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International Journal of Control, Automation, and Systems (2010) 8(2):187-197 DOI 10.1007/s12555-010-0202-z
http://www.springer.com/12555
Stability Analysis of Classic Finite Horizon Model Predictive Control
Wen-Hua Chen
Abstract: This paper revisits the stability issue of earlier model predictive control (MPC) algorithms
where the performance index has a finite receding horizon and there is no terminal penalty in the per-
formance index or other constraints added in online optimisation for the purpose of stability. Stability
conditions are presented for MPC of constrained linear and nonlinear systems, and there is no restric-
tion on the length of the horizon. These conditions can be used to test whether or not desired stability
properties can be achieved under chosen state and control weightings.
Keywords: Constrained control, finite horizon, Lyapunov theory, nonlinear systems, predictive control,
stability.
1. INTRODUCTION
Model predictive control (MPC) is a powerful control
strategy for engineering systems. It has two main
features: one is that an optimisation solver is involved in
feedback loop at each sampling instant, and the other is
that input/state constraints can be dealt with explicitly.
These features enable MPC to fully use available control
authority to achieve best possible performance under
constraints. However, on the other hand, these features
also make analysis of the behaviour of MPC much
difficult. Earlier examples showing the possible
instability of MPC algorithms triggered considerable
research interest in the analysis of stability of MPC; for
example, see [1] and [2]. Due to the efforts over the last
two decades, stability analysis of MPC is now reaching a
preliminarily mature stage [3]. Various stability results
for MPC schemes of different kinds of systems
(linear/nonlinear, unconstrained/constrained, continuous-
time/discrete-time) have been developed; for example
see [4-12]. For state-of-the-art of stability analysis of
MPC, reader can refer to [3].
Although there are a number of approaches to
establish stability of MPC, in general, an extra terminal
penalty is required within the performance index.
Stability can be achieved if the terminal penalty covers
the performance cost beyond the predictive horizon
under a local stabilising control law (although it might be
never implemented). Within this paradigm, there are a lot
of variations; for example, MPC with equality terminal
constraints but without terminal weighting ([7]) can be
considered as a variation of this kind of MPC [3].
Despite widely received success of the above approach,
a few researchers realise the conservativeness of the
above MPC stability theory based on terminal weighting
and terminal constraints, and efforts are made to alleviate
the conservativeness [13-16]. [13] discussed how to
choose the length of a receding horizon and the
weighting for linear systems to achieve stability, and [15]
further explored this concept for nonlinear systems,
where it was shown that asymptotic stability holds for a
sufficiently large predictive horizon. A remarkable effort
was made very recently in [14] to relax the requirement
that the terminal weighting shall be a local control
Lyapunov function in the current MPC stability theory; it
was shown that for an unconstrained discrete-time
system, stability of MPC still can be established if the
value function is bounded by a K∞ 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.
With the same motivation, [16] investigated the stability
of MPC with a general terminal cost (possible zero) and
it was shown that there is always a finite horizon for
which the corresponding receding horizon scheme is
stabilising without the use of a terminal cost or terminal
constraints. It shall be noticed that in all the above
schemes, a very large horizon may be required to achieve
stability.
On the other side, although there are some examples
showing unstable behaviour of earlier MPC [17], most of
the systems under earlier MPC algorithms without
terminal weighting and the extra contractive conditions,
[18], are stable if the design parameters are properly
tuned. This kind of MPC is still now widely used in
industry and many commercial MPC products do not
have terminal weighting; for example, [19]. There is a
clear gap between theoretic research and industrial
applications.
This paper investigates the stability of the classic
MPC algorithms. The classic MPC algorithms stand for
© ICROS, KIEE and Springer 2010
__________
Manuscript received March 19, 2009; accepted November 2,2009. Recommended by Editorial Board member Kwang SoonLee under the direction of Editor Young Il Lee. This work was apart of the output from the project “Model Predictive Control forLow-Earth-Orbiting Spacecraft Using Magnetic Actuators”'. Theauthor would like to express his thanks for the financial supportfrom the European Space Agency (ESA), and for the discussionand comments made by Drs Christian Philippe and Denis Fertin atESA on the earlier report. Wen-Hua Chen is with the Department of Aeronautical andAutomotive Engineering, Loughborough University, Leicester-shire, LE11 3TU, UK (e-mail: [email protected]).
Wen-Hua Chen
188
a class of MPC schemes employing a finite receding
horizon (usually very small for the reason of light on-line
computational burden), no (explicit) terminal weighting
in the performance index and no extra contractive
constraints for on-line optimisation [1,13,20]. We will
try to answer the open question ‘under what condition
are the classic MPC algorithms stable’? This paper
presents new stability results and points out that widely
used terminal weighting or contractive constraint may be
not necessary for many systems for the purpose of
stability.
One of the direct motivations for this work is our
recent work on the application of MPC to three axes
magnetic attitude control for low-Earth-orbiting satellites
with the European Space Agency, [21]. Since the
controllability of the magneto-torquers changes with the
local magnetic filed, MPC is employed to “plan” the
dipole moments in the horizon according to the Earth
magnetic filed model. Due to the periodic orbit
behaviour and the change of the local magnetic field, it is
quite difficult to synthesise a terminal cost and then use
current stability theory to show stability; it is interesting
to point out that this issue was also observed in [14]. The
same difficulties may occur for MPC of nonlinear
systems, i.e. nonlinear MPC (NMPC); for a linear system,
the problem of synthesising a local Lyapunov function
can be alleviated by choosing the terminal weighting as
the solution of the Linear Quadratic Regulation (LQR)
problem with the corresponding state and control
weightings, but it is not easy to solve the similar optimal
control problem for nonlinear systems. Certainly there
are some ways to find a local control Lyapunov function
for a stabilisable system, but this does not necessarily
give desired performance although stability may be
achieved. There might be other reasons for not including
a terminal cost in MPC, for example, reducing the tuning
parameters. It is inevitable that most of the designs end
up with final tuning to achieve best possible trade-off
between different (usually conflict) criteria. Without the
use of a terminal penalty, the parameters for final tuning
are significantly reduced, and this might be one of the
reasons why classic MPC is quite popular in industry.
