International Journal of Automation and Computing
04(2), April 2007, 195-202
DOI: 10.1007/s11633-007-0195-0
Model Predictive Control of Nonlinear Systems: Stability
Region and Feasible Initial Control
Xiao-Bing Hu1∗
1
2
Wen-Hua Chen2
Department of Informatics, University of Sussex, Falmer, Brighton, BN1 9QH, UK
Department of Aeronautical and Automotive Engineering, Loughborough University, LE11 3TU, UK
Abstract: This paper proposes a new method for model predictive control (MPC) of nonlinear systems to calculate stability region
and feasible initial control profile/sequence, which are important to the implementations of MPC. Different from many existing methods,
this paper distinguishes stability region from conservative terminal region. With global linearization, linear differential inclusion (LDI)
and linear matrix inequality (LMI) techniques, a nonlinear system is transformed into a convex set of linear systems, and then the
vertices of the set are used off-line to design the controller, to estimate stability region, and also to determine a feasible initial control
profile/sequence. The advantages of the proposed method are demonstrated by simulation study.
Keywords: Model predictive control (MPC), stability region, terminal region, linear differential inclusion (LDI), linear matrix inequality (LMI).
1
Introduction
Model predictive control (MPC) has been widely adopted
in industry, and stability of MPC has attracted much attention of researchers in the past decades[1−4] . Terminal
penalty is a most widely used technique to guarantee stability of MPC, which introduces a terminal weighting term in
the performance index and (or) imposes extra constraints
on terminal state during on-line optimization. Terminal
penalty technique has achieved a great success in both linear systems, e.g., see [5] and [6], and nonlinear systems,
e.g., see [7-10]. For constrained nonlinear systems, a terminal equality constraint was firstly used to establish stability under some assumptions in [7]. That means the terminal state is required to arrive at a specific point in the
state space during on-line optimization. However, solving a
nonlinear optimization problem with equality constraints is
very time-consuming, and therefore is difficult to finish in a
given time period. Furthermore, the stability region of the
proposed MPC is very small. To avoid this, a dual mode
control scheme was proposed in [8]. This method employs
a local linear state feedback controller and a receding horizon controller, which replaces terminal equality constraints
with terminal inequality constraints. The receding horizon
controller is used to drive the terminal state into a terminal region determined by the terminal inequality, and then
the local linear controller is employed to guarantee stability. Obviously, the advantage of MPC is lost when the
local controller is activated. Recently, Reference [9] proposed a quasi-infinite MPC algorithm. Different from the
dual mode control scheme in [8], the local linear state feedback controller is just used to calculate the infinite horizon
cost of nonlinear system starting from terminal region. To
guarantee stability, a terminal cost which covers this infinite
horizon cost is added into the performance index of MPC.
Manuscript received February 14, 2006; revised November 22, 2006.
This work was supported by an Overseas Research Students Award
to Xiao-Bing Hu.
*Corresponding author. E-mail address:
[email protected]
Therefore, the local controller is called a virtual linear stabilizing controller, and the advantage of MPC never loses
until the system arrives at the equilibrium. Reference [10]
even applied terminal penalty to more complicated nonlinear systems where computational delay and loss of optimality must be considered.
The terminal region discussed in the above papers refers
to a region where once the terminal state arrives under
the control sequence yielded by solving online optimization
problem, there exists a terminal control sequence, MPCbased or not, to steer the system state to the equilibrium.
This is quite different from the definition of stability region,
which is a set of initial states from which the state trajectory, under the control sequence yielded by solving online
optimization problem, will arrive in the terminal region by
the end of receding horizon. Terminal region can be used as
an estimation of stability region, just as the above papers
do, because it is included in the associated stability region,
but usually it is conservative due to the gap between terminal region and stability region. As will be proved later
in this paper, a method which distinguishes stability region
from terminal region can make a better estimation of stability region.
