2 MPC REAL-TIME IMPLEMENTATIONSTATE OF THE PLANT AT THE INSTANTTOR I...
3.2 MPC real-time implementationstate of the plant at the instanttor its estimation based on measurement output.In order for real-time implementation, piece-wise constant functions are used as a pro-This model is used to predict future outputs of the process given the programmed controlgrammed control over the prediction horizon. That is, the programmed control ˜u
(
τ)
is pre-u˜(
τ)
over a finite time intervalτ ∈[
t,t+
Tp
]
. Such a model is calledprediction modelandsented by the sequence{u˜k
, ˜uk+1
, ..., ˜uk+P−1
}, where ˜ui
∈ Em
is the control input at the timethe parameter Tp
is namedprediction horizon. Integrating system (8) we obtain ˜x(
τ) =
interval[
iδ,(
i+
1)
δ]
,δis the sampling interval. Note that,Pis a number of sampling intervalsx˜(
τ,x(
t)
, ˜u(
τ))
—predicted process evolution over time intervalτ∈[
t,t+
Tp
]
.over the prediction horizon, that isTp
=
Pδ. Likewise, general MPC formulation presentedThe programmed control ˜u(
τ)
is chosen in order to minimize quadratic cost functional overabove consider nonlinear prediction model in the discrete formthe prediction horizonx˜i+1
=
f(
x˜i
, ˜ui
)
, i=
k+
j, j=
0, 1, 2, ..., x˜k
=
xk
,t+T
p
y˜i
=
Cx˜i
. (12)J=
J(
x(
t)
, ˜u(
·)
,Tp
) =
t
((
x˜−rx
)
R
(
x˜)(
x˜−rx
) + (
u˜−ru
)
Q
(
x˜)(
u˜−ru
))
dτ, (9)Here ˜yi
∈Er
is the vector of output variables,xk
∈ En
is the actual state of the plant at timewhereR(
x˜)
,Q(
x˜)
are positive definite symmetric weight matrices,rx
,ru
are state and con-instantkor its estimation on the base of measurement output. We shall say that the sequencetrol input reference signals. In addition, the programmed control ˜u(
τ)
should satisfy all of theof vectors{y˜k+1
, ˜yk+2
, ..., ˜yk+P
}represents the prediction of future plant behavior.constraints imposed on the state and control variables. Therefore, the programmed controlSimilar to the cost functional (9), consider also its discrete analog given byu˜(
τ)
over prediction horizon is chosen to provide minimum of the following optimizationJk
=
Jk
(
y, ¯¯ u) =
∑P
j=1
(
y˜k+j
−ry
k+j
)
T
Rk+
j
(
y˜k+j
−ry
k+j
)
problem, (10)J(
x(
t)
, ˜u(·)
,Tp
)
→ min+ (
u˜k+j
−
1
−ru
k+j
−
1
)
T
Qk+j
(
u˜k+j
−
1
−ru
k+j
−
1
)
, (13)u(·)∈Ω
u
˜
whereΩu
is the admissible set given bywhereRk+j
andQk+j
are the weight matrices as in the functional (9),ry
i
andru
i
are the outputand input reference signals,Ωu
=
u˜(·)
∈K0
n
[
t,t+
Tp
]
: ˜u(
τ)
∈U, ˜x(
τ,x(
t)
, ˜u(
τ))
∈X, ∀τ∈[
t,t+
Tp
]
. (11)y¯=
y˜k+1
y˜k+2
... y˜k+P
T
∈ErP
,u¯=
u˜k
u˜k+1
... u˜k+P
−
1
T
Here, K0
n
[
t,t+
Tp
]
is the set of piecewise continuous vector functions over the interval∈EmP
[
t,t+
Tp
]
,U⊂Em
is the set of feasible input values,X⊂En
is the set of feasible state values.are the auxiliary vectors.Denote by ˜u∗
(
τ)
the solution of the optimization problem (10), (11). In order to implementThe optimization problem (10), (11) can now be stated as followsfeedback loop, the obtained optimal programmed control ˜u∗
(
τ)
is used as the input only onthe time interval[
t,t+
δ]
, whereδ<<Tp
. So, only a small part of ˜u∗
(
τ)
is implemented. AtJk
(
xk
, ˜uk
, ˜uk+1
, ... ˜uk+P
−
1
)
→ mintimet+
δthe whole procedure—prediction and optimization—is repeated again to find new{
u
˜
k
, ˜
u
k+1
,..., ˜
u
k+P−1
}∈Ω∈E
mP
, (14)optimal programmed control over time interval[
t+
δ,t+
δ+
Tp
]
. Summarizing, the basicis the admissible set.whereΩ=
u¯ ∈EmP
: ˜uk+j−1
∈U, ˜xk+j
∈X, j=
1, 2, ...,PMPC scheme works as follows:Generally, the functionJ(
xk
, ˜uk
, ˜uk+1
, ... ˜uk+P
−
1