Iterative Distributed Model Predictive Control of Nonlinear Systems: Handling Asynchronous, Delayed Measurements
ABSTRACT In this work, we focus on iterative distributed model predictive control (DMPC) of largescale nonlinear systems subject to asynchronous, delayed state feedback. The motivation for studying this control problem is the presence of asynchronous, delayed measurement samplings in chemical processes and the potential use of networked sensors and actuators in industrial process control applications to improve closedloop performance. Under the assumption that there exist upper bounds on the time interval between two successive state measurements and on the maximum measurement delay, we design an iterative DMPC scheme for nonlinear systems via Lyapunovbased control techniques. Sufficient conditions under which the proposed distributed MPC design guarantees that the state of the closedloop system is ultimately bounded in a region that contains the origin are provided. The theoretical results are illustrated through a catalytic alkylation of benzene process example.

Conference Paper: Control of chaotic systems with uncertain parameters and stochastic disturbance by LMPC
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ABSTRACT: For the chaotic systems with uncertain parameters and stochastic disturbance, in order to satisfy some optimal performance index when chaos control is achieved, the Lyapunovbased model predictive control (LMPC) is introduced. The LMPC scheme is concerned with an auxiliary controller which is constructed in advance. Based on the auxiliary controller and stochastic stability theory, it is shown that the chaotic systems with uncertain parameters and stochastic disturbance are practical stable. With the help of the auxiliary controller, the stability of LMPC can be guaranteed as well as some optimality property. As an example, the unified chaotic system with uncertain parameter and stochastic disturbance is considered and simulation results show the effectiveness of the proposed method.Control (CONTROL), 2012 UKACC International Conference on; 01/2012  [Show abstract] [Hide abstract]
ABSTRACT: A nonlinear networked control system is considered in which the measured values are asynchronously sampled and transmitted over multiple communication links. The effects of communication in each link (transmission delay, packet loss and sampling jitter) are captured by a timevarying delay element. A sufficient condition for asymptotic stability of the resulting nonlinear delayed model is provided using the LyapunovKrasovskii method. This condition is in the form of a compact linear matrix inequality (LMI) which depends on the amount of communication effects in each link. The results are applied to a robot arm networked control system to show the capabilities of the proposed method. Comparison with the previous works indicates that a considerable improvement in the delay bounds for stability is achieved.IEEE Transactions on Automatic Control 01/2014; 59(2):511515. · 2.72 Impact Factor 
Conference Paper: Distributed receding horizon control of constrained linear systems with communication delays
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ABSTRACT: This paper investigates the distributed receding horizon control (DRHC) problem for constrained linear systems subjects to communication delays. Firstly, a novel DRHC scheme is proposed by incorporating the robustness constraint into the optimization problem as well as designing the waiting mechanism for the communication delays. Second, the feasibility and stability properties are studied. We show that, if the communication delays are bounded within a given upper bound, and the cooperation weights are designed appropriately, then the overall system converges to the equilibrium point. Finally, a simulation example is used to verify the theoretical result.American Control Conference (ACC), 2013; 06/2013
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[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 Lyapunovlike functions,” in Proc. Amer. Control Conf.,
Boston, MA, 1991, pp. 1679–1684.
[14] S.SastryandM.Bosdon,AdaptiveControl:Stability,Convergenceand
Robustness. Englewood Cliffs, NJ: PrenticeHall, 1989.
[15] G. Tao, Adaptive Control Design and Analysis.
2003.
[16] K. S. Tsakalis, “Model reference adaptive control of linear
timevarying 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 largescale 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 closedloop 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 Lyapunovbased control techniques. Sufficient conditions under
which the proposed distributed MPC design guarantees that the state of
the closedloop 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 900951592 USA
(email: 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 (email: dmunoz@us.
es).
P. D. Christofides is with the Department of Chemical and Biomolecular En
gineering,UniversityofCalifornia,LosAngeles,CA900951592USAandalso
with the Department of Electrical Engineering, University of California, Los
Angeles, CA 900951592 USA (email: 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
theclosedloopsystemandconvergenceofthecosttoitsoptimalvalue,
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 Lyapunovbased control techniques. Specif
ically, in the sequential DMPC architecture, the distributed controllers
communicate via onedirectional communication, are evaluated in se
quence and once in each sampling time; and in the iterative DMPC
architecture, the distributed controllers communicate via bidirectional
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 timevarying 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 largescale nonlinear systems subject to asynchronous,
delayed state feedback. Under the assumption that there exist upper
boundsonthetimeintervalbetweentwosuccessivestatemeasurements
andonthemaximummeasurementdelay,wedesignaniterativeDMPC
scheme for nonlinear systems via Lyapunovbased control techniques.
Sufficient conditions under which the proposed distributed MPC de
sign guarantees that the state of the closedloop 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
?
00189286/$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 closedloop 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
closedloop system under ????. We denote the region ?? ? ? as the
stability region of the closedloop system under ????. The construc
tion of ? ??? can be carried out in a number of ways using system
atic techniques like, for example, sumofsquares 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.
videsdeterministicclosedloopstabilityproperties.Intheproposedde
sign, we will design ? Lyapunovbased 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 timevarying, 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 openloop 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
systeminopenloopwhennewinformationisnotavailable.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 sampleandhold 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????isthefamilyofpiecewiseconstantfunctions,??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 closedloop stability. The constraint of (8f) is used to guarantee
the closedloop 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 closedloop 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 closedloop 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 closedloop with the controller ????
applied in a sampleandhold fashion and with openloop 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
andhold 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
closedloop system is ultimately bounded in a region that contains the
origin.
