 (cf. [21], [11]): Consider the nominal sampled tra-

jectory ??of the system of (1) in closed-loop with the controller ????

applied in a sample-and-hold fashion and with open-loop state estima-

tion. Let ???? ? ? and ? ? ?? ? ? satisfy:

???????

? ????? ? ??? ? ???

?

(10)

with ??? ?? ?

????? ????? ? ??? ? ? ??????? ? ???, ????? ? ??and ?? ? ? for

all ?, then ? ???????? ? ????? ???????? ?????????.

Proposition 1 ensures that if there is no measurement delay and the

nominal system under the control of ???? implemented in a sample-

and-hold fashion starts in ??, then it is ultimately bounded in ??

Proposition 2 below provides an upper bound on the deviation of the

nominal state trajectory from the actual state trajectory when the same

control actions are applied.

Proposition 2 (cf. [11]): Consider the systems:

?

??????????????? ????????????

?

????? ????

?

. If ???? ? ? where ???? ?

.

? ????? ????????? ?

???

?

? ????? ??????????

???

??????????????

with initial states ?????? ? ?????? ? ??. There exists a class ? func-

tion ????? such that ?????? ? ?????? ? ???? ? ??? for all ?????,

????? ? ??and all ???? ? ? where ????? ? ?????? ?? ?????

with ? being the upper bound of the disturbance ???? and ??, ??

being positive real numbers.

Proposition 3 bounds the difference between the magnitudes of the

Lyapunov function of two states in ??.

Proposition 3 (cf. [21]): Consider the Lyapunov function ? ??? of

the system of (1). There exists a quadratic function ????? such that

? ???? ? ? ???? ? ??????? ???? for all ????? ? ?? with ????? ?

??????

Proposition 4 bounds the difference between the nominal state tra-

jectory (i.e., ????? in Proposition 4) under the optimized control inputs

at the current iteration (i.e., ??

and the predicted nominal state trajectory (i.e., ????? in Proposition 4)

generated in the optimization problem of LMPC ? with ??, ? ?? ?, de-

termined at a previous iteration (i.e., ?? ? ????

calculated at the current iteration (i.e., ?? ? ??

Proposition 4: Consider the systems:

?

???????????

?????? ? ????and ?? ? ?.

????, ? ? ???????, in Proposition 4)

?

?).

, ? ? ?? ?) and ??

? ????? ????????? ?

???

????

? ????? ??????????

?? ????

???

?????????????

?

??? ? ???????????

????

with initial states ?????? ? ?????? ? ??. There exists a class ? func-

tion ??????? such that ?????? ? ?????? ? ?????? ? ??? for all ?????,

????? ? ??, and ??

?

? ??and ???

? ? ???????.

Proof: Define ???? ? ????? ? ?????. The time derivative ? ????

can be calculated as ? ???? ? ? ????? ? ? ?????. Adding and subtracting

?? ????

???

???????????

accountthelocalLipschitzpropertiesassumedforthevectorfields????

and ?????, ? ? ???????, the boundedness of the manipulated inputs,

and the boundedness of the difference between ??

obtain the following inequality:

????,????

?????????

?

???? ? ???with

???? to/from the expression of ? ????and taking into

???? and ????

?

???, we

?? ????? ? ???????? ?

?? ????

???

????????

?

?????? ?

?? ????

???

????????

where ??, ???? and ???? (?

constants. Denoting

????????) are positive

?

???

????

???

?? ????

????????

?

and

with initial

??????

sition 4 with ??????? ? ??????????? ??

To simplifytheproof ofTheorem1, wedefineanew function ?????

based on ????, ? ? ???????, as follows:

????

?

condition

??????????? ??

?? ????

???

?????

???????,

?

???? ?? ? . This proves Propo-

andintegrating

can

?? ?????

?, weobtain that

?

?? ? .

????? ?

?

???

?

???? ?? ????

?

?

??????????? ?????

??????

(11)

It is easy to verify that ????? is a strictly increasing and convex func-

tion of its argument. In Theorem 1 below, we provide sufficient condi-

tions under which the iterative DMPC guarantees that the state of the

closed-loop system is ultimately bounded in a region that contains the

origin.