Nevertheless, this paper is not to advocate that classic
MPC is better than the MPC with a terminal penalty;
each of them may find its own application areas.
This paper is organised as follows: a classic MPC
problem is formulated in Section 2. It is shown that,
although this MPC problem can be reformulated in the
current terminal weighting based MPC framework and
then the existing stability analysis tools can be applied
accordingly, no concrete stability results for classic MPC
can be established by these tools.
A new stability analysis tool for classic MPC is
introduced in Section 3. To illustrate the basic idea and
concept, only an unconstrained linear system with two
steps horizon is first considered. By properly
reformulating the problem, a new performance index is
given. The key idea is the value function under the on-
line optimal control profile is not necessarily the best
choice of the Lyapunov function for proving stability.
New stability results are presented.
The new stability results are extended to general linear
MPC problems, i.e. constrained linear systems with a
receding horizon of arbitrary length, in Section 4. To
develop the results, after the feasibility region of the
MPC algorithms is introduced, the monotonicity property
of the open-loop optimisation problem is established
using Dynamic Programming. The closed-loop stability
is then established. Two special cases, namely uncon-
strained systems and open-loop stable systems, are dis-
cussed. It is shown that for any open-loop stable systems,
there always exist control and state weightings such that
the closed-loop system under classic MPC is globally
stable. For unconstrained systems, a Riccati algebraic
inequality like condition is presented. The results are
further extended to constrained nonlinear systems with a
general performance index in Section 5. The results pre-
sented in this paper are illustrated by a linear system and
a simple constrained nonlinear system in Section 6, and
the paper ends with conclusions in Section 7.
2. CLASSIC MPC PROBLEM AND STABILITY
ISSUE
This paper starts with MPC for constrained linear
systems and the results will be extended to MPC for
nonlinear systems.
A constrained discrete-time linear system to be
considered is given by
0
( 1) ( ) ( )
(0)
+ = +
=
x k Ax k Bu k
x x (1)
with control constraints 1
[ ]= , , ∈…
T
mu u u U
1{ [ ] 1 },T
im iu u u u u i mu∈ = , , :| |≤ , = , ,� … …U (2)
where ∈ n
x R and ∈m
u R are the state and control
vectors, respectively. A classic quadratic predictive
performance index is given by
0
( ( ) ( )) ( ( ) ( )
( ) ( )),
N
T
i
T
J x k U k k x k i k Qx k i k
u k i k Ru k i k
=
, | = + | + |
+ + | + |
∑ (3)
where N is the predictive horizon length, and Q > 0, R > 0
are the state and control weighting, respectively.
( ),x k i k+ | 1 ,i N= , ,… denotes the predicted state at
time instant +k i based on the state measurement at
time instant k, i.e., x(k), and the control sequence
( ) [ ( ), ...,U k k u k k| = | ( )].u k N k+ | When an MPC
algorithm is applied to control of the system (1), at time
instant k, the minimisation problem
( ) ... ( )( ( )) min ( ( ) ( ))
u k k u k N k
J x k J x k U k k∗
| , , + |= , | (4)
subject to
( 1 ) ( ) ( )
( ) ( )
x k i k Ax k i k Bu k i k
x k k x k
+ + | = + | + + |
| = (5)
Stability Analysis of Classic Finite Horizon Model Predictive Control
189
and
( ) 0+ | ∈ , ≤ ≤u k i k i NU (6)
is solved on-line using an optimisation solver, for
example, QP, and a control sequence ( )U k k∗ | =
[ ( ) ( )]u k k u k N k∗ ∗| , , + |… is yielded. The real input
applied on the plant (1) is given by
( ) ( )∗= | .u k u k k (7)
The above process repeats in MPC as time goes.
As shown in [1] and [2], many earlier MPC algorithms
including Generalised Predictive Control (GPC),
Dynamic Matrix Control (DMC) and Model Control
Algorithms (MAC) are fitted into this format. For the
notational simplicity, we refer this kind of MPC with a
finite horizon but without (explicit) terminal penalty and
any extra on-line optimisation constraints introduced for
the purpose of stability as classic MPC. It is assumed
that the optimal value function J*(x) is continuously
differentiable and radically bounded. It has been shown
in [22] that for a constrained linear system, under certain
conditions, the optimal value function is continuously
differentiable, piecewise quadratic if the performance
index is continuously differentiable and quadratic.
This paper firstly reformulates this classic MPC
problem into an MPC problem with special terminal
weighting and then shows that the existing stability
results for MPC are inadequate for stability analysis of
the classic MPC problem.
For the optimisation problem in (4), it can be shown
that the optimal value of ( )+ |u k N k is achieved at 0
since the influence of this control action is not taken into
account in the cost function. In other words, (u k N∗ + |
) 0.k = As a result, the cost function for on-line
optimisation is equivalent to
1
0
( ( ) ( )) ( ) (
) ( ) ,
N
Q
i
Q R
J x k U k k x k N k x k
i k u k i k
−
−
=
, | = + | +
+ | + + |
∑� � �
� � �
(8)
where, for the sake of notation simplicity,
( ) ( ) ( ), ( ) ( ) ( ),T TQ Rx x Qx u u Ru⋅ ⋅ ⋅ ⋅ ⋅ ⋅� � � � � � (9)
and
( ) [ ( ) ( 1 )].−
| = | , , + − |…U k k u k k u k N k (10)
The original optimisation problem with the performance
index (3) is equivalent to the MPC problem with (8).