Theorem 1: Consider the system of (1) in closedloop 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 closedloop 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 openloop operation time
is larger than ? ? ??? ??, we may still apply the proposed DMPC
design but the closedloop stability cannot be guaranteed, depending
on the openloop 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
STEADYSTATE 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 CSTR2 and CSTR3, ?? and ??. The states of the
process consist of the concentrations of benzene (?), ethylene (?),
ethylbenzene (?), and 1,3diethylbenzene (?) 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
steadystate inputs ???, ???, ???, ???, ???, ??? and ???, shown
in Table I. The steadystate 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 openloopoperationinterval(i.e.,???? ? ???).
Note that the maximum possible openloop operation interval only de
pends on the frequency of measurement sampling and the delays in
? ? ???? ??? ???? ? ??????? ?????? ???????,
? ? ???? ???? ? ???? ??? ??? ???? and ??
? ? ???? ???? ?
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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 sampleandhold
fashionandwithopenloopstateestimation,theiterativeDMPCwith?
5 and the centralized LMPC.
? ?,
Fig.4. Totalperformancecostalongtheclosedloopsystemtrajectoriesofcen
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 sampleandhold fashion and with openloop 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
alongtheclosedloopsystemtrajectoriesundertheiterativeDMPCand
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 nonconvexity 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 closedloop performance. The results also imply that the
iterative DMPC may be applicable to processes which require smaller
sampling times to maintain closedloop stability and for which central
ized MPC is not a feasible option due to larger evaluation time.
REFERENCES
[1] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert,
“Constrained model predictive control: Stability and optimality,”
Automatica, vol. 36, pp. 789–814, 2000.
[2] J. B. Rawlings and B. T. Stewart, “Coordinating multiple optimiza
tionbased controllers: New opportunities and challenges,” J. Process
Control, vol. 18, pp. 839–845, 2008.
[3] R.Scattolini,“Architecturesfordistributedandhierarchicalmodelpre
dictive control – A review,” J. Process Control, vol. 19, pp. 723–731,
2009.
[4] E. Camponogara, D. Jia, B. H. Krogh, and S. Talukdar, “Distributed
modelpredictivecontrol,”IEEEControlSyst.Mag.,vol.22,pp.44–52,
2002.
[5] W. B. Dunbar, “Distributed receding horizon control of dynamically
coupled nonlinear systems,” IEEE Trans. Autom. Control, vol. 52, no.
7, pp. 1249–1263, Jul. 2007.
[6] A. Richards and J. P. How, “Robust distributed model predictive con
trol,” Int. J. Control, vol. 80, pp. 1517–1531, 2007.
[7] L. Magni and R. Scattolini, “Stabilizing decentralized model predic
tivecontrolofnonlinearsystems,”Automatica,vol.42,pp.1231–1236,
2006.
[8] B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright, and G.
Pannocchia, “Coorperative distributed model predictive control,” Syst.
Control Lett., vol. 59, pp. 460–469, 2010.
[9] J.M.Maestre,D.M.delaPeña,andE.F.Camacho,“Distributedmodel
predictive control based on a cooperative game,” Optim. Control Appl.
Methods, vol. 32, pp. 153–176, 2011.
[10] J. Liu, D. M. de la Peña, and P. D. Christofides, “Distributed model
predictive control of nonlinear process systems,” AIChE J., vol. 55, pp.
1171–1184, 2009.
[11] J. Liu, X. Chen, D. M. de la Peña, and P. D. Christofides, “Sequential
and iterative architectures for distributed model predictive control of
nonlinear process systems,” AIChE J., vol. 56, pp. 2137–2149, 2010.
[12] Y.Tipsuwan andM.Chow,“Controlmethodologies in networkedcon
trol systems,” Control Eng. Pract., vol. 11, pp. 1099–1111, 2003.
[13] G.P. Liu, Y. Xia, J. Chen, D. Rees, and W. Hu, “Networked predic
tive control of systems with random networked delays in both forward
and feedback channels,” IEEE Trans. Ind. Electron., vol. 54, no. 3, pp.
1282–1297, Jun. 2007.
[14] J. Liu, D. M. de la Peña, P. D. Christofides, and J. F. Davis, “Lya
punovbased model predictive control of nonlinear systems subject
to timevarying measurement delays,” Int. J. Adaptive Control Signal
Processing, vol. 23, pp. 788–807, 2009.
[15] L. Grüne, J. Pannek, and K. Worthmann, “A networked unconstrained
nonlinear MPC scheme,” in Proc. ECC’09, Budapest, Hungary, 2009,
pp. 371–376.
Page 7
534IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 2, FEBRUARY 2012
[16] E. Franco, L. Magni, T. Parisini, M. M. Polycarpou, and D. M. Rai
mondo, “Cooperative constrained control of distributed agents with
nonlinear dynamics and delayed information exchange: A stabilizing
recedinghorizon approach,” IEEE Trans. Autom. Control, vol. 53, no.
1, pp. 324–338, Feb. 2008.
[17] J. Liu, D. M. de la Peña, and P. D. Christofides, “Distributed model
predictive control of nonlinear systems subject to asynchronous and
delayed measurements,” Automatica, vol. 46, pp. 52–61, 2010.
[18] Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse Lyapunov
theorem for robust stability,” SIAM J. Control Optim., vol. 34, pp.
124–160, 1996.
[19] P. D.Christofides andN. H. ElFarra,Control of Nonlinearand Hybrid
Process Systems: Designs for Uncertainty, Constraints and TimeDe
lays.Berlin, Germany: SpringerVerlag, 2005.
[20] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Compu
tation.Belmont, MA: Athena Scinetific, 1997.
[21] D. M. de la Peña and P. D. Christofides, “Lyapunovbased model pre
dictivecontrolofnonlinearsystemssubjecttodatalosses,”IEEETrans.
Autom. Control, vol. 53, no. 9, pp. 2076–2089, Oct. 2008.
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 (email: 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)
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