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: W.Chen@lboro.ac.uk).

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.