546
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
Model Predictive Control: For Want of a Local
Control Lyapunov Function, All is Not Lost
Gene Grimm, Michael J. Messina, Sezai E. Tuna, and Andrew R. Teel, Fellow, IEEE
Abstract—We present stability results for unconstrained discrete-time nonlinear systems controlled using finite-horizon model
predictive control (MPC) algorithms that do not require the
terminal cost to be a local control Lyapunov function. The two key
assumptions we make are that the value function is bounded by a
function of a state measure related to the distance of the state
to the target set and that this measure is detectable from the stage
cost. We show that these assumptions are sufficient to guarantee
closed-loop asymptotic stability that is semiglobal and practical
in the horizon length and robust to small perturbations. If the
functions,
assumptions hold with linear (or locally linear)
then the stability will be global (or semiglobal) for long enough
horizon lengths. In the global case, we give an explicit formula
for a sufficiently long horizon length. We relate the upper bound
assumption to exponential and asymptotic controllability. Using
terminal and stage costs that are controllable to zero with respect
to a state measure, we can guarantee the required upper bound,
but we also require that the state measure be detectable from the
stage cost to ensure stability. While such costs and state measures
may not be easy to construct in general, we explore a class of systems, called homogeneous systems, for which it is straightforward
to choose them. In fact, we show for homogeneous systems that the
associated
functions are linear, thereby guaranteeing global
asymptotic stability. We discuss two examples found elsewhere
in the MPC literature, including the discrete-time nonholonomic
integrator, to demonstrate our methods. For these systems, we give
a new result: They can be globally asymptotically stabilized by
a finite-horizon MPC algorithm that has guaranteed robustness.
We also show that stable linear systems with control constraints
can be globally exponentially stabilized using finite-horizon MPC
without requiring the terminal cost to be a global control Lyapunov function.
Index Terms—Discrete-time systems, homogeneous systems,
nonlinear model predictive control (MPC).
systems initially required terminal equality constraints to ensure stability of closed-loop systems employing the MPC algorithm [11]. The computational complexity associated with such
constraints led to the use of inequality constraints [16]. Drawbacks of these methods include the computational complexity
associated with the terminal constraints (equality or, less so, inequality) and their potential lack of robustness in general (see
[4]). This complexity was ameliorated by imposing properties
on the terminal cost. In [3], a general framework is presented
that balances the need to impose terminal inequality constraints
with requirements on the terminal cost in order to achieve global
asymptotic stability for nonlinear systems without stabilizable
linearizations. In [9], terminal constraints are ignored and the
terminal cost is taken to be a local control Lyapunov function
(CLF) for the system, whose linearization is assumed to be stabilizable. With such a choice and for a horizon length chosen to
be sufficiently long, the terminal constraints of [16] are automatically satisfied, robust semiglobal stability is guaranteed, and the
computational complexity is substantially reduced. A drawback
of this approach is its need for an a priori computation of a CLF,
though it is trivial to find a CLF when the linearization is stabilizable. More recent research has sought to obviate the need for
a CLF or any terminal cost at all. In [8], it is shown, in the case
of unconstrained continuous-time systems with stabilizable linearizations, that when MPC is employed with a general terminal
cost (including a terminal cost of zero) there is a sufficiently long
horizon length for which exponential convergence is guaranteed
from any compact set of initial conditions.
B. Contribution
I. INTRODUCTION
A. Background
T
HE stability of systems that employ finite-horizon model
predictive control (MPC) has been the subject of significant research; see [14] for a survey. Results for general nonlinear
Manuscript received November 1, 2002; revised November 26, 2003 and January 22, 2005. Recommended by Associate Editor A. Bemporad. This work was
supported in part by the Air Force Office of Scientific Research under Grants
F49620-00-1-0106 and F49620-03-1-0203, and by the National Science Foundation under Grants ECS9988813 and ECS0324679.
G. Grimm is with the Raytheon Company, Space and Airborne Systems, El
Segundo, CA 90245 USA (e-mail: [email protected]).
M. J. Messina, S. E. Tuna, and Dr. A. R. Teel are with the Center for
Control Engineering and Computation, Department of Electrical and Computer
Engineering, University of California at Santa Barbara, CA 93106-9560 USA
(e-mail: [email protected]; [email protected]; [email protected]).
Digital Object Identifier 10.1109/TAC.2005.847055
This work provides insights into the stability properties of discrete-time closed-loop systems resulting from unconstrained1 finite-horizon MPC without terminal constraints using semidefinite terminal and stage cost functions. When stabilizing an equilibrium point, we do not assume that a linearization exists or is
stabilizable. Our results can then be seen as discrete-time counterparts of those in [8] for the class of systems investigated in
[3]. Moreover, we provide stability assurances when stabilizing
general sets, such as periodic orbits, where it can be quite challenging to synthesize a local CLF. Since the stage cost need not
be positive definite, the value function cannot be used directly
as a Lyapunov function. Indeed, it typically increases along trajectories for values of the state that make the stage cost small.
1Explicit state constraints are not considered here, but control constraints are
allowed. We do not consider state constraints because it would obscure the results presented here since it is difficult to guarantee that closed-loop asymptotic
stability is robust when using MPC with state constraints; see [4] for illustrative
examples. State constraints will be considered in future work.
0018-9286/$20.00 © 2005 IEEE
GRIMM et al.: MPC: FOR WANT OF A LOCAL CONTROL LYAPUNOV FUNCTION
547
Nevertheless, a continuous closed-loop Lyapunov function can
be constructed by combining the value function with a function that characterizes the assumed detectability property. Since
the Lyapunov function is continuous, the closed-loop asymptotic stability has some robustness. For more insight into the relationship between continuous Lyapunov functions and robustness, see [12].