Another issue worthy of investigation is the importance
of initial control profile/sequence to start online optimization. Simply speaking, a properly chosen initial control
profile/sequence can help to make good estimation of stability region, and also to improve computational efficiency
of MPC. For MPC of nonlinear systems, due to the heavy
computational burden of solving online optimization problem, computational delay is normally too large to be ignored. Sometimes, a sampling time interval runs out even
before any feasible solution has been found, let alone optimal ones. In such a case, a properly chosen feasible initial
control profile/sequence is crucial to successful implementations of MPC to nonlinear systems.
Actually, the idea of distinguishing stability region from
terminal region and the importance of initial control pro-
196
International Journal of Automation and Computing 04(2), April 2007
file/sequence have already been studied in some papers on
MPC for linear systems, e.g., see [11] and [12]. This paper
aims to address the same issues of MPC for nonlinear systems, where both estimation of stability region and feasible
initial control profile/sequence are practically more important than in the case of linear systems. In the proposed
method, instead of using terminal region as estimation of
stability region, an offline algorithm is introduced to estimate stability region, and at the same time, to find a series
of state feedback control laws which are used to calculate
feasible initial control profile/sequence. The remainder of
this paper is organized as follows. Constrained MPC problem for nonlinear systems is formulated in Section 2. The
new method is described in Section 3. Stability and feasibility are analyzed in Section 4. Section 5 reports some
simulation results. The paper ends with some conclusions
in Section 6.
2
Problem formulation
Consider a nonlinear system
ẋ(t) = f (x(t), u(t)), x(t0 ) = x0
(1)
subject to control constraints
u(t) ∈ U
n
(2)
m
where x ∈ R and u ∈ R are state and control vectors,
respectively, and 0 ∈ U ⊂ Rm is a compact and convex
set. In this paper, hatted variables are used in the receding
horizon time frame, in order to distinguish the real variables. In general, a nonlinear MPC problem can be stated
as: for any state x at time t, find a continuous function
û(τ ; x(t)) : [t, t + T ] → U , in a receding horizon time frame
T , such that the performance index
Z T
J = g(x̂(t + T )) +
(x̂(t + τ )T Qx̂(t + τ )+
0
û(t + τ ; x(t))T Rû(t + τ ; x(t)))dτ
(3)
is minimized, where Q ≥ 0 and R > 0 are weighting matrices, and û(·; x(t)) is the control profile. û depends on
the state measurement x(t) at time t. It is required that
g(x) should be a continuous differentiable function of X,
g(x) = 0 and g(x) > 0 for all 0 6= x ∈ Rn . A typical choice
of g(x) is given by
g(x(t)) = x(t)T P x(t)
(4)
where P ∈ Rn×n is a positive definite matrix. The above
MPC problem can be mathematically formulated as
min
û(τ,x(t)):(t,t+T )
J
(5)
subject to the system dynamics (1), input constraints (2)
and terminal state constraint x̂(t + T ) ∈ v, where ν is a
terminal region.
Let the optimal solution to the optimization problem
(OP) (5) be denoted as û∗ . Then the nonlinear MPC law
is determined by
u(t) = û∗ (t, x(t)).
(6)
Similar to [10], the following assumptions on the system (1)
are imposed:
Assumption 1. f : Rn × Rm → Rn is twice continuously differentiable and f (0, 0) = 0, 0 ∈ Rn is an equilibrium of the system with u = 0.
Assumption 2. System (1) has a unique solution for
any initial condition x0 ∈ Rn and any piece-wise continuous and right-continuous u(·) : [0, ∞) → U .
Assumption 3. The nonzero state of system (1) is detectable in the cost. That is, Q1/2 x 6= 0 for all nonzero X
such that f (x, 0) = 0[13] .
Assumption 4. All states, x(t), are available.
Basically, to solve the OP (5), a feasible initial control
profile needs to be determined online, such that terminal
constraint will be satisfied and stability can then be guaranteed. Let δ denote a sampling time interval and δ < T .