Theorem 1: Consider the system of (1) in closed-loop with the

DMPC design of (8)–(9) based on the controller ???? that satisfies

the conditions of (2) with class ? functions ?????, ? ? ?, 2, 3, 4. Let

???? ? ?, ? ? ????? ?, ? ? ?? ? ?, ? ? ? and ? ? ? satisfy the

condition of (10) and the following inequality:

?????? ??????? ? ??????????? ? ????????? ? ? (12)

with ?? being the smallest integer satisfying ??? ? ?? ? ? and

??being the smallest integer satisfying ??? ? ??. If ????? ? ??,

? ? ?? and ?? ? ?, then ???? is ultimately bounded in ?? ? ??

where ?? ? ????? ??????? ? ??????????? ? ?????????.

Proof: We assume that at ??, a delayed measurement ????????

containing new information is received, and that the next measurement

with new state information is not received until ????. This implies that

???? ? ???? ? ?? ? ?? and that the iterative DMPC of (8)–(9) is

solved at ??and the optimal input trajectories ??

are applied from ?? to ????. In this proof, we will refer to ? ???? for

? ? ????????? as the state trajectory of the nominal system of (1) under

the control of the iterative DMPC with ? ????? ? ??????.

PartI: Inthispart,weprovethatthestabilityresultsstatedinThe-

orem 1 hold for ????? ?? ? ???? (recall that ???is the smallest

integer satisfying ???? ? ???????) and all ?? ? ?. By Propo-

sition 1 and taking into account that ?????? ? ??????, the following

inequality can be obtained:

???????, ? ? ???????,

? ????????? ??????? ? ????? ???????? ? ????????????

(13)

By Proposition 2 and taking into account that ?????? ??? ? ?????

???, ? ????? ? ?????? and??? ? ???????,thefollowinginequal-

ities can be obtained:

??????? ? ?????? ???????

?? ??????? ? ???????? ?????????

(14)

When ???? ? ?? for all times (this point will be proved below), we

can apply Proposition 3 to obtain the following inequalities:

? ???????? ?? ???????? ??????????

? ????????? ?? ?? ???????? ? ????????????

(15)

From (13) and (15), the following inequality is obtained:

? ????????? ??????? ? ????? ???????? ???????????

????????????

(16)

The derivative of the Lyapunov function of the nominal system of (1)

under the control of the iterative DMPC from ?? to ???? is expressed

as follows for ? ? ????????:

?? ?? ????? ??? ?? ?????

??

??? ????? ?

?

???

???? ???????

????????

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532IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 2, FEBRUARY 2012

Adding the above equation and the constraints of (8f) in each LMPC

together, and reworking the resulting inequality, we can obtain the fol-

lowing inequality for ? ? ???????? by accounting for (3) and Propo-

sition 4:

?? ?? ????? ??? ???????????

?

???? ?? ????

?

???? ?? ????

?

??????? ? ???

?

?

????????

Integrating the above inequality from ? ? ? to ? ? ???? and taking

into account that ? ????? ? ??????, ????? ??? ???? and the defini-

tion of ?????, the following inequality can be obtained:

? ?? ???????? ? ? ????????? ??????? ? ?????????

(17)

From (15), (16) and (17), the following inequality is obtained:

? ????????? ? ????? ???????? ??????????? ? ??????????

???????????? ? ?????????

(18)

In order toprovethatthe Lyapunov function is decreasing between two

consecutive measurements, the following inequality must hold:

????? ? ??????????? ??????????? ? ????????

(19)

for all possible ? ? ??? ?. Taking into account that ??, ?? and ??

arestrictlyincreasingfunctionsoftheirarguments,???isadecreasing

function of the delay ?? and that if ?? ? ? then ??

if the condition of (12) is satisfied, the condition of (19) holds for all

possible ??and there exists ?? ? ? such that the following inequality

holds:

? ??, then

? ????????? ? ????? ???????? ??????

(20)

which implies that if ????? ? ????? , then ? ????????? ? ? ???????,

and if ????? ? ?? , then ? ????????? ? ??. Because the upper bound

on the difference between the Lyapunov function of the actual trajec-

tory ? and the nominal trajectory ? ? is a strictly increasing function of

time, the inequality of (20) also implies that:

? ?????? ? ????? ???????????? ?? ? ??????????

(21)

Using theinequalityof(21) recursively,itcanbe provedthatif????? ?

??, then the closed-loop trajectories of the system of (1) under the

proposed iterativeDMPCstay in??foralltimes (i.e.,???? ? ??? ??).

Moreover, it can be proved that if ????? ? ??, ??????

???

? ?????? ? ??.

Thisprovesthat???? ? ??foralltimesand????isultimatelybounded

in ?? when ????? ?? ? ????.