Then the following existing stability result for MPC
with terminal weighting can be directly applied, which is
modified from [3] and [12].
Lemma 1: Consider the MPC scheme for the con-
strained system (1) and (2) with the performance index
�( ( ) ( )) ( )P
J x k U k k x k N k, | = + |� � (11)
1
0
( ) ( ) ,N
Q R
i
x k i k u k i k
−
=
+ + | + + |∑ � � � �
where P is positive definite and ,Q R are as defined
before. Suppose that there exists ( )u k satisfying (2)
such that for all ( )∈x k V
( 1) ( ) ( ) ( ) 0P P Q Rx k x k x k u k+ − + + <� � � � � � � � (12)
and
( 1) .+ ∈x k V (13)
Then the set V is a terminal region of the MPC
algorithm. Moreover, if the MPC scheme is feasible at
time instant 0 in the sense that there exists a control
sequence such that the terminal state, i.e., ( ),x k N+
falls into the terminal set ,V it is feasible for all
0,k ≥ and the closed-loop system under the MPC law
is asymptotically stable about the origin.
Equation (12) implies that in order to guarantee
stability, the reduction in the terminal penalty (the first
two terms) shall cover the stage cost (the last two terms)
under control action generated by minimising the cost
function. More explanation of this condition can be
found in Remark 4. Compared with the standard
formulation of the terminal weighting based MPC having
the performance index (11), the MPC problem with the
performance index (8) can be considered as an MPC
problem with special terminal weighting .P Q= Then
condition (12) becomes
( 1) ( ) ( ) ( ) 0,Q Q Q Rx k x k x k u k+ − + + <� � � � � � � � (14)
i.e.,
( 1) ( ) 0,Q Rx k u k+ + <� � � � (15)
which is impossible to meet for any nonzero x and u
since both Q and R are positive definite.
This clearly indicates that no concrete stability results for
classic MPC can be established by directly applying the
current stability analysis methods for MPC with terminal
weighting, although it is evident that in many
engineering applications, classic MPC is indeed stable.
This highlights, as the example in Section 6, that the
current stability analysis methods in MPC might be quite
conservative.
Remark 1: Similar attempt of directly applying
stability results for receding LQ control to GPC has been
made in [1]. For linear unconstrained systems, the
stability issue for receding horizon LQ problem, i.e.
MPC with terminal penalty, was carefully examined
using the monotonicity of the Riccati equation, and the
so called Fake Riccati Algebraic Equation (FRAE)
stability condition was established. This approach was
then applied in investigating stability for GPC, and it was
found that, similar to the results above, no concrete
stability result can be obtained since the monotonicity
goes the wrong way in GPC (see Theorem 4.13 in [1]). It
was suggested that to guarantee stability, either infinite
horizon, or different state and terminal weighting
matrices shall be employed.
Wen-Hua Chen
190
3. NEW STABILITY RESULTS
To clearly demonstrate the idea of this paper, first a
classic MPC problem with two steps horizon and without
control constraints is considered in this section and then
the established result will be extended to general
constrained linear MPC problems in Section 4 and
nonlinear MPC problems in Section 5 respectively.
The performance cost function with two steps receding
horizon can be written as
1
0
( ( ) ( ) ( 1 ))
( ) ( ) .Q R
i
J x k u k k u k k
x k i k u k i k
=
, | , + |
= + | + + |∑ � � � � (16)
Two observations can be made for this optimation
problem.
• Since the first term ( ) ( )Tx k Qx k is a constant after
( )x k is measured, it has no influence on the
solution of the on-line optimisation and the only
difference is the optimal value of the cost function.
• The minimum of the performance cost is always
attained at ( 1 ) 0.u k k∗ + | =
Based on these two observations, the following stability
result can be established:
Theorem 1 (unconstrained system): Consider the
unconstrained system (1) with the performance index
(16). The closed-loop system under the classic MPC is
stable if for all ( ) ,nx k R∈ there exists a control ( )u k
such that
( 1) ( 1) ( ) ( ) ( ) ( ) 0.+ + − + <T T Tx k Qx k x k Qx k u k Ru k (17)
Proof: With these two observations, it can be shown
that on-line MPC problem
( ) ( 1 )min ( ( ) ( ) ( 1 ))
| , + |, | , + |
u k k u k k
J x k u k k u k k (18)
is equivalent to the following on-line optimisation
problem
1( )
( )
min ( ( ) ( ))
min ( 1 ) ( )
u k k
Q Ru k k
J x k u k k
x k k u k k
|
|
, |
= + | + |� � � � (19)
with ( 1 ) 0.u k k∗ + | = Furthermore, since only the first
part of the optimal control sequence in the receding
horizon is implemented, i.e., ( ) ( ),u k u k k∗= | the
closed-loop behaviour under these two MPC algorithms
are exactly the same. Now we only need to show that the
closed-loop system for the second MPC problem is
stable about the origin.
For the sake of space, only the outline of the proof for
the stability of the second MPC problem is given below;
for more detail about the proof, please refer to the proof
of Theorem 4 in Section 4. After choosing an Lyapunov
function candidate as
1( )
( ( )) min ( ( ) ( )),|
= , |u k k
V x k J x k u k k (20)
it can be shown that if there exists a control ( )u k
satisfying (17), the Lyapunov function decreases along
the state trajectory under the classic MPC, i.e.,
( ( 1)) ( ( )) ( ) 0.R
V x k V x k u k k∗+ − < − | ≤� � (21)
Therefore, the stability can be established using the
monotonicity of the associated Lyapunov function. �
Remark 2: For this MPC problem with a two steps
receding horizon, it is clear that the stability condition
(17) given in Theorem 1 is much less conservative than
that in Lemma 1. If the state weighting Q in the
performance index (16) is chosen as ,P by comparing
these two conditions, i.e., (17) and (12), one can
conclude that even better stability can be achieved by
classic MPC; this is in contrast with the perception that
stability of the terminal weighting based MPC is better
than that without terminal weighting. This highlights that
the current stability analysis tools might be too
conservative, as indicated in [14] and [16].