We make two key assumptions. We assume that the value
function, for all horizon lengths, is globally bounded by a
function of a state measure related to the distance of the state
from the target set. We also assume this state measure to be
detectable from the stage cost. In general, we provide closedloop stability results that are semiglobal and practical in the
horizon length, that is, as the horizon length increases, the basin
of attraction becomes arbitrarily large and the neighborhood
of the target set to which trajectories converge becomes arbifunctrarily small. When the assumptions hold with linear
tions, the results are global, that is, there exists a sufficiently
long horizon length such that global asymptotic stability results.
When the state measure from the assumptions is a -norm, the
result is global exponential stability. If the assumptions hold
functions, then the result is semiglobal
with locally linear
in general with local exponential stability when -norm state
measures are used. For the global case in general, we provide
an explicit formula for horizon lengths sufficient to guarantee
stability. This result applies to general nonlinear homogeneous
systems, which include the nonholonomic integrator and the
main example of [15]. Our results guarantee that the origin of
these systems can be globally asymptotically stabilized by a
finite-horizon MPC algorithm that has guaranteed robustness.
Using a quadratic state measure for exponentially stabilizable
linear systems with bounded inputs, we provide a result that
seems to be missing from the literature: for a sufficiently long
horizon length, finite-horizon MPC results in global exponential stability—not just semiglobal asymptotic stability—even
when the terminal cost is not a global CLF. We also recover
a semiglobal asymptotic stability result for linear systems for
which the system matrix has no eigenvalues with a modulus
greater than one.
The horizon lengths we report for guaranteed stability are typically longer than those that would be required by methods that
use terminal constraints (for example, compare our Example 1
to how this example is treated in [15]). There are several reasons why one might want to use these longer horizon results.
First, it can be difficult to satisfy the assumptions required to
use terminal constraints. Second, the optimization problem that
arises from using terminal constraints may be more conservative, since modifying the costs to ensure stability may compromise performance. Perhaps most significant, and certainly least
well understood, is the fact that the asymptotic stability that results from using terminal constraints and short horizon lengths
may have absolutely no robustness; see [4] for examples.
The paper is organized as follows. The next subsection gives
definitions required for our results. Section II begins with the
definition of the problem under investigation including our assumptions and then states our main results and gives examples.
Section III has a more in-depth discussion of the main upper
bound assumption. Section IV contains the proofs of our main
results and is followed by our conclusions in Section V. An Appendix is included that gives technical lemmas used to prove
our main results.
C. Preliminaries
, we use the notation
(
) to refer to
For any
(
).
the subset of real numbers
The obvious parallel notation applies to the set of integers .
for
.
We define as
A function
is said to belong to class
if it is continuous, nondecreasing, and zero at zero. It is said
) if it belongs to class and is
to belong to class (
(
) if
strictly increasing. It is said to belong to class
it belongs to class and is unbounded. A function
is said to belong to class
(
) if for each
is of class and for each fixed
tends to
fixed
zero at infinity.
via
We measure the distance to a set
, where
is any norm. This allows us to study
regulation to sets more general than a point.
be closed. A function
is said to
Let
be a proper indicator function for if there exist
such that
for all
.
II. PROBLEM STATEMENT AND MAIN RESULTS
A. The MPC Problem
We consider the discrete-time system
or, more succinctly
(1)
where the state
and the control input
.
We consider nonlinear in general and continuous in both arguments and the set closed. We denote by a control input
; if
for all
sequence enumerated as
, then is said to be an admissible control input sequence. All of the input sequences we consider are
assumed to be admissible and we drop this term to avoid clutter.
The solution of (1) steps into the future, starting at initial condition , and under the influence of a control input sequence is
. It follows that
. We similarly
denoted
denote the solution of a closed-loop system with
. On
occasion, we need infinite-length control input sequences and
; we always assume that
denote these as
these sequences are admissible.
We use the cost function
(2)
and the stage cost
where the terminal cost
can be semidefinite. We define the value
function, which represents the optimal value of (2) for a given
initial condition, as
(3)
which is to be understood as a constrained optimization problem
over the set of admissible control input sequences. Whenever the
infimum is achieved by an admissible control input sequence
then the MPC feedback law
is a function that
548
returns the first control input of such an admissible control input
, where
sequence given the current state, that is,
.
We now discuss our standing assumptions (SAs) which will
hold throughout this paper.
SA1: The functions and are continuous.
SA2: Either is bounded or for each compact set , real
number , and positive integer , there exists
such that
, all satisfying
satisfy
for each
for
.
, the infimum in
SA1 and SA2 guarantee that for each
(3) is achieved by some admissible control input sequence, that
such that
(see
is, there exists
is contin[10]). Moreover, the assumptions guarantee that
uous (see [22, Th. 1.17]).
We use a continuous, positive–definite function
as a state measure. If we are interested in stabilizing the
are
or
. A sucorigin, then possible choices of
cessful control design will drive solutions to (or nearly to) the
. If we are interested in stabilizing the
set
set
, then possible choices of
are
or
.
Since may be semidefinite, we make an assumption about the
detectability of from , a concept we define now.
Definition 1: Consider the system (1), and functions
,
,
, and
. The function is said to be detectable from with
if there exists a continuous function
respect to
such that
for all
and
.
SA3: For (1), the continuous function is detectable from
with respect to some
.
Remark 1: If there exists
such that
for all
and
, then we can choose
to show that is detectable from with respect
for any pair
satisfying
to
for all
. One such pair is
.
SA4: There exists
such that
for
and all
.
all
Conditions sufficient to guarantee that SA4 holds are provided in Section III. We note that our standing assumptions
are global and consequently our control results are global or
semiglobal. The standing assumptions can be made regional resulting in regional control results.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
, where the function
is
nondecreasing and unbounded. Furthermore, there exist funcsuch that for all
,
tions
and there exists
such that
and
.