Part 2: In this part, we extend the results of Part 1 to the general

case, that is, ???????? ????. Taking into account that ??, ?? and

?? are strictly increasing functions of their arguments and following

similar steps as in Part 1, it can be readily proved that the inequality of

(19) holds for all possible ?? ? ? and ????? ?? ? ????. Using

this inequality and following a similar line argument as in Part 1, the

stability results stated in Theorem 1 can be proved.

Remark 2: Note that in the case that the open-loop operation time

is larger than ? ? ??? ??, we may still apply the proposed DMPC

design but the closed-loop stability cannot be guaranteed, depending

on the open-loop process dynamic behavior.

IV. APPLICATION TO AN ALKYLATION OF BENZENE PROCESS

We consider an alkylation of benzene with ethylene process which

consists of four continuously stirred tank reactors (CSTRs) and a

flash tank separator and is modeled by 25 nonlinear ordinary dif-

ferential equations. Please see [11] for the detailed modeling of the

process. Each of the tanks has an external heat/coolant input. The

TABLE I

STEADY-STATE INPUT VALUES FOR ?

TABLE II

MANIPULATED INPUT CONSTRAINTS

manipulated inputs to the process are the heat injected to or removed

from the five vessels, ??, ??, ??, ?? and ??, and the feed stream

flow rates to CSTR-2 and CSTR-3, ?? and ??. The states of the

process consist of the concentrations of benzene (?), ethylene (?),

ethylbenzene (?), and 1,3-diethylbenzene (?) in each of the five

vessels and the temperatures of the vessels. We consider a steady

state (operating point), ??, of the process which is defined by the

steady-state inputs ???, ???, ???, ???, ???, ??? and ???, shown

in Table I. The steady-state temperatures in the five vessels are the

following: ??? ? ?????? ?, ??? ? ?????? ?, ??? ? ?????? ?,

??? ? ???????, ??? ? ???????. The control objective is to regulate

the system from an initial state to the steady state. The initial tem-

peratures of the five vessels are: ??? ? ?????? ?, ??? ? ?????? ?,

??? ? ?????? ?, ??? ? ?????? ?, ??? ? ?????? ?. The first dis-

tributed controller (LMPC 1) will be designed to compute the values

of ??, ??and ??, the second distributed controller (LMPC 2) will be

designed to compute the values of ??and ??, and the third distributed

controller (LMPC 3) will be designed to compute the values of ??and

??. Taking this into account, the process model belongs to the class

of nonlinear systems: ? ???? ? ???? ? ???????? ???????? ???????

where the state ? is the deviation of the state of the process from the

steady state, ??

??

??? ? ??? ?? ? ???? are the manipulated inputs which are sub-

ject to the constraints shown in Table II. We use the same de-

sign of ???? as in [11] based on a quadratic Lyapunov function

? ??? ? ???? with ? being the following weighting matrix: ? ?

??????? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ???. Based on

????,we design the iterative DMPC with the weighting matrices being

?? ? ??????? ? ? ? ???? ? ? ? ????? ?? ?? ?? ???? ? ? ? ?

???? ? ? ? ?????, ??? ? ??????? ? ????? ? ????? ? ??????,

??? ? ??????? ? ????? ? ?????? and ??? ? ???????? ????. The

sampling time of the LMPCs is chosen to be ? ? ?? ?. ???is chosen

to be ????????

?

for the distributed LMPCs and maximum iteration

number (i.e., ? ? ????) is used as the termination condition. In the

simulations, bounded process noise is considered.

We consider that the state of the process is sampled at asynchronous

timeinstants??????with?? ? ???.Moreover,weconsiderthatthere

are delays involved in the measurements with ? ? ?? ?. Measure-

ment delays can naturally arise in the context of species concentration

measurements. We will compare the proposed iterative DMPC with a

centralized LMPC which takes into account delayed measurements ex-

plicitly [14]. The centralized LMPC uses the same weighting matrices,

sampling time and prediction horizon as used in the DMPC. In order

to model the sampling time instants, a bounded Poisson process (see

[17]) is used to generate ?????? and another bounded random process

is used to generate the associated delay sequence ??????. We choose

thehorizonofalltheLMPCstobe? ? ?sothatthehorizoncoversthe

maximumpossible open-loopoperationinterval(i.e.,???? ? ???).

Note that the maximum possible open-loop operation interval only de-

pends on the frequency of measurement sampling and the delays in

? ? ???? ??? ???? ? ??????? ?????? ???????,

? ? ???? ???? ? ???? ??? ??? ???? and ??

? ? ???? ???? ?