Remark 3: The widely used receding LQ like
performance index (3) is adopted in this paper. It is
worth noting that J1 was also widely used in adaptive
control such as generalised minimum variance (GMV)
control [23] from which GPC was developed [22]. It was
also used as the performance index in some earlier MPC
[2]. In that case, this is just one step ahead predictive
control. However, in this paper, 1( ( ) ( )), |J x k u k k is
introduced only for the purpose of the stability proof.
4. STABILITY ANALYSIS: GENERAL CASE
Section 3 presents a new result for classic MPC with a
two-steps horizon and without control constraints, and a
much less conservative stability condition is presented
(see the first order example in Section 6). This section
develops this idea for the classic MPC problem with a
receding horizon of arbitrary length and control
constraints. In order to present the results, some
preliminaries are necessary.
First since the control constraints are presented, it is
unlikely to achieve global stability and feasibility for
MPC of general systems. The feasibility region in this
paper is defined as below:
Definition 1: A feasibility region X for MPC
problem (1) with the performance index (3) is defined as
a set such that for any ( ) ,x k ∈X there exists control
( )∈u k U such that
Item 1
( 1)+ ∈x k X (22)
and Item 2
( 1) ( ) ( ) 0.Q Q Rx k x k u k+ − + <� � � � � � (23)
Condition 2 in the above Definition implies that there
exists a control such that the decrease in the state cost
Stability Analysis of Classic Finite Horizon Model Predictive Control
191
(the first two items) covers the control cost. This is quite
reasonably assumption as otherwise, the overall stage
cost (including both the state and control cost) would
increase, no matter what control is taken.
We now choose a special form of the feasibility set as
{ there exists ( )
such that (23) holds},
nQx R x u kα∈ : ≤ , ∈� � �X U
(24)
where 0.α > It shall be noticed that condition (22) is
not required in this special form of the feasibility set.
This is because once condition (23) is met, condition
(22) is satisfied for the set defined in (24).
Definition 2: Suppose that, at time instant k, the
solution of the on-line optimisation in MPC (4) is
denoted by ( ) ( 1 )u k k u k N k∗ ∗| , , + − | ,… ( ),u k N k
∗ + |
and ( 1 ) ( )∗ ∗+ | , , + |…x k k x k N k are the state trajectory
accordingly. We say the MPC problem is feasible at time
k if ( ) 0 1∗+ | ∈ , = , , −…x k i k i NX where ( )∗ |x k k
( ).x k=
To simplify the notation, let +k i stage cost be
denoted by
( 1 ) ( )
0 1
i Q Rl x k i k u k i k
i N
= + + | + + | ,
= , , −
� � � �
� (25)
and +k i stage cost under the open-loop optimal
control be denoted by
( 1 ) ( )
0 1.
i Q Rl x k i k u k i k
i N
∗ ∗ ∗= + + | + + | ,
= , , −
� � � �
� (26)
Cost-to-go at stage +k i with initial state ( )+ |x k i k
is defined as
( )
1
( ( ))
min ( ( ) ( ))
( 1 ) ( )
0 1,
i
iU k i k
N
Q R
j i
L x k i k
L x k i k U k i k
x k j k u k j k
i N
∗
+ | ∈
−∗ ∗
=
+ |
= + | , + |
= + + | + + | ,
= , , −
∑ � � � �
�
U
(27)
where
( )
[ ( ) ( 1 ) ( 1 )]
U k i k
u k i k u k i k u k N k
+ |
+ | , + + | , , + − |� … (28)
and ( ( ))∗ + |iL x k i k explicitly indicates the cost-to-go is
the function of the state ( ).x k i k+ | It can be seen that
1
( ( )) 0 1.−
∗ ∗
=
+ | = , = , , −∑ �
N
i j
j i
L x k i k l i N (29)
Furthermore, taking into account the fact that (u k N∗ + |
) 0,k = the optimal value of the performance index in
(4) can be expressed as
1
0
0
( ( )) ( ) ( ( )) ( ) .N
Q Q i
i
J x k x k L x k x k l
−
∗ ∗ ∗
=
= + = +∑� � � � (30)
Lemma 2: Consider the MPC problem for the
constrained systems (1) with the performance index (3).
Suppose that the optimisation problem (4) at time k is
feasible as defined in Definition 2. The following
properties hold:
( ( )) ( ) ( )
0 1,
i QL x k i k N i x k i k
i N
∗ ∗ ∗+ | < − + | ,
= , , −
� �
� (31)
( ) ( 1 )
( ) 0 0 1.
Q R
Q
x k N k u k N k
x k i k i N
∗ ∗
∗
+ | + + − |
− + | < , = , , −
� � � �
� � … (32)
Proof: The proof is based on the backward time and
forward time properties for the open-loop optimisation
problem (4). The on-line optimisation problem can be
solved by N-stage Dynamic Programming. Using inverse
time technique, the optimisation starts from the last stage,
i.e., time ,k N+ and then recursively solves the
optimisation problem for each stage. It is trivial to show
( ) 0.u k N k∗ + | = Now consider stage 1.k N+ − The
stage cost is given by
1( ) ( 1 ) .N Q Rl x k N k u k N k
∗ ∗ ∗
−= + | + + − |� � � � (33)
Following the assumption that the optimisation problem
is feasible, ( 1 )∗ + − |x k N k belongs to the set X and
there exists a control effort ( 1 )+ − |u k N k such that
(23) holds. It follows from the principle of optimality
that
1
( 1 )min ( ) ( 1 )
N
Q Ru k N k
l
x k N k u k N k
∗−
+ − | ∈= + | + + − |� � � �
U
( ) ( 1 )
( 1 ) .
