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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004
Model Predictive Control of Continuous-Time
Nonlinear Systems With Piecewise
Constant Control
Lalo Magni and Riccardo Scattolini
Abstract—A new model predictive control (MPC) algorithm for
nonlinear systems is presented. The plant under control, the state
and control constraints, and the performance index to be minimized are described in continuous time, while the manipulated
variables are allowed to change at fixed and uniformly distributed
sampling times. In so doing, the optimization is performed with
respect to sequences, as in discrete-time nonlinear MPC, but the
continuous-time evolution of the system is considered as in continuous-time nonlinear MPC.
Index Terms—Nonlinear control, nonlinear model predictive
control (MPC), sampled systems.
I. INTRODUCTION
T
HE extraordinary industrial success of model predictive
control (MPC) techniques based on linear plant models,
see, e.g., the survey paper [20], motivates the development of
MPC algorithms of nonlinear systems. Nowadays, there are
many theoretical results, see [17] and [12], as well as industrial
applications, see [21], which witness that MPC for nonlinear
systems is going to have a diffusion and popularity similar to
the one achieved by MPC algorithms for linear systems.
MPC methods for nonlinear systems are developed by assuming that the plant under control is either described by a
continuous-time model, see [3], [9], [10], [15], [16], and [18], or
by a discrete-time one, see [5], [11], and [13]. A continuous-time
representation is much more natural, since the plant model is
usually derived by resorting to first principles equations, but it
results in a more difficult development of the MPC control law,
which in principle calls for the solution of a functional optimization problem. As a matter of fact, the performance index to be
minimized is defined in a continuous-time setting and the overall
optimization procedure is assumed to be continuously repeated
after any vanishingly small sampling time, which often turns
out to be a computationally intractable task. On the contrary,
MPC algorithms based on a discrete-time system representation
are computationally simpler, but require the discretization of
the model equations, so that they rely from the very beginning
on an approximate system representation. Moreover, the performance index to be minimized as well as the state constraints
Manuscript received February 27, 2002; revised May 7, 2003 and January 30,
2004. Recommended by Associate Editor A. Bemporad. This work was supported in part by the MURST Project ”New techniques for identification and
adaptive control of industrial systems.”
L. Magni is with the Dipartimento di Informatica e Sistemistica, Universita’
di Pavia, 27100 Pavia, Italy (e-mail: [email protected]).
R. Scattolini is with the Dipartimento di Elettronica e Informazione, Politecnico di Milan, 20133 Milan, Italy (e-mail: [email protected])
Digital Object Identifier 10.1109/TAC.2004.829595
only consider the system behavior in the sampling instants, so
ignoring the intersample behavior, which in some cases could be
significant in the evaluation of the control performance.
In this paper, a different approach is taken, which accounts
fully for the hybrid nature of sampled data control systems.
The plant under control, the state and control constraints and
the performance index to be minimized are described in continuous-time, while the manipulated variables are allowed to
change at fixed and uniformly distributed sampling times. In so
doing, one has to deal with the optimization with respect to sequences, as in discrete-time nonlinear MPC, while taking into
account the continuous-time evolution of the system. The stability properties of the algorithm are established by including in
the cost function a penalty and a terminal constraint on the state
at the end of the prediction horizon, according to well-known
results, see [17]. However, the proof of stability is not trivial
due to the sampled-data nature of the problem and the use of
piece-wise constant signals. A similar approach was already
taken in [7], where the sampling mechanism was explicitly considered, for the solution of a regulation problem, but a functional optimization problem had to be solved iteratively. Piecewise constant signals have also been used in [1] and [8] for the
design of reference governors. Preliminary results concerning
the hybrid approach considered in this paper are reported in [14].
All the proofs are gathered in an Appendix to improve readability.
II. PROBLEM STATEMENT AND PRELIMINARY RESULTS
,
denotes the EuIn the paper, for any vector
clidean norm in
,
, where
is an arbitrary Hermitian matrix, denotes the weighted norm. For any
and
denote the largest
Hermitian matrix ,
and the smallest real part of the eigenvalues of the matrix ,
stands for the induced 2-norm of .
respectively and
denotes the closed ball of radius defined with the weighted
,
.
norm , i.e.,
Consider a plant described by the nonlinear continuous-time
dynamic system
(1)
where
is the state,
is the input,
and
is a
function of its arguments. The state and control
variables are restricted to fulfill the following constraints:
0018-9286/04$20.00 © 2004 IEEE
(2)
MAGNI AND SCATTOLINI: MODEL PREDICTIVE CONTROL OF CONTINUOUS-TIME NONLINEAR SYSTEMS
where
and are compact subsets of
and
, respectively, both containing the origin as an interior point. The movefor a conment of (1) from the initial time and initial state
trol signal
is denoted by
.