Page 6

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 2, FEBRUARY 2012 533

Fig. 2. Asynchronous time sequence ??

quence ??

? with ?

and the ?–axis indicates the size of ? .

? and corresponding delay se-

? ?? ? and ? ? ?? ?: the ?–axis indicates ??

?

Fig. 3. Trajectories of ? ??? under ???? implemented in a sample-and-hold

fashionandwithopen-loopstateestimation,theiterativeDMPCwith?

5 and the centralized LMPC.

? ?,

Fig.4. Totalperformancecostalongtheclosed-loopsystemtrajectoriesofcen-

tralized LMPC (dashed line) and iterative DMPC (solid line).

the measurements and is not related to the dynamics of the chemical

plant. Note also that, in terms of practical considerations, it is possible,

particularly in the context of species concentration measurements, for

the measurement delays to exceed 30 s and the use of a 40 s delay

upper bound for species concentration measurements is realistic from

a practical standpoint. Fig. 2 shows the time instants when new state

measurements are received and the associated delay sizes

Fig. 3 shows the trajectory of ? ??? under different control designs.

FromFig.3,weseethatboththeproposediterativeDMPCandthecen-

tralized LMPC are able to drive the system state to a region very close

to the desired steady state (? ??? ? ???); the trajectories of ? ???

generated by the iterative DMPC design are bounded by the corre-

sponding trajectory of ? ??? under the controller ???? implemented in

a sample-and-hold fashion and with open-loop state estimation. From

Fig. 3, we can also see that the centralized LMPC and the iterative

DMPC with ???? ? ? give very similar ? ??? trajectories.

Next, we compare the centralized LMPC and the iterative DMPC

from a performance index point of view. To carry out this comparison,

the same initial condition and parameters were used for the different

controlschemesand thetotalcost undereachcontrolschemewascom-

puted as follows: ? ?

?

????????

?? ? ???? ? is the final simulation time. Fig. 4 shows the total cost

alongtheclosed-loopsystemtrajectoriesundertheiterativeDMPCand

the centralized LMPC. For the iterative DMPC design, different max-

imum numbers of iterations, ????, are used. From Fig. 4, we can see

?

?

???????????

?? where

that as the iteration number ? increases, the performance cost given by

the iterative DMPC design decreases and converges to a value which

is very close to the cost of the one corresponding to the centralized

LMPC. However, we note that there is no guaranteed convergence of

the cost of iterative DMPC to the cost of a centralized MPC because

of the non-convexity of the LMPC optimization problems, and the dif-

ferent stability constraints imposed in the centralized LMPC and the

iterative DMPC (Remark 1).

Finally, we compare the evaluation times of the various control de-

signs. The simulations are carried out by Java programming language

inaPentium3.20GHzcomputer.Theoptimizationproblemsaresolved

by the interior point optimizer Ipopt. We evaluate the LMPC optimiza-

tionproblemsfor100runs.Themeanevaluationtimeofthecentralized

LMPCisabout23.7s.ThemeanevaluationtimeoftheiterativeDMPC

with????? ? is6.3 swhichisthe largesttime amongthe threeLMPC

evaluation times (1.6 s, 6.3 s and 4.3 s). The mean evaluation time of

the iterative DMPC with ???? ? ? is 18.7 s with the evaluation times

of the three LMPCs being 6.9 s, 18.7 s and 14.0 s, respectively. From

the results, we see that the proposed DMPC leads to a reduction in the

evaluation time compared to the centralized LMPC though both pro-

vide a similar closed-loop performance. The results also imply that the

iterative DMPC may be applicable to processes which require smaller

sampling times to maintain closed-loop stability and for which central-

ized MPC is not a feasible option due to larger evaluation time.

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Integrated Design of Symbolic Controllers

for Nonlinear Systems

Giordano Pola, Member, IEEE, Alessandro Borri, Member, IEEE,

and Maria Domenica Di Benedetto, Fellow, IEEE

Abstract—Symbolic models of continuous and hybrid systems have been

studied for a long time, because they provide a formal approach to solve

control problems where software and hardware interact with the physical

world. While being powerful, this approach often encounters some limi-

tations in concrete applications, because of the large size of the symbolic

modelsneededtobeconstructed.Inspiredbyon–the–flytechniquesforver-

ification and control of finite state machines, in this note we propose an al-

gorithm that integrates the construction of the symbolic models with the

design of the symbolic controllers. Computational complexity of the pro-

posed algorithm is discussed and an illustrative example is included.