Q R
Q
x k N k u k N k
x k N k∗
≤ + | + + − |
< + − |
� � � �
� � (34)
It also can be written as
1 1( ( 1 ))
( 1 ) .
N N
Q
L x k N k l
x k N k
∗ ∗ ∗
− −
∗
+ − | =
< + − |� � (35)
At the stage 2,k N+ − one obtains
2
1
( 2 )2
2 1( 2 )
( 2 )
1
( ( 2 ))
min
min
min ( ( 1 ) (
2 ) ( ( 1 ))),
N
N
jU k N k
j N
N Nu k N k
Qu k N k
R N
L x k N k
l
l l
x k N k u k N
k L x k N k
∗ ∗−
−
+ − | ∈= −
∗− −
+ − | ∈
+ − | ∈
∗−
+ − |
=
= +
= + − | + +
− | + + − |
∑
� � �
�
U
U
U
(36)
where the second equality follows from the Bellman
equation.
Since the state ( 2 )∗ + − |x k N k is within the set ,X
there exists a control effort such that (23) holds and
denotes such a control as ( 2 ).u k N k+ − |� The corres-
Wen-Hua Chen
192
ponding state is given by
( 1 ) ( 2 )
( 2 )
x k N k Ax k N k
Bx k N k
∗+ − | = + − |
+ + − |
�
�
(37)
and furthermore, from (23),
( 1 ) ( 2 )
( 2 ) .
Q R
Q
x k N k u k N k
x k N k∗
+ − | + + − |
< + − |
� �� � � �
� � (38)
Following from the principle of optimality and invoking
(35) and (38) in (36), one concludes
2
( 2 )
1
1
( ( 2 ))
min ( 1 )
( 2 ) ( ( 1 ))
( 1 )
( 2 ) ( ( 1 ))
( 1 ) ( 2 )
( 1 )
2( ( 1 ) ( 2
N
Qu k N k
R N
Q
R N
Q R
Q
Q
L x k N k
x k N k
u k N k L x k N k
x k N k
u k N k L x k N k
x k N k u k N k
x k N k
x k N k u k N
∗ ∗−
+ − | ∈
∗−
∗−
+ − |
= + − |
+ + − | + + − |
≤ + − |
+ + − | + + − |
< + − | + + − |
+ + − |
< + − | + + −
� �
� �
�� �
� �� �
� �� � � �
�� �
� �� � �
U
) )
2 ( 2 ) .
R
Q
k
x k N k∗
|
< + − |
�
� �
(39)
In the above process, the fact that ( 1 )x k N k+ − |�
belongs to the set X is employed. This is because
( 2 )∗+ − | ∈x k N k X and (38) implies that
( 1 )
( 2 ) ( 2 )
( 2 )
.
Q
Q R
Q
x k N k
x k N k x k N k
x k N k
α
∗
∗
+ − |
< + − | − + − |
< + − |
<
�� �
�� � � �
� �
(40)
Hence (35) holds when replacing ( 1 )∗ + − |x k N k by
( 1 ).x k N k+ − |�
Repeating the above process, one can prove Item 1 for
the stage +k i until 0,i = which is given by
0 0( ( )) ( ( )) ( )
( ) .
Q
Q
L x k L x k k N x k k
N x k
∗ ∗ ∗ ∗= | < |
=
� �
� � (41)
We are now in the stage to prove the Item 2.
It is straightforward to show that Item 2 holds for
1.i N= − Actually it has been shown in (35). Now we
first prove that it is also true for 2.i N= − It follows
from (31) that
2
1 2
( 2 ) ( ( 2 )) 2
( ) 2
1( ( 1 ) ( 2 )
2
( ) ( 1 ) )
Q N
N N
Q R
Q R
x k N k L x k N k
l l
x k N k u k N k
x k N k u k N k
∗ ∗ ∗
−
∗ ∗
− −
∗ ∗
∗ ∗
+ − | > + − | /
= + /
= + − | + + − |
+ + | + + − |
� �
� � � �
� � � �
1(2 ( ) 2 ( 1 )
2
( 2 ) )
( ) ( 1 ) .
Q R
R
Q R
x k N k u k N k
u k N k
x k N k u k N k
∗ ∗
∗
∗ ∗
> + | + + − |
+ + − |
> + | + + − |
� � � �
� �
� � � �
(42)
So Item 2 is proven for 2.i N= − Similarly, one has
1
1
2
( ) ( ( )) ( )
( )
( 1 ) ( )
( ) ( 1 )
( ( ) ) ( ))
( ) ( 1 ) .
Q i
N
j
j i
NQ R
j i
Q R
N
R
j i
Q R
x k i k L x k i k N i
l N i
x k j k u k j k
N i
x k N k u k N k
u k j k N i
x k N k u k N k
∗ ∗ ∗
−
∗
=
∗ ∗−
=
∗ ∗
−
∗
=
∗ ∗
+ | > + | / −
= / −
+ + | + + |=
−
> + | + + − |
+ + | / −
> + | + + − |
∑
∑
∑
� �
� � � �
� � � �
� �
� � � �
(43)
Therefore, (32) is proved. �
We are now ready to present one of our main results.
Theorem 2: Consider MPC problem for the system
(1) and (2) with the performance index (3). The closed
loop system under the MPC law stemming from on-line
optimisation, i.e., solving the optimisation problem (4)
subject to (5) and (6), is asymptotically stable about the
origin if there exists an 0α > such that it is feasible at
time 0=k with respect to the feasibility region X
(24).