,
Given a suitable sampling period , and letting
nonnegative integer, be the sampling instants, the goal is to determine a “sampled” feedback control law for the computation
of a piecewise constant control signal which exponentially stabilizes the origin of the associated closed-loop system.
For the solution of this regulation problem, the following preliminary assumption is introduced.
Assumption 1: Letting
901
be called feasible hereafter. Besides, one can also wish to find
a control law (3) in order to enlarge the maximal output admissible set guaranteed by a given feasible control law and to optimize a given performance index.
Let us now suppose that a feasible control law (3) satisfying
the following assumption is known.
is a
function
Assumption 2: The feasible control law
with Lipschitz constant .
For this control law, an associated sampled output admissible
set can be computed as follows. First, define the linearization of
(1) at the origin
(5)
Then, introduce the discretization of (5) given by
(6)
the pair
is stabilizable.
Given a generic sampled feedback control law
with
(3)
with
, the description of the hold mechanism implicit
in (3) calls for a state augmentation. Letting
, the closed-loop system (1)–(3) is
(4)
and its movement from the initial time and initial state
is denoted by
With reference to the closed-loop system (4), define the following sets.
Definition 1: A sampled output admissible set associated
such that for all
,
to (4) is a set
,
,
,
,
. In
is a state invariant set, associated to the
other words,
closed-loop system (4), defined at the sampling instants and
such that: i) the state and control constraints (2) are satisfied in
all the future continuous-time instants, and ii) the regulation
problem is asymptotically solved. The (unique) maximal
is defined as the union of
sampled output admissible set
all sampled output admissible sets.
Definition 2: An output admissible set associated to (4)
such that for all
,
is a set
, where is the closest sampling time in
,
,
. In
the future,
other words,
is a set, defined at any continuous-time
instant , of states of the closed-loop system (4) such that:
i) the state of (1) at the closest sampling time in the future
, and ii) the state and control constraints (2)
belongs to
are satisfied in all the future continuous-time instants. The
is defined as
(unique) maximal output admissible set
the union of all output admissible sets.
The regulation problem can now be formally stated as the
problem of finding a sampled control law (3) such that its maximal output admissible set is non empty. Such a control law will
Finally, let
In view of Assumption 2, it is then easy to show that the
of the linearized
closed-loop matrix
discrete-time system (6) is Hurwitz and the following result
holds.
be a feasible control law, suppose that
Lemma 1: Let
Assumptions 1 and 2 are satisfied and consider a positive–defsuch that
inite matrix and two real positive scalars and
. Define by the unique symmetric positive–definite solution of the following Lyapunov equation:
(7)
where
and
(8)
Then, there exist two constants
specifying a neighborhood
and
of the origin of the form
(9)
such that
i)
ii)
,
,
;
(10)
iii)
is a sampled output admissible set for (4);
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004
iv)
is a positive–definite function decreasing
in the sampling times along the trajectory of (4).
Remark 1: An obvious way to determine a feasible sampled
control law is to choose a suitable , to consider the linearization of (1) around the origin and the sampled linear model described by (6) and to synthesize, with any standard linear control
synthesis technique, a linear control law
(11)
such that
is discrete-time Hurwitz. Provided that
defined in (6) is discrete-time Hurwitz, an even simpler
choice is to set equal to zero in (11). However, the maximal
output admissible set associated to these linear sampled control
laws is in general quite small due to their local nature and their
performance are likely to be improved by taking into account
the nonlinearity of the system under control.
where the terminal penalty
is selected as
The minimization of (13) must be performed under the following constraints:
;
i) the state dynamics (1) with
with given by (12);
ii) the constraints (2),
iii) the terminal state constraint
.
According to the receding horizon approach, the state-feedback MPC control law is derived by solving FHOCP at every
sampling time instant , and applying the constant control
,
where
is the
signal
. In so doing,
first column of the optimal sequence
one implicitly defines the sampled state-feedback control law
(14)
III. SAMPLED MPC CONTROL LAW
,
Let us suppose that a feasible sampled control law
henceforth called the auxiliary control law, is known together
with an associated sampled output admissible set and the Lyapunov function both given in Lemma 1. It is now shown how
MPC allows one to extend the maximal output admissible set of
and to improve the control performance by minimizing a cost
function suitably chosen by the designer.
and the control seTo this end, given the sampling time
quence
with
, define the Finite Horizon piece-wise constant
control signal
(12)
where
and
.
the signal
in the
Moreover, denote by
.
interval
For (1), the MPC control problem here considered is based
on the solution of the following.