Index Terms—Approximate bisimulation, digital control systems, non-

linear systems, on–the–fly design, symbolic models.

I. INTRODUCTION

Symbolic models of continuous and hybrid systems have been

studied for a long time, because they provide a formal approach to

solve control problems where software and hardware interact with the

physical world. Symbolic models are abstract descriptions of control

systems in which a symbolic state corresponds to an aggregate of

states. Several classes of dynamical and control systems that admit

symbolic models were identified during the last few years, see, e.g.,

[1], [12] and the references therein. In particular, incrementally

stable [2] nonlinear control systems were shown in [7], [10] to admit

symbolic models. This last result has been further generalized to

Manuscript received June14, 2010; accepted August 07, 2011. Dateof publi-

cation August 15, 2011; date of current version January 27, 2012. The research

leading to these results has been supported in part by the Center of Excellence

for DEWS and received funding from the European Union Seventh Framework

Programme[FP7/2007–2013]underGrantAgreementn.257462HYCON2Net-

work of Excellence. Recommended by Associate Editor A. Chiuso.

The authors are with the Department of Electrical and Information

Engineering, CenterofExcellence

of L’Aquila, L’Aquila 67040, Italy (e-mail: giordano.pola@univaq.it;

alessandro.borri@univaq.it; mariadomenica.dibenedetto@univaq.it).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2011.2164740

forResearch DEWS,University

incrementally stable nonlinear switched systems in [6], incrementally

stable nonlinear time–delay systems in [8], [9] and incrementally

forward complete nonlinear control systems in [15]. The use of

symbolic models for the control design of continuous and hybrid

systems has been investigated in [11], [14]. As discussed in [12], this

approach provides the designer with a systematic method to address a

wide spectrum of novel specifications, that are difficult to enforce by

means of conventional control design paradigms. Examples of such

specifications include logic specifications expressed in terms of linear

temporal logic formulae or automata on infinite strings. The use of

these specifications has been shown to be relevant in the control design

of important domains of application, including robot motion planning

and systems biology (see, e.g., [14] and the references therein). While

being powerful, this approach often encounters some limitations in

concrete applications, because of the large size of the symbolic models

needed to be constructed. In this note we propose one approach to cope

with this drawback. We consider a symbolic control design problem

for nonlinear control systems. Given a nonlinear control plant and

a specification expressed in terms of a finite automaton on infinite

strings, we face the problem of designing a symbolic controller that

implements the specification with arbitrarily good accuracy. The sym-

bolic controller is furthermore requested to avoid blocking behaviors,

when interacting with the plant. This problem can be viewed as an

approximate version of similarity games, as discussed in [12]. Related

control design problems have been studied in [11] and [14]. The first

contribution of this note lies in the derivation of an explicit solution to

the control problem under study. The symbolic controller is proven to

be the non–blocking part [3] of the approximate parallel composition

[12] between the specification automaton and the symbolic model of

the plant. The synthesis of such a controller requires the preliminary

construction of the symbolic model of the plant, which is generally

demanding from the computational complexity point of view. Inspired

by the research line on on–the–fly verification and control of finite

state machines (see e.g., [4], [13]), we give the second contribution

of this note consisting in an efficient algorithm that integrates the

construction of the symbolic model of the plant with the design of

the symbolic controller. Computational complexity of the proposed

algorithm is discussed and an illustrative example is included.

II. PRELIMINARY DEFINITIONS

Notation

The symbol ??? denotes the cardinality of a finite set ?. The identity

map on a set ? is denoted by ??. Given a relation ? ? ? ? ?, the

symbol ???denotes the inverse relation of ?, i.e., ???? ?????? ?

? ? ? ? ????? ? ? ? ??. The symbols

the set of integer, real, positive real, and nonnegative real numbers,

respectively. The symbol ??? denotes the infinity norm of ? ?

Given a measurable function ? ?

??

of ? is denoted by ????. Given ? ?

????? and ??????? denote the set ?? ?

??????????????????????????????????????????, respectively.

Given ? ?

????? ? ? ???. For any ? ?

the unique vector in ?

,,

?and

?

?denote

?.

?

?, the (essential) supremum

?and ? ?

????? ? ?? and the set

?, the symbols

?and ? ?

?, we denote by ?? the set ?? ?

?and ? ?

?such that ? ? ????????????.

???? ?

?the symbol ????denotes

A. Control Systems

In this note we consider the nonlinear control system

? ?

? ???? ? ?????????????? ?

???? ? ???

?

??

(1)

0018-9286/$26.00 © 2011 IEEE