Proof: At time k, after on-line solving the MPC
problem (4), the optimal control sequence within the
receding horizon is given by ( )∗ | =U k k [ ( )u k k∗ | , ,…
( )].u k N k∗ + |
Define a new performance index as
2
1
0
( ( ) ( ))
( 1 ) ( ) ,N
Q R
i
J x k U k k
x k i k u k i k
−
−
=
, |
= + + | + + |∑ � � � � (44)
where ( )−
|U k k consists of first N components of
( ),U k k| i.e., (10). A Lyapunov function candidate
along the state trajectory under MPC is chosen as
2
2
( ( )) ( ( )) ( )
min ( ( ) ( ))
( ( ) ( ))
Q
U
V x k J x k x k
J x k U k k
J x k U k k
−
∗
−
∗
−
= −
= , |
= , |
� �
(45)
1
0
( 1 ) ( ) .N
Q R
i
x k i k u k i k
−
∗ ∗
=
= + + | + + |∑ � � � �
To establish the stability, first we need to prove that if
the MPC problem is feasible at time k, there exists a
control sequence � ( 1 1) [ ( 1 1))k k u k kU −
+ | + = + | + , ,� …
( 1))]u k N k+ | +� such that
Stability Analysis of Classic Finite Horizon Model Predictive Control
193
�2( ( 1) ( 1 1)) ( ( )) 0.
−
+ , + | + − <J x k k k V x kU (46)
Substituting (44) with �( 1 1)+ | +U k k and (45) into (46)
obtains
�2
1
0
1
0
( ( 1) ( 1 1)) ( ( ))
( 2 1) ( 1 1)
( 1 ) ( ) ,
N
Q R
i
N
Q R
i
J x k k k V x kU
x k i k u k i k
x k i k u k i k
−
−
=
−
∗ ∗
=
+ , + | + −
= + + | + + + + | +
− + + | − + |
∑
∑
� �� � � �
� � � �
(47)
where ( 1) 1 1,x k i k i N+ | + , = , , +� … are the state
sequence under the control � ( 1 1).k kU −
+ | +
At time instant 1,k + since the optimal control
( )∗ |u k k is implemented, one has
( 1) ( 1 ).∗+ = + |x k x k k (48)
Now choosing the first 1−N components in the control
sequence � ( 1 1)−
+ | +k kU as
( 1) ( ) 1 1u k i k u k i k i N∗+ | + = + | , = , , −� � (49)
the corresponding state trajectory is given by
( 1) ( ) 2 .x k i k x k i k i N∗+ | + = + | , = , ,� � (50)
It can be shown by Item 2 in Lemma 2 that
( )∗ + |x k N k belongs to the set X if ( ) .x k ∈X This
is implied by
( )
( ) ( 1 )
Q
Q R
x k N k
x k u k N k α
∗
∗
+ |
< − − + − | <
� �
� � � � (51)
and thus ( 1) ( ) .x k N k x k N k∗+ | + = + | ∈� X Therefore,
there exists a control ( 1)u k N k+ | + ∈� U such that
( 1 1) ( 1)
( ) 0.
Q R
Q
x k N k u k N k
x k N k∗
+ + | + + + | +
− + | <
� �� � � �
� � (52)
Invoking (49), (50) and (52) into (47), simple
manipulation yields
�2( ( 1) ( 1 1)) ( ( ))
( 1 1) ( 1)
( 1 ) ( )
( ) ( 1 )
( ) .
Q R
Q R
Q Q
R
J x k k k V x kU
x k N k u k N k
x k k u k k
x k N k x k k
u k k
−
∗ ∗
∗ ∗
∗
+ , + | + −
= + + | + + + | +
− + | − |
< + | − + |
− |
� �� � � �
� � � �
� � � �
� �
(53)
Item 2 in Lemma 2 implies that
( ) ( 1 ) .Q Qx k N k x k k∗ ∗+ | < + |� � � � (54)
Combining (54) with (53) gives
�2( ( 1) ( 1 1)) ( ( )) 0.
−
+ , + | + − <J x k k k V x kU (55)
Recalling the definition of the associated Lyapunov
function, one has
2( 1 1) ( 1)
( ( 1))
min ( ( 1) ( 1 1)).−+ | + , , + | + ∈
+ =
+ , + | +�u k k u k N k U
V x k
J x k U k k (56)
At time instant 1,k + the MPC on-line calculates the
optimal control sequence in receding horizon by
minimising the performance index ( ( 1)+ ,J x k ( 1U k + |
1)).k + Similar to the two observations made in Section
3 for two steps receding horizon, it is important to notice
that this is equivalent to the minimisation of the
performance index 2( ( 1)J x k + , ( 1 1)U k k
−
+ | + in (44),
with the same optimal control sequence but different
optimal value of the cost function.
Combining (56) with (53), one concludes
( ( 1)) ( ( )) 0+ − < .V x k V x k (57)
Following the monotonicity of the Lyapunov function, it
can be shown that the closed-loop system under the MPC
is asymptotically stable about the origin. �
Remark 4: In the existing methods, stability is
established using the terminal penalty covering the cost-
to-go for remaining horizon. Recursively applying (12)
or
( 1) ( ) ( )
( )
P Q R
P
x k N x k N u k N
x k N
+ + + + + +
≤ +
� � � � � �
� � (58)
gives
1
( ) ( ) ( ) .Q R P
i N
x k i u k i x k N
∞
= +
+ + + ≤ +∑ � � � � � � (59)
The closed-loop system under MPC is stable if there
exists local control satisfying the constraints such that
the terminal penalty in the performance index can cover
the cost-to-go for the remaining horizon. It is impossible
to establish the stability of classic MPC in this way as
shown in Section 2. Instead, as indicated in (53), the
stability is established in this paper by requiring that
( 1) ( )
( 1) ( ) .
