3 LINEAR MPCFEATURE OF CORRESPONDING ALGORITHMS IS THAT THEY GUARANT...

3.3 Linear MPCfeature of corresponding algorithms is that they guarantee linear closed-loop system stabilityIn this particular case, MPC is based on the linear prediction model. These algorithms areat each sampling period. It is necessary to remark that in the case of traditional MPC algorithmcomputationally efficient which is especially important from the real-time implementationimplementation, described above, closed-loop system stability can be provided only for thepoint of view.simplest case when we have a linear prediction model, quadratic cost functional and withoutGenerally, linear prediction model is presented byconstraints.Let us assume that the mathematical model of the plant to be controlled is described by thex˜

_i+1

=

Ax˜

_i

+

Bu˜

_i

, i

=

k

+

j, j

=

0, 1, 2, ..., x˜

_k

=

x

_k

,following system of difference equationsy˜

_i

=

C˜x

_i

. (15)xˆ

_k+1

=

F

(

xˆ

_k

, ˆu

_k

, ˆϕ

_k

)

,Suppose ¯u

=

u˜

k

u˜

k+1

... u˜

k+P

−

1

T

is the programmed control over the predictionyˆ

k

=

Cxˆ

k

. (21)horizon. Then, integrating (15) we obtain future outputs of the plant in the formHere ˆy

k

∈E

^s

is the vector of output variables, ˆx

k

∈E

ⁿ

is the state space vector, ˆu

k

∈E

^m

is they¯

=

Lx

k

+

Mu,¯ (16)vector of controls, ˆϕ

_k

∈E

^l

is the vector of external disturbances.whereEquations (21) can be used as a basis for nonlinear prediction model construction. SupposeCB 0 . . . 0CAthat obtained prediction model is given byCA

²

CAB ....L

=

...x˜

_i+1

=

f

(

x˜

_i

, ˜u

_i

)

, i

=

k

+

j, j

=

0, 1, 2, ..., x˜

_k

=

x

_k

,, M

=

... ...y˜

i

=

Cx˜

i

. (22)CA

^P

CA

^P

⁻

¹

B . . . CAB CBPlasma stabilization system design on the base of model predictive control 205