Usually, the sub-profile ûinitial (τ ; x(t+δ)) : [t+δ, t+T ] → U
of an initial control profile can simply inherit the subprofile û∗ (τ ; x(t)) : [t + δ, t + T ] → U of the last optimal solution to the OP (5), and only the sub-profile
ûinitial (τ ; x(t + δ)) : [t + T, t + T + δ] → U , i.e., the initial
terminal control, needs to be determined online according
to stability requirements. Starting from this initial control
profile, the result of online optimization can usually steer
the state trajectory into the terminal region ν by the end
of receding horizon. In most literature, terminal region is
directly used as an estimation of stability region, but little information is given about which state out of terminal
region can be chosen as initial state. Clearly, it could be
very conservative to use terminal region, which is related to
terminal state, to estimate stability region, which is defined
as a set of initial states.
The basic idea of the new method proposed in this paper
is to distinguish estimated stability region from terminal
region, and then calculate a feasible initial control profile
ûinitial (τ ; x(t))[t, t + T ] → U which will lead to a feasible
solution to the OP (5) to steer any state from estimated
stability region into terminal region by the end of receding
horizon. However, it is not an easy task to online calculate
feasible initial control profile, particularly for nonlinear systems. To avoid this, global linearization and linear differential inclusion (LDI) techniques are adopted to transform
the nonlinear MPC problem given by (1)∼(6), to make it
possible to offline determine feasible initial control laws.
Firstly, a LDI is defined as
h x i » x(0) – » x –
0
ẋ ∈ Θ
,
=
(7)
u
u(0)
u0
where Θ ⊆ Rn×(n+m) . Consider system (1). Suppose for
each [x(t), u(t)] and t, there is a matrix G(x(t), u(t)) ∈ Θ
such that
»
–
x(t)
f (x(t), u(t)) = G(x(t), u(t))
.
(8)
u(t)
Then every trajectory of the nonlinear systems (1) is also
a trajectory of the LDI defined by Θ. If we can prove
that every trajectory of the LDI defined by Θ has a certain
property (e.g., reduces into the terminal region), then every trajectory of the nonlinear system (1) has this property.
Conditions that guarantee the existence of such a matrix G
197
X. B. Hu et al./ Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control
are f (0, 0) = 0 and
»
–
∂f ∂f
∈ Θ for all x(t), u(t) and t.
∂x ∂u
such that Condition (9) is satisfied.
3) Suppose C0 Θ has l vertices, and they are
(9)
By the Relaxation Theorem[14] , one may also assume Θ
is a convex set for each x(t), u(t) and t. The LDI given by
h x i » x(0) – » x –
0
=
(10)
,
ẋ ∈ C0 Θ
u
u(0)
u0
is called the relaxed version of LDI (7). Since C0 Θ ⊇ Θ,
every trajectory of the nonlinear system (1) is also a trajectory of relaxed LDI (10). Actually, we will not need the
Relaxation Theorem, or rather, we will get it “for free” in
this paper. The reason is that if a quadratic Lyapunov function, e.g., a quadratic performance index as used by MPC
in this paper, is adopted to establish some properties for
the LDI (7), then the same Lyapunov function establishes
the same properties for the relaxed LDI (10)[14] .
The properties of every point in C0 Θ, a convex set, can be
revealed by studying the properties of the vertices. If for all
vertex systems, there exists an initial control profile which
can steer any state from estimated stability region into terminal region, then it is feasible to any system within C0 Θ.
For more details about the problem formulation, readers
are suggested to refer to [10], [14] and [15].
3
Stability region and new MPC
Definition 1. Terminal region ν is defined as a region
where once the state x̂(t + T ) arrives, under the control
û∗ (τ ; x(t)) : [t, t + T ] → U yielded by solution to the OP
(5), there exists a control û : [t+T, ∞] → U which can steer
the state to the origin.
Definition 2. Stability region M refers to a set of initial
states from which the optimal state trajectory x̂(·)∗ : [t, t +
T ], under the optimal open-loop control profile û∗ (τ ; x(t)) :
[t, t + T ] → U yielded by solving the OP (5), will arrive in
the terminal region ν by terminal time t + T .
As mentioned above, many existing MPC methods simply use terminal region as an estimation of stability region.