Theorem 1: Let
,
be con, there exstructed according to Algorithm 1. For each
ists a continuous function
such that for the
and all
closed-loop system
(4)
(5)
If Assumption 5 holds, then for each
, there exists a
continuous function
that for the closed-loop system
and all
, satisfies (4) and
(6)
Algorithm 1: Construction of Theorem 1 functions. Given
,
, and
from SA3, from SA4, and
functions
and
(if applicable) from Assumption 5.
Step 1. If
select
for all
for
lowing.
Step 2.
. Otherwise, define the fol-
B. Main Results
Although one of the major points of this paper is that stability can be guaranteed with no particular requirements on the
terminal cost, we will consider the case when the terminal and
state costs satisfy the following CLF-like assumption. Note that
it is not a standing assumption and will be explicitly invoked
when necessary.
Assumption 5: For the system (1) and a continuous function
, the terminal cost can be decomposed as
for
, then
.
GRIMM et al.: MPC: FOR WANT OF A LOCAL CONTROL LYAPUNOV FUNCTION
549
According to Theorem 1,
will serve as a Lyapunov function for the closed-loop system if, by picking the horizon length
sufficiently large, we can make
or
sufficiently small.
Since
and
are
functions, this can always be done.
Remark 2: If and are not continuous, then the bounds
is not guaranteed to be continuous.
in Theorem 1 hold but
Since the MPC feedback law may be discontinuous, the importance of having a continuous Lyapunov function for the closedloop system cannot be overstated; it guarantees nominal robustness [12]. See [4] for examples where the MPC feedback law is
discontinuous, but no continuous Lyapunov function exists, and
the system is, therefore, nonrobust. The results in [12] are significant for MPC in particular, as previous robustness results, such
as [13] and [19], have restrictive continuous differentiability requirements on the control law and value function.
Using Theorem 1, we can assert semiglobal practical stability
in general, as the following corollary states.
and for each
Corollary 1: There exists
, there exists
such that for all
and
, the solutions
of the system
satisfy
, then the set
is globally
if is of the form
exponentially stable.
,
Corollary 3: Suppose SA3 holds with
, and
and SA4 holds with
, where
and
. Suppose either
1)
, or
2)
Assumption 5 holds with functions
,
, and
is such that
for
.
,
such that the solutions
Then, there exist
of
satisfy
for all
. Moreover, given , let be a proper indicator
and for each
function for . Then, there exists
there exists
such that for all
and
, the solutions
of the system
satisfy
for all
.
It is natural to ask whether using a finite horizon will ever result in semiglobal or global asymptotic stability. The next corollary states that the set is semiglobally asymptotically stable
functions and
when SA3 and SA4 hold with locally linear
is a proper indicator function for .
that
,
Corollary 2: Given , suppose that for
,
(from SA3) satisfy
,
,
and (from SA4) satisfies
, where
and
and
. Then, there exists
and for each
, there exists
such that for all
and
, the solutions
of the system
satisfy
for all
. Moreover, given , let be a proper indicator
function for . Then, there exists
and for each
there exists
such that for all
and
, the solution
of the system
satisfies
for all
.
If the detectability and upper bound assumptions hold with
linear
functions and is a proper indicator function for ,
then the set is globally asymptotically stable. Furthermore,
for all
. If, given
, is a proper indicator function
is globally asymptotically stable, that is, there
for , then
exists
such that
for all
and
. Moreover, if
and
then is
and
globally exponentially stable, that is, there exist
such that
for all
and
.
Corollary 4: Consider the system
,
,
where is compact and contains a neighborhood of the origin.
Let
,
, and
,
, and either
and
or
and
where
, with
detectable. Then, the following
hold.
1)
If
is stabilizable and has no eigenvalues with
modulus greater than one, then there exists
such that for all
, the MPC algorithm locally
exponentially and semiglobally asymptotically stabilizes the origin.
2)
If all of the eigenvalues of have modulus strictly less
, then the
than one and
MPC algorithm globally exponentially stabilizes the
origin.
Remark 3: The first case of Corollary 4 recovers the result
[20] of semiglobal asymptotic stability (although we allow for
semidefinite or ). The result can also be seen as allowing
performance improvement over the result of [1], which requires
a control Lyapunov function terminal cost. In regards to the
second case of Corollary 4, other authors, such as [21], have established global exponential stability for stable linear systems
with input constraints using finite-horizon MPC with a terminal
cost that is a global control Lyapunov function. To the authors’
knowledge, this case is the first result to establish global exponential stability for these systems employing MPC with a finite
horizon length and any quadratic semidefinite terminal cost.
C. Examples
Example 1: The discrete cubic integrator. Consider the
system (also considered in [15])
(7)
where
and
. Results in Section III-B.1 based on homogeneity constructively suggest that
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
desirable selections for , , and are
,
,
, with yet to be
determined but assumed to be nondecreasing and unbounded.
We note that and are continuous (for a given horizon), As, and is detectable
sumption 5 holds with functions
. From this choice of and
from with respect to
, we also see that SA2 holds, since for any compact set and
, all for which
, with
, are such
are bounded. Therefore, a control input sequence
that the
exists such that
. Given any ini, define the open-loop control
tial condition
input sequence
It can be verified that this control sequence drives the system to
if
and
the origin in four time steps. If we let
if
, where
, then it can be verified
for all
that
. Since all the standing assumptions hold with linear
functions, Corollary 3 (with
) guarantees that if
,
then the MPC feedback law globally asymptotically stabilizes
the origin. Moreover, since Assumption 5 holds, Corollary 3
) guarantees that if
, then the MPC feedback
(with
law globally asymptotically stabilizes the origin.