Q R
Q R
x k N u k N
x k u k
+ + + +
< + +
� � � �
� � � � (60)
In other words, stability for classic MPC is established
in this paper by requiring that the cost at stage +k N is
less than that at stage k.
With Theorem 2 in hand, we are now ready to derive
some interesting results for special classes of systems.
Corollary 1 (stable systems): For an open-loop stable
system, the closed-loop system under the classic MPC is
always asymptotically stable if the state weighting Q is
chosen to satisfy
0.− <T
A QA Q (61)
Furthermore, the feasibility region for the classic MPC is
the whole state space irrespective of control constraints.
Remark 5: One may expect that stability can always
Wen-Hua Chen
194
be achieved when a classic MPC algorithm is applied to
an open-loop stable system. Unfortunately, this is not
true and there is a very interesting counterexample in
[17]. This Corollary states that the stability can be
guaranteed if the state weighting satisfies condition (61).
Proof: This result can be easily derived from Theorem
4. For an open-loop stable system, when the state
weighting is chosen to satisfy (61), it suffices to choose
( ) 0=u k such that (23) holds for all ( )x k ∈ Rn.
Therefore, following Theorem 4, the closed-loop system
under the MPC is stable about the origin. Furthermore
since 0=u provides a feasible sequence for all
,
n
x R∈ this implies that the feasible region is the whole
space irrespective of control constraints. �
In the absence of control constraints (the open-loop
system might be unstable), the following result can be
established:
Corollary 2 (unconstrained): Consider a classic MPC
problem for linear systems (1) without control con-
straints and with the performance index (3). The closed-
loop system under MPC is globally asymptotically stable
if the state and control weightings ( ),Q R satisfy
1( ) 0.−
− + − <T T T T
A QA A QB R B QB B QA Q (62)
Proof: When there are no control constraints, it
follows from Theorem 2 that the closed-loop system
under MPC is stable about the origin with the attraction
region of the whole space if there exists control u such
that (23) holds for any initial state ( ) ,nx k R∈ which
can be written as
( )min ( 1) ( ) ( ) 0.+ − + <� � � � � �Q Q Ru k
x k x k u k (63)
After substituting the system dynamics (1), it can be
found that the optimum on the left side in (63) is
achieved at
1( ) ( ) ( ).∗ −= − +
T Tu k R B QB B QAx k (64)
Invoking (64) into (63) yields
1( ) ( ( )
) ( ) 0.
T T T T Tx k A QA A QB R B QB B QA
Q x k
−− +
− < (65)
One can conclude that the closed-loop system under
MPC is stable if Q and R are chosen such that (62)
holds. �
5. STABILITY OF MPC FOR NONLINEAR
CONSTRAINED SYSTEMS
Consider a constrained nonlinear system
( 1) ( ( ) ( ))+ = ,x k f x k u k (66)
with the input constraints (2) and the receding quadratic
horizon performance index
( ( )) ( ( ))
0
( ) ( ) ( ) .N
Q x k i R x k i
i
J k x k i u k i+ +
=
= + + +∑ � � � � (67)
Assumptions on the nonlinear system (66) and the
performance index (67) are made as follows:
A1: (0 0) 0;f , =
A2: ( ) 0>R x and Q(x) 0> for all ∈ nx R and 0≠x
A3: The optimal cost function is continuously
differentiable, decrescent and radically bounded.
A4: There exists an 0α > such that for all ( )x k ∈
( ( )){ ( ) },nQ x kx R x k α∈ : <� � �X there exists control
( )∈u k U satisfying
( ( 1)) ( ( )) ( ( ))( 1) ( ) ( ) 0.Q x k Q x k R x kx k x k u k+
+ − + <� � � � � �
(68)
It can be seen that the nonlinear system (66) and
associated performance index (67) satisfying the
assumptions A1-A2 are rather general. A3 is introduced
for the stability purpose.
Theorem 3: Suppose that the nonlinear system (66)
and its predictive control performance index (67) satisfy
the assumptions A1-A4. The MPC scheme stemming
from online minimisation of (67) subject to (2) and (66)
asymptotically stabilises the system about origin with the
stability region X if the MPC problem is feasible as
defined in Definition 2.
Proof: The proof of Theorem 3 is similar to that of
Theorem 2. It is omitted.
6. EXAMPLES
6.1. Extensive first-order system
First consider an unconstrained first order linear
system
( 1) ( ) ( )+ = +x k ax k bu k (69)
and a two steps horizon performance index
1
2 2
0
( ) ( ) ( )=
= + | + + |∑i
J k qx k i k ru k i k (70)
is employed.
For the purpose of comparison, first the existing
stability results are applied on this MPC problem [3].
Since this is an unconstrained linear system, the stability
condition reduces to the Fake Algebraic Riccati Equation
(FARE) [1]
1( ) 0.−
− + + − <T T T T
A PA A PB R B PB B PA Q P (71)
Using the similar technique in Section 2, one has P = q.
Substituting this and other parameters of this system into
(71) yields
2 2 1( )−− + + <a q aqb r qb bqa q q (72)
i.e., the closed-loop system under MPC stemming from
the minimisation of the performance index (70) is stable
if the weighting q, r are chosen such that
2
20.<
+
a r
r qb (73)
Stability Analysis of Classic Finite Horizon Model Predictive Control
195
It is impossible to meet this stability requirement for any
a and b since q > 0, r > 0.
We now apply new stability results developed in this
paper. For this unconstrained linear system, directly
applying Corollary 2 gives
2 2 1( ) ,−
− + <a q aqb r qb bqa q (74)
which is implied by
2
21<
+
a r
r qb (75)
i.e.,
2
21.