The new MPC proposed in this paper will distinguish estimated stability region from terminal region, particularly
for nonlinear systems. The new method is composed of two
parts: offline algorithm and online algorithm. The offline
algorithm, which is the core of the new method, aims to
make an estimation of stability region as large as possible,
and at the same time, to find a series of feasible initial control laws which can improve the computational efficiency of
online optimization. The online algorithm is used to calculate optimal control profile over receding horizon.
3.1
Offline algorithm
1) Make a global linearization of system (1):
»
–
∂f ∂f x(t)
]
.
f (x(t), u(t)) = [
∂x ∂u u(t)
∂f ∂f
]
∂x ∂u
Then, we can construct l linear vertex systems
ẋ(t) = A(r)x(t) + B(r)u(t), u(t) ∈ U ; r = 1, . . . , l. (13)
Given a sampling time interval δ , discretize the above
continuous-time vertex systems. Suppose the corresponding discrete-time vertex systems are
x(k + 1) = Ā(r)x(k) + B̄(r)u(k), u(k)2 ≤ ū2 , r = 1, . . . , l
(14)
with discrete-time performance index
¯
J(k)
= x(k + N |k)T P x(k + N |k)+
N
−1
X
(x(k + i|k)T Qx(k + i|k) + u(k + i|k)T Ru(k + i|k)).
i=0
(15)
4) Determine a terminal region ν according to any existing
MPC method. In this paper, the method reported in [6]
is used to determine the terminal region ν based on those
vertex systems given in (14):
ν = {x ∈ Rn : xT P x ≤ 1}
(16)
where the matrix P is optimized such that the terminal
region ν is as large as possible. For the sake of identification, hereafter, TMPC is used to denote the MPC method
adopted to determine terminal region, and SMPC denotes
the new MPC proposed in this paper. Based on ν, an estimation of stability region and a feasible initial control profile will be determined in the following steps.
5) Solve the following optimization problem
min log(det(S −1 ))
(17)
S,S̄
subject to
"
"
S
Ā(r)N S + Γ̄ (r)S̄
Y
(Si )T
Si
S
(Ā(r)N S + Γ̄ (r)S̄)T
W
#
≥ 0, r = 1, . . . , l
(18)
#
2
j
≥ 0, Yjj ≤ ū , i = 0, . . . , N − 1; j = 1, . . . , m
(19)
where
W = P −1
S̄
= [S0T
N −1
...
(20)
T
T
SN
−1 ]
(21)
Ā(r)0 B̄(r)]
(22)
M = {x ∈ Rn : xT Zx ≤ 1}, where Z = S −1 .
(23)
Γ̄ (i) = [Ā(r)
B̄(r)
...
n×n
and 0 < S ∈ R
.
6) Estimate the stability region M as
7) Calculate the discrete-time initial control sequence as
(11)
K(k + i) = Si Z, u(k + i|k) = K(k + i)x(k),
i = 0, . . . , N − 1; k ≥ 0
(24)
and the associated continuous-time initial control profile is
2) Choose a relaxed LDI defined by
C0 Θ = [
V (r) = [A(r) B(r)], r = 1, . . . , l.
(12)
û(τ ) = u(k + i|k), τ ∈ [t + iδ, t + iδ + δ), i = 0, . . . , N − 1.
(25)
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International Journal of Automation and Computing 04(2), April 2007
3.2
Online algorithm
1) Measure the state x(t). Let x(k) = x(t). If x(k) ∈ M but
x(k) ∈
/ ν, then calculate the initial control profile according
to (24) and (25), in order to steer the system state from
M into ν; otherwise, determine the initial control profile
according to the TMPC.
2) If x(k) ∈ M but x(k) ∈
/ ν, set the performance index
as (15); otherwise, set it according to the TMPC. Then
solve the OP (5). Basically, the OP (5) is a quadratic programming problem, which can be solved by some standard
algorithms such as active set methods and interior point
methods[2] .
3) Let t = t + δ and go to 1).
4
Stability and feasibility
Stability is guaranteed by the TMPC which is used to determine the terminal region ν. Based on this ν, the SMPC
is implemented to estimate the stability region M as large
as possible, and also to guarantee that the system state can
be driven from M into ν. The following theorem establishes
the feasibility of the SMPC.