Example 2: The discrete nonholonomic integrator. Consider
the system (related to a system considered in [2])
(8)
and
.
where
System (8) is the exact sampled version (with period 1) of the
continuous time nonholonomic integrator
under a piecewise-constant control
input. The methods in Section III-B.1 constructively suggest
,
and
selecting
, where is yet to be determined but assumed to be nondecreasing and unbounded. We note that Asand and are consumption 5 holds with functions
tinuous. This system satisfies SA2, since for any compact set
and
, all for which
, with
,
are such that the are bounded, since from (8), bounded states
imply bounded controls. Therefore, there exists a control input
such that
. Remark 1
sequence
. As
shows is detectable from with respect to
before, we aim to establish an upper bound on the value func, we use
tion, given any initial condition
the control input sequence given in Table I.
if
and
if
, where
If we again let
, then it can be verified that
for all
. Since all of our standing assumpfunctions, Corollary 3 (with
)
tions hold with linear
then the MPC feedback law globally
guarantees that if
TABLE I
TRAJECTORY OF SYSTEM (8) USED TO DETERMINE A FUNCTION asymptotically stabilizes the origin. Moreover, since Assump) guarantees that if
tion 5 holds, Corollary 3 (with
then the MPC feedback law globally asymptotically stabilizes the origin.
Before moving on, we compare our MPC controller for the
discrete nonholonomic integrator with the contractive MPC
(CNTMPC) feedback law proposed by Kothare and Morari [2]
for the continuous time nonholonomic system
(9)
Using the globally invertible transformation
we can convert (9) into the continuous time nonholonomic integrator
. Since in
our example we use the exact sampled version of the continuous time nonholonomic integrator, the comparison of the two
examples is valid. The CNTMPC algorithm successfully drives
the state to a very small neighborhood of the origin, the desired
equilibrium point. However, due to lack of controllability at the
origin, the control input that the CNTMPC algorithm generates
gets larger as the states get closer to the origin. Hence, the algorithm must be terminated at some distance from the origin.
The problem stems from the fact that the stage cost used in
, a weighted norm of the state
the CNTMPC algorithm is
alone. Such a problem does not occur when the costs are chosen
according to our MPC methodology, which exploits the homogeneity properties of the system, as discussed in Section III-B.1.
Since we pick a homogeneous stage cost, the exponential decay
of the states passed through a homogeneous function is guaranteed with a fixed length of horizon; additionally, the control
input passed through a homogeneous function also decays exponentially (see Proposition 1 in Section III-B.1). Therefore, the
control input gets smaller as the state gets closer to the origin. In
fact, the contraction that is enforced by the CNTMPC algorithm
is guaranteed automatically for the horizon length we use when
we use homogeneous costs.
III. GUARANTEEING THE UPPER BOUND ON THE VALUE
FUNCTION
In this section, we discuss some of the technical issues that
arise due to SA4 and aim to interpret them in a natural system
theoretic way. We pay specific attention to the case when SA4
functions, which allows the use of Corolholds with linear
lary 3 to guarantee global stability.
GRIMM et al.: MPC: FOR WANT OF A LOCAL CONTROL LYAPUNOV FUNCTION
A. Exponential Controllability
We now introduce a generalized form of exponential controllability for functions.
,
. The
Definition 2: Consider the system
is said to be globally exponentially
function
if there exists
controllable to zero with respect to
a pair
such that for any initial condition
, there exists an infinite-length control input sequence
such that
(10)
for all
.
When the stage cost is globally exponentially controllable
, then SA4
to zero with respect to and
.
holds with linear function
B. Asymptotic Controllability
Definition 3: Consider the system
,
. The
is said to be globally asymptotically
function
if there exists
controllable to zero with respect to
a
such that for each initial condition
, there
such that
exists an infinite-length control input sequence
for all
.
Based on one of the main results in [12], for any system that
is globally asymptotically controllable to the set , there always
exists a that is a proper indicator function for and a function
such that is globally exponentially controllable to zero with
respect to ; furthermore, is detectable from with respect to
linear functions in
. In particular, the results in [12] show
that is a proper indithat given , there exists
and
cator for such that letting
for our MPC problem implies that (that is, ) is globally exponentially controllable to zero with respect to . Trivially, is
.
also detectable from (that is, ) with respect to
The following lemma will show that it is possible to construct a
function that is exponentially controllable to zero from a function that is asymptotically controllable to zero.
,
Lemma 1: [6] Suppose for the system
, that
is asymptotically controllable
and that
to zero with respect to
satisfies the conditions of Definition 3. Then, given
,
such that for each
, there
there exist
such that
exists an infinite-length control input sequence
for all
.
Lemma 1 guarantees that given any function
that is
asymptotically controllable to zero with respect to
, there
and
in
such that with
exists a pair of functions in
and
(with
) then is exponentially controllable to zero with
respect to , and the value function satisfies
for all horizon lengths
. Moreover, if
is
, then is dedetectable from with respect to
.
tectable from with respect to
551
In the next section, a systematic construction of , , and
is offered for homogeneous nonlinear systems. With this confunctions and thus
struction, SA3 and SA4 hold with linear
lead to globally stabilizing MPC feedback laws.
1) Homogeneity: In this section we show, for asymptotically
controllable homogeneous systems, how to construct a stage
functions recost and a terminal cost that yield the linear
quired to employ Corollary 3 and, thus, yield global stability of
the desired set. The stability of homogeneous systems has been
studied by numerous researchers; see, for instance, Grüne [5]
and references therein for continuous-time systems, and Hammouri and Benamor [7] and references therein for discrete-time
systems. Before stating our main result pertaining to homogeneous systems, we introduce some basic terminology regarding
homogeneity and dilations.
is an operator
Definition 4: A dilation
,
such that given any
with
,
.
is homogeneous of degree
Definition 5: A map
with respect to
if
.