1 ( )<
+ /
a
b q r (76)
To give more insight into condition (76), the following
observations are made:
Case 1 (Open-loop stable systems): It is always stable
under the proposed MPC, no matter what control and
state weighting are chosen. This is because 21 ( )b q r+ /
> 1 for any q > 0 and r > 0. It follows from the stability
of the open-loop system that 1.a| |< Combining them
together implies that condition (76) is always satisfied
and hence the stability is guaranteed.
Case 2 (Open-loop unstable systems): In this case,
1.a| |≥ It can be derived from the condition (76) that the
closed-loop system under the MPC is stable if the control
and state weightings are chosen to satisfy
2
2
1.
−>
q a
r b (77)
This example clearly demonstrates that stability of the
closed -loop system of MPC can be achieved without
terminal weights but by properly choosing the state and
control weightings. If the open-loop system is stable, full
degree of freedom of the design parameter are preserved
for performance requirements. For an unstable open-loop
system, a certain condition, i.e., (77), shall be imposed
for the purpose of stability.
Since this is an unconstrained linear system, it is
possible to work out an analytic solution for the
associated MPC problem and thus the closed-loop
system. Then the stability of the closed system can be
analysed. This study serves two purposes: to verify the
proposed stability method and to investigate possible
conservativeness of the new method for this example.
For each time instant k, the optimal solution for the
control sequence is obtained as
2( ) ( ) ( 1 ) 0.∗ ∗| = ; + | =
+
bqau k k x k u k k
r qb (78)
After applying ( ) ( ) ,u k u k k∗= | the closed-loop system
is given by
2
2 2( 1) ( ) ( ) ( ).+ = − =
+ +
b q rax k a a x k x k
r b q r b q (79)
It is stable if and only if the eigenvalue satisfies
21
ra
r b q<
+
(80)
or
21.
1 ( )
a
b q r<
+ / (81)
If the open-loop system is stable, (81) holds for any pair
of ( 0 0),r q> , > which confirms the result given by
the new stability method proposed in paper.
For an open-loop unstable system, the closed-loop
system is stable if and only if the state and control
weightings satisfy
2
1.
| | −>
q a
r b (82)
It can be seen that (77) implies (82) since 1| |≥a for
open-loop unstable systems. This verifies the new
stability result in this paper. For this example, sufficient
stability condition proposed in this paper is close to the
sufficient and necessary condition for stability of the
classic MPC.
6.2. Nonlinear system
The following system is adopted from [5]
2
( )( 1) ( )
1 ( )+ = +
+
x kx k u k
x k (83)
with the control constraint
( ) 0 2.| |≤ .u k (84)
The same design parameters are chosen as the MPC
algorithm in [5], i.e., 1=Q and 1=R with receding
horizon 3.N = It shall be noticed that an equality
terminal constraint was added for the on-line
optimisation in the MPC algorithm proposed and then its
stability was established in [5]. We now remove the
terminal constraint and investigate the stability of the
MPC scheme using the results presented in this paper. It
follows from Theorem 3 that the closed-loop system
under the MPC scheme is asymptotically stable if there
exists ( ) 0 2| |≤ .u k such that (68) holds, i.e.,
2
2 2
2
( )( ) ( ) ( ) 0.
1 ( )
+ − + < +
x ku k x k u k
x k (85)
This can be shown that this condition holds for all
.
n
x R∈ This can be proved by letting ( ) 0,u k = (85)
reduces to
2
2
2
( )( ) 0,
1 ( )
− < +
x kx k
x k (86)
which holds for all ∈ n
x R since 21 ( ) 1+ >x k for
Wen-Hua Chen
196
nonzero x(k). This implies that the nonlinear system
under the MPC scheme without terminal constraint is
asymptotically stable about the origin with the attraction
region of the whole state space.
7. CONCLUSION
Although there is criticism for earlier MPC for
possible loss of stability, and various modified MPC
algorithms have been proposed for guaranteeing stability,
earlier MPC algorithms still offer many features such as
simplicity and transparency in tuning which are attractive
to practitioners, and are still widely used in industry.
Sufficient stability conditions are presented in this paper
for earlier MPC algorithms for unconstrained/constrained
linear and nonlinear systems. Furthermore, some
interesting results are provided for special classes of
linear MPC problems. It is shown that when the open-
loop system is stable, global stability can be achieved by
properly choosing the state and control weightings even
in the presence of control constraints. For unconstrained
linear systems, the presented stability conditions reduces
to an algebraic Riccati inequality like condition and
global stability is also achieved. This might, to a certain
extent, explain why earlier MPC algorithms work well
for many engineering systems by properly tuning design
parameters. To establish the results, a new approach for
the stability proof is developed in this paper and this tool
may help to establish new stability conditions for MPC
with terminal penalty. This will help to relax the
requirements for MPC with a terminal penalty and
provide more degrees of freedom for performance
consideration. The stability analysis tool presented in this
paper has been successfully applied in developing of
stability guaranteed classic MPC algorithm for spacecraft
attitude control problem, which will be reported in
another paper. It also provides a method of choosing the
state and control weighing to satisfy the proposed
conditions so as to guarantee stability of satellite attitude
control systems under magnetic actuators.
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Wen-Hua Chen currently holds a Senior Lectureship in fight control systems in Department of Aeronautical and Automo-tive Engineering at Loughborough Uni-versity, UK since 2000. From 1991 to 1996, he was a Lecturer in Department of Automatic Control at Nanjing University of Aeronautics and Astronautics, China. He held a research position and then a
Lectureship in control engineering in Center for Systems and Control at University of Glasgow, UK, from 1997 to 2000. Dr. Chen has published one book and more than 100 papers on journals and conferences. He is a Senior Member of IEEE. His research interests include autonomous aerial vehicles, the de-velopment of advanced control strategies, and engineering applications in particular in aerospace, space and automotive engineering.