Finite Horizon Optimal Control Problem (FHOCP)
Given the sampling time , the control horizon , the pre, two positive–definite matrices
diction horizon ,
and , a feasible auxiliary control law
satisfying Assumpgiven in Lemma 1
tion 2, the matrix and the region
with
and
, at every sampling
, the perfortime instant minimize, with respect to
mance index
(13)
In order to establish the properties of the control law (14),
first let
be the movement of (4) with
. Then, define the
following sets.
be the set of states
Definition 3: Let
of system (1) at the sampling times such that there exists a
for FHOCP.
feasible control sequence
Definition 4: Let
be the set of states
such that for all
,
, where is the closest sampling time in the future,
,
,
.
The main stability results of the proposed MPC algorithm can
now be stated.
Theorem 2: Under Assumptions 1 and 2
i) the origin is an exponentially stable equilibrium point
for the closed-loop system formed by (1) and (14) with
;
maximal output admissible set
,
;
ii)
iii)
,
;
such that
,
iv) there exist a finite
.
Remark 2: The use of different control ( ) and prediction
( ) horizons is already well known in MPC for linear systems,
see the popular GPC algorithm [4]. As discussed in [13], it is
even more important in nonlinear MPC, where the need to reduce the size of the optimization problem is of crucial importance for computational reasons.
Remark 3: In Remark 1, it has been suggested to derive the
auxiliary control law by means of linearization and Lemma 1
provides a systematic tool to derive the terminal penalty and
the terminal inequality constraint. However, it is possible to
relax Assumptions 1 and 2 and to consider any other feasible
MAGNI AND SCATTOLINI: MODEL PREDICTIVE CONTROL OF CONTINUOUS-TIME NONLINEAR SYSTEMS
piecewise constant control law provided that an associated possuch that
itive–definite function
903
out to be substantially greater than the one implied by the proposed solution. In fact, the maximum integration time used
in the integration of the state (1) can be (and is usually) chosen
much smaller than the sampling time required for the solution
of the MPC optimization problem.
IV. CONCLUSION
and a sampled output admissible set associated to it are known.
In particular, this allows one to include in this framework the
results of [7], which show how MPC can be used to control
nonholonomic systems. On the contrary, in [7] it is required that
As is well know for linear systems too, this condition is not
fulfilled by the terminal penalty derived in Lemma 1.
Remark 4: According to [5], the terminal weight in (13) can
be dropped with the terminal inequality constraint iii) in FHOCP
provided that
. This can appear to be only of theoretical interest, however it is in practice sufficient to consider
such that the settling time of the system (1) is
a value of
completely included into the prediction horizon. This is what is
usually done in many practical implementations of linear and
nonlinear MPC.
In the FHOCP optimization problem, continuous-time state
constraints are considered. It can appear that this approach is
only conceptual, because any numerical implementation needs
a time discretization and the constraints satisfaction can be
checked only in the integration time instants. However, this is
not a significant limitation; in fact, letting be the (maximum)
integration step used in the optimization phase to simulate the
plant (1) with the control signal (12), the following result holds.
Theorem 3: Let
if: a)
and b)
,
then
.
From this result it is clear that one can choose the maximum
integration step and a more conservative discrete-time state
so as to guarantee continuous-time
constraint (defined by
state constraint satisfaction. More precisely, given and such
, a nonnegative integer , a constant integration
that
, condition ii) in the FHOCP can be replaced by
step
The intrinsic characteristics of sampled data control systems
have been explicitly considered in this paper for the development of an MPC algorithm for nonlinear systems. The main
peculiarities of the proposed method are related to the piecewise nature of the control signal and to the use of different
control and prediction horizons. These two features lead to a
tractable optimization problem, where the cost minimization is
performed with respect to sequences and the number of future
control moves to be selected can be small.
Notably, once a stabilizing auxiliary control law is known,
the method here proposed can only improve its performance
with respect to the adopted cost function. This renders more
attractive the approach also when a stabilizing control law is
already available.
APPENDIX
A. Proof of Lemma 1
i) From the smoothness of and and recalling that
,
, it follows that there exist
and
,
such that
,
,
. Thus, let
.
, the inii) Letting
equality (10) is equivalent to
(15)
From (7), it is easy to see that inequality (15) is equivalent to
Moreover, from the definition of
The use of a more conservative constraints set has already been
proposed for linear systems in [2].