Definition 6: A transition map
is homoif
geneous of degree with respect to the dilation pair
.
Notice that if the homogeneity of
is of degree 0
then
with respect to the dilation pair
. It follows that, given an infinite-length control
, the solution
for the system
input sequence
satisfies
for
.
all where
A general way to construct nonnegative homogeneous funcis as follows. For
tions of degree with respect to a dilation
define
,
. Then,
is homogeneous of degree
where
with respect to
, where
and
, that
. The following propois,
sition relates such homogeneous functions and the exponential
controllability discussed in Section III-A.
has the
Proposition 1: Suppose the system
following properties.
1)
is homogeneous of degree 0 with respect to
.
is asymptotically controllable to the
2)
origin.
for all
, where
.
3)
Then, given a positive–definite function
that is homogeneous of degree
with respect
, there exist constants
and
such
to
that for each there exists an infinite-length control
such that
input sequence
(11)
. Moreover, if Properties 1)–3) hold and
implies that
, then given
that is also homogeneous of degree with respect
such that the infiniteto , there exists
length control input sequence
also satisfies
for all
4)
(12)
for all
.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
Proposition 1 states that given an asymptotically controllable
homogeneous system, a homogeneous positive–definite function and a nonnegative homogeneous function , both of degree
, the function
is globally
exponentially controllable to zero with respect to . Therefore,
, then SA4 holds with
for some
if
and SA3 holds since is detectable from with respect to
.
2) Homogeneity Property of the Value Function and Optimal
Input Sequence:
Proposition 2: Consider the system (1). Suppose that is homogeneous of degree 0 with respect to the dilation pair
.
that is positive defThen given a function
inite in its first argument and homogeneous of degree with
, that is,
respect to the dilation pair
for all
, and a function
that
, let the state
is homogeneous of degree with respect to
and teminal costs of the cost function (2) be
and
. Then,
is homogeneous of degree with
; that is
. Morerespect to the dilation
over, if is a minimizing control input sequence for initial con,
is a minimizing control input
dition then for any
sequence for the initial condition
.
Proposition 2 is of practical importance. It implies that the
optimization algorithm can always be run on a constant closed
for instance, and the resultant control can then
set,
be dilated to make it correspond to the actual states. This could
be useful to prevent the need for truncation within computing
environments while dealing with very small (large) numbers.
that
notation
For all
. In what follows, we use the shorthand
and
.
, we write
(15)
SA3 requires that
is detectable from
with respect to
. We will use the corresponding
to guarantee
the desired properties on
.
for All
): We first write
Case 1 (
(16)
Since
IV. PROOFS
A. Proof of Theorem 1
Before we prove Theorem 1, we state and prove a similar
. Since may be semidefinite, we are prevented
result for
as a Lyapunov function, however the properties
from using
stated will be useful in the proof of Theorem 1.
,
be constructed acTheorem 2: Let
cording to Algorithm 1 and consider the closed-loop system
. For each
and all
,
satisfies
and
(13)
Moreover, if Assumption 5 holds, then for each
,
satisfies the previous bounds and
and all
(14)
Proof:
Bounds: The lower bound comes by the definition of
since
is the first term of a sum of nonnegative terms.
The upper bound is SA4.
. Given any
Statement (13): Here, we assume that
, let be any admissible control input sequence such
satisfies Definition 1,
and
hold. SA4 guarantees that
.
term on the left-hand
We rearrange (16) (and drop the
side, which is possible since all terms are positive) to get that
. Thus, there exists
such that
a
Now, choose
and note that for
, ranges from 2 to , as desired. Then
(17)
as
Combining (17) with (15) and comparing the terms to
constructed in Algorithm 1, shows that (13) is satisfied.
for Some
): Let
,
,
Case 2 (
,
, and
be defined as in Step 2 of Algorithm
and
, we can
1. Noting that
use Lemma 4 in the Appendix to guarantee
for all
, and, therefore
GRIMM et al.: MPC: FOR WANT OF A LOCAL CONTROL LYAPUNOV FUNCTION
SA4 guarantees
. Therefore
Moreover, if Assumption 5 holds, then for each
Additionally, the inequalities
and
hold since
satisfies the conditions of Definition
term on the left-hand side)
1. Thus, (again dropping the
Therefore, there exists a
553
and all
Thus,
and
constructed according to Step 1 of Algorithm
1 satisfy (4) and (5).
for Some
): Define
, ,
Case 2 (
according to Step 2 of Algorithm 1. Define
and
. Then,
and
. If
, then
. If
, then
.
Therefore
such that
. As before, choose
. Then, we have that
Combining (18) with (15) and comparing the terms to
as
constructed in Algorithm 1, shows that (13) is satisfied.
Statement (14): We will first establish a bound on the ingiven a bound on
and then show
cremental change in
. Using Assumption 5, we have that for
a bound on
It is evident that
and
. With
the aforementioned definitions, the proof of Theorem 2, Lemma
3 in the Appendix , and the definition of the functions in Step 2
and all
of Algorithm 1, it follows that for each
Moreover, if Assumption 5 holds, then for each
and all
(18)
Moreover
Thus,
. Thus with
defined
in Algorithm 1, (14) holds.
We now proceed with the proof of the main result.