Theorem 1. For those discrete-time vertex systems
given in (14), suppose there exist S and s̄ such that conditions (18) and (19) hold. Then the SMPC proposed in
Section 3 is feasible to steer any initial state from M into
ν.
Proof. Set the initial control sequence according to (24),
i.e.,
u(k +i|k) = K(k +i)x(k), K(k +i) = Si Z, i = 1, . . . , N −1.
(26)
Then one has
N
x(k + N |k) = (Ā(r) + Γ̄ (r)KN (k))x(k), r = 1, . . . , l (27)
where
2
6
KN (k) = 6
4
3
K(k)
7
..
7.
.
5
K(k + N − 1)
Using the transforms (20), (23) and (26), one can see
that (29) is equivalent to condition (18). Similar to [6], it
is easy to prove that condition (19) guarantees that input
constraints are satisfied.
According to the LDI theory, when the vertex system has
a certain property, then any system within the C0 Θ has the
same property.
¤
Remark 1. Theorem 1 gives an estimation of stability
region for initial state instead of terminal region ν for terminal state. It also gives a feasible initial control sequence
to steer any state from M into ν.
Remark 2. Theorem 1 can not guarantee that starting
from the initial control sequence, the solution to the OP
(5) will always drive the system state from M to ν. There
are two ways to improve Theorem 1. One way is to impose
a terminal constraint x(k + N |k) ∈ ν on the OP (5). The
other is to modify the offline algorithm of SMPC such that
the optimal control sequence yielded by solving the OP (5)
automatically steers terminal state into ν. In general, a
larger stability region is achieved by the former, but at the
cost of heavier online computational burden. Theorem 2
gives the set by which the system state automatically arrives in ν under the control sequence yielded by solving the
OP (5).
Theorem 2. Suppose there exist matrices S > 0 and S̄
such that (30) and (31) are obtained.
#
"
Y
(Si )T
Si
S
≥ 0, Yjj ≤ ū2j , i = 0, . . . , N −1; j = 1, . . . , m
(31)
hold where
1/2
1/2
Φ̄(r)Q
Φ̄(r)N , Γ̄ (r)Q
Γ̄ (r)N
N = (QN )
N = (QN )
(32)
QN = diag {Q, . . . , Q}, RN = diag {R, . . . , R}
|
|
{z
}
{z
}
(33)
N
2
(28)
Γ̄ (r)N
If
Z ≥ (Ā(r)N + Γ̄ (r)KN (k))T P (Ā(r)N + Γ̄(r)KN (k)) (29)
6
6
6
=6
6
6
4
0
B̄(r)
Ā(r)B̄(r)
..
.
Ā(r)N −2 B̄(r)
1 ≥ x(k)T (Ā(r)N + Γ̄ (r)KN (k))T P (Ā(r)N +
Φ̄(r)N
T
Γ̄ (r)KN (k))x(k) = x(k + N |k) P x(k + N |k).
This means: if Condition (29) is satisfied, there is at last
one control sequence to steer any initial state within M into
ν.
6
6
6
4
−S
Ā(r)N S + Γ̄ (r)S
Q
Φ̄(r)Q
N S + Γ̄ (r)N S̄
S̄
(Ā(r)N S + Γ̄ (r)S)T
−W
0
0
0
0
B̄(r)
..
.
Ā(r)N −3 B̄(r)
···
···
···
..
.
···
2
then because x(k)T Zx(k) ≤ 1, one has
2
N
3
I
6
7
..
7
=6
.
4
5
Ā(r)N −1
0
0
0
..
.
B̄(r)
3
0
0 7
7
7
0 7
.. 7
7
. 5
0
(34)
(35)
where S̄ and Γ̄ (r) are given in (21) and (22) respectively.
Then the optimal control sequence of SMPC is feasible to
automatically steer any initial state from M into ν.