Remark 5: An alternative approach to the solution of the
MPC problem consists in reformulating it in a completely discrete-time framework using the sampling time . In this way,
also the state constraints (2) should be modified according to
the results of Theorem 3, but the conservatism introduced turns
it follows that
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004
Define
, then (16)
becomes (17), as shown at the bottom of the page. Define
and
, given in (8),
as
. Then, by the
of
and there exists
definition of ,
such that the inequality (18) holds
[see (9)],
which implies that (10) holds as well. Moreover, from (10) it
follows that,
(19)
Note that
and
are finite because
(17) is satisfied provided that
. Then,
so that
implies
. This fact with ( ) means that
is a sampled
output admissible set associated to (4). Finally, from (19) it
iv) is satisfied.
follows that
B. Proof of Theorem 2
(18)
In fact
is the maximal
In view of Definitions 2 and 4, if
then
sampled output admissible set of (4) with
is the maximal output admissible set of (4) with
. Note in fact that, from Definition 4 it follows
the constraints are not satisfied for
that
and/or
. But in view of
cannot belong to any output
Definition 2 this means that
.
admissible set of (4) with
First of all, from Lemma 1 it follows that
is
and
are
nonempty, then also
is the maximal
nonempty. Let us now show that
sampled output admissible set for (1) and (14). In fact, letting
and the associated solution
of the FHOCP at time , a feasible solution at
for the FHOCP is
time
(20)
where
Moreover, in view of the zero-order-hold sampling of (5), given
in (6),
is formed by terms of of order higher than one so
as
. Similarly, considering the definition
that
,
is the optimal control signal
computed at time and
is the value of the state of the MPC closed-loop system at time
. Then, by definition,
and, in view of
constraints ii) of the FHOCP, (2) are satisfied along the trajec. Finally,
tory of (4) with
the MPC control law is not defined so that a sample output
cannot exist.
admissible set larger than
Let us now show that the origin is an asymptotically stable
equilibrium point for the closed-loop system (1), (14). To this
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MAGNI AND SCATTOLINI: MODEL PREDICTIVE CONTROL OF CONTINUOUS-TIME NONLINEAR SYSTEMS
end, define
that
•
, as shown at the top of the page, and note
is bounded
. Moreover
and, since
905
and are positive–definite matrices, if
, both
and
are bounded. As shown in [19], these facts prove that
(21)
• At time
given by (20) is a (suboptimal) feasible solution for the new FHOCP so that
As for the exponential stability property, it easily follows by
analyzing the linearized closed-loop system, see [6].
The proof of ii)–iv) can be derived as in [13, Th. 6].
C. Proof of Theorem 3
First, note that
exists in view of the smoothness of and
,
and
the compactness of ,
because
. Finally, suppose that there exist some times
such that
and call
the first time such that
. Then,
,
and then
(22)
where
and this contradicts the assumption that there exists some times
such that
.
REFERENCES
From Lemma 1 and the conditions
,
, introduced in the formulation of the FHOCP, it
follows that
(23)
In conclusion, using (21) when
(23) when
,
,
,
, and
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 2004
Lalo Magni was born in Bormio (SO), Italy, in 1971.
He received the Laurea degree (summa cum laude) in
computer engineering from the University of Pavia,
Pavia, Italy, in 1994 and the Ph.D. degree in electronic and computer engineering in 1998.
Currently, he is an Assistant Professor with the
University of Pavia. From October 1996 to February
1997, and in March 1998, he was with CESAME,
Universitè Catholique de Louvain, Louvain La
Neuve, Belgium. From October to November 1997,
he was at the University of Twente, Twente, The
Netherlands, with the System and Control Group in the Faculty of Applied
Mathematics. In 2003, he was a Plenary Speaker at the 2nd IFAC Conference
“CONTROL SYSTEMS DESIGN” (CSD’03). His current research interests
include nonlinear control, predictive control, receding-horizon control, robust
control and process control. His research is testified by about 25 papers
published in the main international journals of the field.
Dr. Magni was Guest Editor for the Special Issue ”Control of Nonlinear Systems with Model Predictive Control” in the International Journal of Robust
and Nonlinear Control. He also serves as an Associate Editor for the IEEE
TRANSACTIONS ON AUTOMATIC CONTROL.
Riccardo Scattolini was born in Milan, Italy, in
1956. He received the Laurea degree in electrical
engineering from the Politecnico di Milan, Milan,
Italy, in 1979.
He is currently a Professor of Process Control
with the Politecnico di Milan. During the academic
year 1984/1985, he was with the Department of
Engineering Science, Oxford University, Oxford,
U.K. He also spent one year working in industry
on the simulation and control of chemical plants.
His current research interests include predictive and
robust control theory, with applications to automotive engine control and power
plants control.
Dr. Scattolini was awarded the Heaviside Premium of the Institution of Electrical Engineers (IEE) in 1991.