Proof of Theorem 1: As before, we will use the
cor.
responding to SA3 to guarantee the desired properties on
. The
Also, we will use the shorthand notation
proof of this theorem is split into two cases: Case 1)
for all
, and Case 2)
for some
.
for All
): Define
Case 1 (
. Then,
and
. Using the results of Theorem
and all
2, we can write that for each
Thus,
and
constructed according to Algorithm 1 satisfy
(4) and (5).
is
Continuity: Since the value function is continuous,
and
are also continuous,
continuous by assumption, and
constructed according to either of the previous two cases is
continuous.
B. Proofs of Corollaries 1–3
For the proofs of Corollaries 1 and 2, we will make use of the
following lemma ([18, Prop. 2]).
and
Lemma 2: Consider the system
. Let
and
be such that
for all . Suppose
satisfy
,
for all
, and
for all
. Then, for all
554
the difference equation
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
the solution
satisfies
of
for all
, where
is the maximal solution of the
differential equation
with
.
Proof of Corollary 1: Consider
of
satisfy
Theorem 1. Let
,
, and
. Since
, we know that there exists
such that for all
,
. If Assumption 5
be such that
were to hold, we would let
and let
.
Otherwise, we just let
. Note that the particular choice
does not affect the functions used to bound trajectories
of
is useful for com(see later), but the choice of the lowest
putational reasons. From (5) (or (6), if applicable) we have that
and all
,
for all
Since
,
,
we have that for all
. Then, from Lemma 2,
whenever
, the solution
of the system
satisfies
and
From the supposition, we know that there exists
such that
. Then,
(since it must be an integer) and using (20) we have
Defining
that
and noting from the lower bound on
, we have that
Using the bounds on
yields that for any
(21)
. Choosing
and
in (21),
for all
we have the decrease condition for .
Case 2: Since Assumption 5 holds, Theorem 1 guarantees
there exists a
that satisfies (20) and for all
that for
(19)
for all
, where
with
defined in Lemma 2.
Now, consider the case that is a proper indicator function
. Since
were arbitrary
for with functions
in the previous calculations, the bound (19) also holds by approand replaced by
priate choice of with replaced by
. Then using the proper indicator bounds, we have that
, then
for
when
, where
with
as in
all
(19).
Proof of Corollary 2: Given
and , let
satisfy
. Let
be such that for all
,
.
. For
Now let
, we have, using the assumed linear functions,
and
that
. Lettherefore that
, we have that for
,
ting
for all
. Therefore, we can apply
Lemma 2 using
and the results follow as in the proof
. Note that, as in the proof of
of Corollary 1, but with
Corollary 1, if Assumption 5 were to hold, we may have been
, but the bounding functions would
able to choose a shorter
be the same.
Proof of Corollary 3:
Case 1: Since
, Theorem 1 guarantees that for
there exists a
such that for any
(20)
From the supposition, we then know that there exists
such that
. Therefor all
.
fore, we have that
By the same argument as before, this is sufficient to guarantee
the desired exponential property of .
Thus,
satisfies the decrease condition. If
is a proper
with functions
, then
indicator function for
for all
and
.
for all
Therefore,
and
. Letting
,
and, therefore, that the set
we have that
is globally asymptotically stable. If
, then
for all
and
and the set
is globally exponentially stable with constants
and
.
C. Proof of Corollary 4
SA1, SA2, and SA3 hold under the general assumptions of
the corollary and we prove this first.
SA1 Holds: The costs are quadratic functions.
SA2 Holds: The set is compact.
SA3 Holds: We show that there exist real numbers
and
such that is detectable from with respect to
.
: Let
be the smallest eigenvalue of (which
For
is positive since
). Since
, for all
, the inequality
holds.
Therefore, is detectable from with respect to
.
GRIMM et al.: MPC: FOR WANT OF A LOCAL CONTROL LYAPUNOV FUNCTION
For
and
: Since
is detectable, we can choose
that satisfy
, where
. Let
, where
is to be
determined. Then
(22)
where we have thrice used Young’s inequality,
for any
. We can choose
in (22) such that
,
555
for all
. Suppose instead that
. Let
satbe the smallest integer such that
isfy (23) for . Let
and let
. Note that, by
definition,
. Let
satisfy (24) for . Since is
such that
implies that
bounded, there exists
. Then, for
and any
we have
. If we
choose such that
, then we can state that for
For
, we have that
,
Now, pick
such that
. We then can
there exists an infinite-length control
state that for any
input sequence
such that
where
. Now, let
,
, and
be times
the maximum eigenvalue of . It follows that
.
.
Thus, is detectable from with respect to
For SA4, we prove each case separately.
SA4 Holds: Case 1: Using a result from [23], the assumpand for
tions allow us to guarantee that there exists
there exists an infinite-length control input seeach
such that
quence
for all
. We can then define
such that
and
for
. We thus satisfy our claim, and can state that there
such that
exists an infinite-length control input sequence
. Let
for
. Then
some
(23)
. Also from the assumptions, there exist
,
,
, and for each
there exists an infinitesuch that for all
length control input sequence
for all
where
for all
(24)
. Given , , ,
Let
such that (23) and (24) are satisfied for some infinite-length
control input sequences, we claim that there exists a
functhere
tion , that is locally linear, such that for each
such that
exists an infinite-length control input sequence
, for all
. Let
be
such that
. Let
be an integer-valued
for all
function with the property that
and
for all
. Note that, without loss of genfor all
and, since
erality, we can assume
there is a locally linear stabilizing controller, we can assume that
such that
for all
.
there exists
, suppose that
. Then, there exists
Now, given
such that
. Then, there exists
and
such that
for some
and all
.
SA4 Holds: Case 2: The eigenvalues of have modulus
exists such that
less than one, so a matrix
and
. Let be the maximum eigenvalue of
and
denote the state at time . Then
(25)
be such that
. Then, using (25) we can state
for all
and
.