Q
T
(Φ̄(r)Q
N S + Γ̄ (r)N S̄)
0
−I
0
S̄ T
0
0
−(RN )−1
3
7
7
7 ≤ 0; r = 1, . . . , l,
5
(30)
X. B. Hu et al./ Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control
Proof. According to (26)∼(28), one can re-write the
performance index (15) as
¯
J(k)
= x(k)T ((Ā(r)N + Γ̄ (r)KN (k))T P ((Ā(r)N
+ Γ̄ (r)KN (k)) + (Φ̄(r)N + Γ̄ (r)N KN (k))T QN (Φ̄(r)N
+ Γ̄ (r)N KN (k)) + KN (k)T RN KN (k))x(k)
(36)
If
Z ≥ (Ā(r)N + Γ̄ (r)KN (k))T P ((Ā(r)N + Γ̄ (r)KN (k))+
(Φ̄(r)N + Γ̄ (r)N + Γ̄ (r)N KN (k))T QN (Φ̄(r)N +
Γ̄ (r)N KN (k)) + KN (k)T RN KN (k)
(37)
¯
¯ ∗≥
then because x(k)T Zx(k) ≤ 1, one has 1 ≥ J(k)
≥ J(k)
¯ ∗ and x∗ (·) reprex∗ (k + N |k)T P x∗ (k + N |k), where J(k)
sent the optimal performance index and the associated state
respectively.
One can see that if condition (37) is satisfied, the solution to the OP (5) can automatically steer any terminal
state from M into ν.
By using the transforms (20), (23) and (26), one can see
that condition (37) is equivalent to condition (30). Condition (31) guarantees that input constraints are satisfied.
According to the LDI theory, when the vertex system has
a property, then any system within the C0 Θ has the same
property.
¤
Remark 3. Theorem 2 also gives an estimation of stability region M, but it is more conservative than Theorem
1. The advantage of Theorem 2 is that it gives a feasible
initial control sequence, by which the SMPC can automatically steer the system state trajectory from M to ν.
Remark 4. According to Theorem 2, Step 5 in the offline algorithm needs to be modified, i.e., when the optimization problem (17) is to be solved, conditions (30) and
(31) instead of (18) and (19) must be satisfied.
5
Simulation results
Consider the following system
(
ẋ1 = x2 + u(µ + (1 − µ)x1 )
ẋ2 = x1 + u(µ − 4(1 − µ)x2 )
or ẋ = f (x, u)
(38)
which is borrowed from [7] and is unstable for any µ ∈ (0, 1).
Assume µ = 0.5 in this simulation. The performance index
is chosen as (3) with g(x) defined by (4). Weighting matrices are
#
"
0.5
0
, R = 1.0.
(39)
Q=
0
0.5
Input constraint is given as
U = {u ∈ R| − 1 ≤ u ≤ 1}.
(40)
A sampling time interval δ = 0.1 time-units. The length
of receding horizon is 1.5 time-units. In other words, the
receding horizon is 15 steps long.
In the following simulation study, the MPC in [6] is used
to determine terminal region and the new MPC aims to estimate stability region as large as possible. For the sake of
identification, the new MPC based on Theorem 1 is denoted
199
as SMPC1, the one on Theorem 2 as SMPC2, while the
MPC in [6] as TMPC. Four methods are used to set initial
control profiles for TMPC in order to investigate the feasibility of TMPC under computational time limit, and they
are denoted as TMPC1, TMPC2, TMPC3 and TMPC4 respectively. Table 1 explains these methods as well as the
methods for SMPC1 and SMPC2 to set up initial control
profile. In Table 1, Kterm is the terminal gain determined
offline by TMPC, û∗ (·; x(t0 )) is the optimal control profile
at time t = t0 , Kstab1 (·) and Kstab2 (·) are feasible control
gain sequences determined offline by SMPC1 and SMPC2
respectively, and Inheriting means to inherit the optimal
solution of last run of online optimization except its first
element.
5.1
Estimation of stability region
Firstly, global linearization technique is used to determine a C0 Θ:
#
»
– "
0.5u
1
0.5 + 0.5x1
∂f ∂f
C0 Θ =
=
. (41)
∂x ∂u
1
−2u
0.5 − 2x2
For the sake of simplification, we assume
xi ∈ [−1 1], i = 1, 2.