Concluding the Proof of Case 1: Since is locally linear,
the detectability and upper-bound assumptions of Corollary
such that for all
3 hold locally. Therefore, there exists
, the closed-loop system is locally exponentially
stable. Since the conditions of Corollary 2 are met, we conclude
Let
that
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
that there exists and for each
there exists an
such that for all
and
, the
for all
closed-loop trajectories satisfy
. Thus, the origin of the closed-loop system is semiglobally asymptotically stable.
Concluding the Proof of Case 2: Since SA3 and SA4 hold
with linear
functions, Corollary 3 guarantees that if
, then there exist
and
such
satisfy
that the trajectories of
for all
and all
. Taking the square
. Thus, the
root of each side reveals
origin of the closed-loop system is globally exponentially stable.
D. Proof of Homogeneity Propositions 1 and 2
Proof of Proposition 1: The second assumption implies
that there exists a function
such that for each
,
there exists an infinite-length control input sequence
such that
for all
. Define
and
.
and
, there exists
With
an
such that
. Therefore, for each
there exists an infinite-length control sequence
such that
whenever
.
. We claim for each
Define
that there exists an infinite-length control input sequence
such that
for all
. We first show this for initial conditions in
and
then generalize the result using homogeneity. We have that
there exists a control input sequence of
for each
such that
for
length
and
. Given
all
, let a control input sequence
of length
some
be such that
for all
and
. If
we know that there exists an input sequence that will keep
the state at the origin for all times in the future and, hence,
is satisfied trivially.
the exponential bound on
For nonzero define
and
. Note
. Now, let
be the input sequence of length
that
such that
for
and
. Since
all
we have for all
the upper bound on
for all
one gets
. In addition
By induction, given
, for each
there
exists a control input sequence
of length
such that
for all
and
. Therefore, for each
there
such that
exists an infinite-length control input sequence
for all
.
Finally, we show that the exponential decay can be
attained for any initial condition. Given some nonzero
let
be such that
. Then, there exists an infinite-length control input sequence
such that
. To show this,
. Then
define
Therefore, (11) follows with
To prove (12) we begin by defining
. Define
. Since
implies
,
and
.
. Therefore
(26)
whenever
all
Suppose
. We now show that (26) is satisfied for
. It is trivially satisfied when
.
. Define
and note that
. Then
Therefore,
for all
. If
is an infinite-length control input sequence such that
for all
, then
Then, (12) follows with
.
Proof of Proposition 2: Consider the initial condition for
be
which a minimizing control input sequence is . Let
given. Then
If we define the input sequence
length
for all
which is of
, we have
. Combining this with
GRIMM et al.: MPC: FOR WANT OF A LOCAL CONTROL LYAPUNOV FUNCTION
Consider the initial condition
trol input sequence is
557
for which a minimizing con-
• {Case 1:
}
• {Case 2:
}
In each case, the final inequality in Lemma 3 holds.
,
,
Lemma 4: Let
and , ,
such that
Hence, the result follows.
V. CONCLUSION
We have presented unconstrained nonlinear MPC results for
closed-loop stability of general attractors. We have shown that
with assumptions of detectability and boundedness of the value
function, there is a finite horizon length sufficient to guarantee
stability. Throughout, we have not required that a local control Lyapunov function be available for use in the algorithm, although we have shown that making additional assumptions on
the terminal cost can result in a shorter horizon length that is
sufficient for stability. Generally the stability is semiglobal and
practical in horizon length, but when detectability and the upper
bound on the value function are guaranteed with linear
functions, global results can be asserted. A semiglobal result
functions are locally linearly bounded.
is given when these
In particular, we have shown that the MPC algorithm can globally exponentially stabilize the origin of a stable linear system
with control constraints as well as stabilize two classic examples using relatively short horizons.
APPENDIX
CHANGING SUPPLY FUNCTIONS
The results here are based on the calculations in [17]; however, we consider the case when the value function is not necessarily positive definite and radially unbounded. Note that for
, the
a continuous nondecreasing function
for
mean value theorem states that
all
.
,
,
Lemma 3: Let
and ,
such that
(28)
for all
and
. Let
be such that
is well defined, continuous, and nonde-
creasing. Then
for all
and
Proof: We use
and
.
,
,
. From (28),
and
. We cover all possible values of and
with four cases.
• {Case 1:
} In this case,
. Then
and
• {Case 2:
} First note that
and
• {Case 3:
} Note that
. Then
and
and
. Then
(27)
. Let
be such that
for all
well defined, continuous, and nondecreasing. Then
for all
.
Proof: We use
,
and
. Note that
from (27). We cover all possible values of
is
,
, which comes
with two cases.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 5, MAY 2005
• {Case 4:
} First note that
implies both
sion of Case 2. Then
and
and the first conclu-
In each case, the final inequality in Lemma 4 holds.
REFERENCES
[1] A. Casavola, M. Giannelli, and E. Mosca, “Global predictive regulation
of null-controllable input-saturated linear systems,” IEEE Trans. Autom.
Control, vol. 44, no. 11, pp. 2226–2230, Nov. 1999.
[2] S. L. de Oliveira Kothare and M. Morari, “Contractive model predictive
control for constrained nonlinear systems,” IEEE Trans. Autom. Control,
vol. 45, no. 6, pp. 1053–1071, Jun. 2000.
[3] F. A. C. C. Fontes, “A general framework to design stabilizing nonlinear model predictive controllers,” Syst. Control Lett., vol. 42, no. 2,
pp. 127–143, 2001.
[4] G. Grimm, M. J. Messina, S. E. Tuna, and A. R. Teel, “Examples when
nonlinear model predictive control is nonrobust,” Automatica, vol. 40,
no. 10, pp. 1729–1738, 2004.