(42)
Therefore, C0 Θ has 8 vertices. Terminal region, estimated
stability region and feasible initial control laws are then
calculated based on the vertices of C0 Θ.
Fig. 1 gives the terminal regions. Each dashed terminal
region is related to a certain vertex system, while the solid
region is calculated for all 8 vertex systems. All terminal
regions are calculated by TMPC. The stability regions estimated by SMPC1 are illustrated in Fig. 2, and the stability
regions estimated by SMPC2 in Fig. 3.
From Figs. 1∼3, one can make the following observations:
1) All estimated stability regions are larger than the associated terminal regions. 2) The terminal region or estimated
stability region for a certain vertex system is much larger
than that for all 8 vertex systems. Actually, the latter is
just a subset of the intersection of all formers. 3) The stability regions estimated by SMPC1 are larger than those
by SMPC2. 4) The stability regions estimated by SMPC 2
are similar to the associated terminal regions. This implies
that the conditions in Theorem 2 are restrictive.
To further enlarge estimated stability region, we introduce an iteration process which repeatedly applies SMPC1
or SMPC2. This means when a new estimation of stability
region is made, it is then used as a terminal region to calculate another larger new estimation of stability region. In
other words, increasing the length of receding horizon can
effectively enlarge estimated stability region. However, for
TMPC in [6], the length of receding horizon has no influence on terminal region. This implies TMPC is somehow
conservative and unreasonable. Fig. 4 gives the result of
repeating SMPC1.
5.2
Online performance
The following simulation study is conducted with two
different initial states: Case 1, x0 = (0.2, 0.2); Case 2,
x0 = (0.5, −0.6). The control performances are given in
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Figs. 5 and 6, where dashed lines are related to Case 1,
while solid lines to Case 2.
Suppose a sampling time interval (0.1 time-units) is long
enough for solving the online optimization problem, i.e.,
feasible global-optimal solution can always be found within
a sampling time interval. We then have Fig. 5 and Table 2.
Fig. 5 gives the online control performances of TMPC and
SMPC, which are almost the same. This is understandable:
if a sampling time interval is long enough, because the same
performance index is applied to all controllers, they should
find the identical optimal control profile, and then achieve
the same control performance. Actually, the main difference between these controllers is that they adopt different
initial control profiles to start online optimization.
Table 2 gives the computational burdens under the assumption that a sampling time interval is long enough.
For SMPC, the online optimization starts from a series of
fixed initial control laws, which are determined offline in
advance, while for TMPC, it starts by following some experiential guidelines, as listed in Table 1. From Table 2, one
can reach the following conclusions: 1) TPMC1, TPMC2
and TMPC4 take relatively less computational time than
TMPC3, SMPC1 and SMPC2; the possible reason is because TPMC1, TPMC2 and TMPC4 inherit the optimal
solution of last run of online optimization. 2) Generally,
SMPC2 completes online optimization faster than SMPC1,
due to the reason discussed in Remark 2 in Section 4. 3)
The average computational time associated with a long simulation time is less than that with a short one, because at
the beginning of simulation, the system state is far away
from the origin and consequently it is more difficult to find
out the optimal solution.
A more general situation for nonlinear systems is that
the online optimization can probably not be completed in
a sampling time interval. For example, if a time-unit is 10
seconds, then a sampling time interval is just 1 second, less
than those maximum computational times given in Table
2. If the online optimization can not be completed in time,
because no feasible initial control profile is available, TMPC
just simply adopts the latest solution to determine the actual control signal, no matter whether it is feasible or not.
SMPC1 uses the latest solution if and only if this solution
is feasible, i.e., it can drive the state trajectory into the terminal region at the end of receding horizon. Otherwise, the
associated initial control laws, which have been determined
offline, are used to calculate the actual control signal. For
SMPC2, starting from the offline-determined feasible initial
control laws, any half-done solution to the OP (5) is feasible to stabilize the system. As shown in Fig. 6, TMPC can
not stabilize the system when the sampling time interval is
1 second and the initial state is out of the terminal region,
while both SMPC1 and SMPC2 still have good online control performances. Fig. 6 demonstrates that a feasible initial
control profile is very important to the implementations of
MPC for nonlinear systems.