[5] L. Grüne, “Homogeneous state feedback stabilization of homogeneous
systems,” SIAM J. Control Optim., vol. 38, no. 4, pp. 1288–1308, 2000.
[6] L. Grüne, E. D. Sontag, and F. R. Wirth, “Asymptotic stability equals
exponential stability, and ISS equals finite energy gain-if you twist your
eyes,” Syst. Control Lett., vol. 38, no. 2, pp. 127–134, 1999.
[7] H. Hammouri and S. Benamor, “Global stabilization of discrete-time homogeneous systems,” Syst. Control Lett., vol. 38, no. 1, pp. 5–11, 1999.
[8] A. Jadbabaie and J. Hauser, “On the stability of unconstrained receding
horizon control with a general terminal cost,” in Proc. Conf. Decision
and Control, Orlando, FL, Dec. 2001, pp. 4826–4831.
[9] A. Jadbabaie, J. Yu, and J. Hauser, “Unconstrained receding-horizon
control of nonlinear systems,” IEEE Trans. Autom. Control, vol. 46, no.
5, pp. 776–783, May 2001.
[10] S. S. Keerthi and E. G. Gilbert, “An existence theorem for discrete-time
infinite-horizon optimal control problems,” IEEE Trans. Autom. Control,
vol. AC-30, no. 9, pp. 907–909, Sep. 1985.
[11]
, “Optimal infinite-horizon feedback laws for a general class of constrained discrete time systems: Stability and moving-horizon approximations,” J. Optim. Theory Appl., vol. 57, no. 2, pp. 265–293, 1988.
[12] C. M. Kellett, “Advances in converse and control Lyapunov functions,”
Ph.D. dissertation, Univ. California, Santa Barbara, CA, 2002.
[13] L. Magni and R. Sepulchre, “Stability margins of nonlinear receding
horizon control via inverse optimality,” Syst. Control Lett., vol. 32, no.
4, pp. 241–245, 1997.
[14] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica,
vol. 36, no. 6, pp. 789–814, 2000.
[15] E. S. Meadows, M. A. Henson, J. W. Eaton, and J. B. Rawlings, “Receding horizon control and discontinuous state feedback stabilization,”
Int. J. Control, vol. 62, no. 5, pp. 1217–1229, 1995.
[16] H. Michalska and D. Q. Mayne, “Robust receding horizon control of
constrained nonlinear systems,” IEEE Trans. Autom. Control, vol. 38,
no. 11, pp. 1623–1633, Nov. 1993.
[17] D. Nešić and A. R. Teel, “Changing supply functions in input to state
stable systems: The discrete-time case,” IEEE Trans. Autom. Control,
vol. 46, no. 6, pp. 960–962, Jun. 2001.
[18]
, “A framework for stabilization of nonlinear sampled-data systems
based on their approximate discrete-time models,” IEEE Trans. Autom.
Control, vol. 49, no. 7, pp. 1103–1122, Jul. 2004.
[19] G. De Nicolao, L. Magni, and R. Scattolini, “On the robustness of receding-horizon control with terminal constraints,” IEEE Trans. Autom.
Control, vol. 41, no. 3, pp. 451–453, Mar. 1996.
[20] J. A. Primbs and V. Nevistić, “Feasibility and stability of constrained
finite horizon control,” Automatica, vol. 36, no. 7, pp. 965–971, 2000.
[21] J. B. Rawlings and K. R. Muske, “The stability of constrained receding
horizon control,” IEEE Trans. Autom. Control, vol. 38, no. 10, pp.
1512–1516, Oct. 1993.
[22] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis. New York:
Springer-Verlag, 1998.
[23] Y. Yang, E. D. Sontag, and H. J. Sussman, “Global stabilization of linear
discrete-time systems with bounded feedback,” Syst. Control Lett., vol.
30, no. 5, pp. 273–281, 1997.
Gene Grimm received the Ph.D. degree in electrical
and computer engineering from the University of
California, Santa Barbara, in 2003.
Since joining the Space and Airborne Systems
business unit of Raytheon, El Segundo, CA, he has
worked on line-of-sight control design and analysis.
Michael J. Messina received the B.S. degree in engineering from Harvey Mudd College, Claremont, CA,
in 2001, and the M.S. degree in electrical and computer engineering in 2002 from the University of California, Santa Barbara, where he is currently working
toward the Ph.D. degree.
Sezai E. Tuna received the B.S. degree in electrical and electronics engineering from Orta Dogu
Teknik Universitesi, Ankara, Turkey, in 2000. He
is currently working toward the Ph.D. degree in
electrical and computer engineering at the University
of California, Santa Barbara.
Andrew R. Teel (S’91–M’92–SM’99–F’02) received the A.B. degree in engineering sciences from
Dartmouth College, Hanover, NH, in 1987, and the
M.S. and Ph.D. degrees in electrical engineering
from the University of California, Berkeley, in 1989
and 1992, respectively.
After receiving the Ph.D., he was a Postdoctoral Fellow at the Ecole des Mines de Paris,
Fontainebleau, France. In September 1992, he joined
the Faculty of the Electrical Engineering Department, the University of Minnesota, Minneapolis,
where he was an Assistant Professor until September 1997. In 1997, he joined
the Faculty of the Electrical and Computer Engineering Department, the
University of California, Santa Barbara, where he is currently a Professor.
Dr. Teel has received National Science Foundation Research Initiation and
CAREER Awards, the 1998 IEEE Leon K. Kirchmayer Prize Paper Award, the
1998 George S. Axelby Outstanding Paper Award, and was the recipient of the
first SIAM Control and Systems Theory Prize in 1998. He was also the recipient
of the 1999 Donald P. Eckman Award and the 2001 O. Hugo Schuck Best Paper
Award, both given by the American Automatic Control Council.