6
ferent from many existing MPC methods, the new method
distinguishes estimated stability region from conservative
terminal region, and stability region can be estimated offline as large as possible; 2) a feasible initial control profile
can be determined off-line to guarantee stability when computational time limit presents; 3) global optimum to online
nonlinear optimization is not necessary to establish stability.
Fig. 1
Terminal regions
Fig. 2
Stability regions under SMPC 1
Fig. 3
Stability regions under SMPC 2
Conclusions
A practicable MPC scheme for general nonlinear systems
is proposed in this paper. Simulation study illustrates that
the main advantages of the new method include: 1) dif-
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X. B. Hu et al./ Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control
Table 1
Methods to set initial control profiles
Beginning
Initial control profile ûinitial (·; x(t))
time
TMPC 1
TMPC 2
TMPC 3
TMPC 4
SMPC 1
SMPC 2
t = t0
[0, . . . , 0]
[0, . . . , 0]
[0, . . . , 0]
û∗ (·; x(t0 ))
Kstab1 (·) × x(t0 )
Kstab2 (·) × x(t0 )
t > t0
[Inheriting, Kterm x(t)]
[Inheriting, 0]
[0, · · · , 0]
[Inheriting, Kterm x(t)]
Kstab1 (·) × x(t)
Kstab2 (·) × x(t)
Table 2
Comparison of computational time (the sampling time is long enough)
Simulation time (time-unite)
Maximum
Case 1: x0 = (0.2, 0.2);
0.5
5.0
10.0
computational time(s)
Case 2: x0 = (0.5, −0.6).
Case 1
Case 2
Case 1
Case 2
Case 1
Case 2
Case 1
Case 2
Average
TMPC 1
2.362 0
2.866 0
1.440 2
1.339 2
1.077 6
0.870 0
3.130 0
5.440 0
computational time of
TMPC 2
2.504 0
1.878 0
1.443 4
1.239 0
1.057 3
0.873 3
2.690 0
2.860 0
a run of online OP
TMPC 3
3.604 0
3.078 0
1.844 4
1.542 4
1.322 6
1.134 3
3.970 0
4.610 0
solver(s)
TMPC 4
1.878 0
1.912 0
1.414 8
1.268 8
1.062 8
0.838 8
2.090 0
4.280 0
SMPC 1
3.218 0
3.658 0
2.286 0
2.245 4
1.993 2
1.623 0
5.000 0
5.820 0
SMPC 2
3.516 0
3.350 0
2.018 0
1.956 4
1.549 4
1.461 6
3.710 0
3.620 0
Fig. 4
Stability regions under iteration of SMPC 1
Fig. 6 Control/state profiles under different MPC schemes
with a sampling time of 1 second
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Xiao-Bing Hu received his B. Sc. degree in aviation electronic engineering at
Civil Aviation Institute of China, Tianjin, China, in 1998, the M. Sc. degree in
automatic control engineering at Nanjing
University of Aeronautics & Astronautics,
Nanjing, China, in 2001, and the Ph.D. degree in aeronautical and automotive engineering at Loughborough University, UK,
in 2005. He is currently a research fellow in
Department of Informatics at Sussex University, UK.
His research interests include predictive control, artificial intelligence, air traffic management, and flight control.
Wen-Hua Chen received his M. Sc and
Ph. D. degrees from Department of Automatic Control at Northeast University,
China, in 1989 and 1991, respectively.
From 1991 to 1996, he was a lecturer in Department of Automatic Control at Nanjing
University of Aeronautics & 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. He holds a senior lectureship in flight control systems in Department of Aeronautical and
Automotive Engineering at Loughborough University, UK.
He has published one book and more than 80 papers on journals and conferences. His research interests include the development of advanced control strategies and their applications in
aerospace engineering.