AS A RESULT WE OBTAIN VECTORS SEQUENCE{X˜I}, (I = K+1, ...,K+P), WH...

1. As a result we obtain vectors sequence{x˜

i

},

(

i

=

k

+

1, ...,k

+

P

)

, which represents theϕ.¯

=

0−B

c

(

h

)

C E

ν

z−A

c

(

h

)

ξprediction of future plant behavior over the prediction horizonP. It must be noted, that thecontrol sequence ˜u

k

, ˜u

k+1

, ... ˜u

k+P

−

1

over this horizon is determined uniquely by the choiceTherefore, the closed-loop system characteristic polynomial∆

(

z,h

)

is given byof parameter vectorh. So, in this case the problem of control is reduced to the problem of E

n

z−A−BD

c

(

h

)

C −BC

c

(

h

)

parameters vectorhtuning..∆

(

z,h

) =

detThe controlled processes quality over the prediction horizonPcan be presented by the fol-lowing cost functionalLet us denote the degree of the polinomial∆

(

z,h

)

byn

_d

.Let find parameter vectorh, which provide a given roots for the system (28) characteristicJ

_k

=

J

_k

(

{x^˜

i

}^,{u^˜

i

}

) =

J

_k

(

W

(

q,h

)) =

J

_k

(

h

)

≥^{0, (26)}polynomial. In other words, it is nesessary to find such parameter vectorhthat provide thewhere{x^˜

_i

}^, i

=

k

+

1, ...,k

+

P, {u^˜

_i

}^, i

=

k, ...,k

+

P−1 are the state and control vectorsfollowing identitysequences correspondently, which satisfies the system of equations (25). It is easy to see, that∆

(

z,h

)

≡∆^˜

(

z

)

,the cost functional (26) is reduced to the function of parameter vectorh.where ˜∆

(

z

)

is a given polynomial with degreen

_d

, having desired roots. In order to find vectorLet us consider the following optimization problemh, equate the correspondent coefficients for the same degrees of z-variable. As a result obtainthe system of

(

n

_d

+

1

)

nonlinear equations withr unknown components of vectorhin the, (27)J

_k

=

J

_k

(

h

)

→ ^infform

h∈Ω

H

L

(

h

) =

γ. (29)where Ω

H

is a set of parameter vectors providing that the eigenvalues of the closed-loopsystem (23), (24) are placed in the desired areaC

_∆

inside a unit circle.It is evident that the controller (24) has a full structure if and only if the system of equations(33) has a solution for any vectorγ.It is necessary to remark that the problem (27) is a nonlinear programming problem with anextremely complicated definition of the cost function, which, in generall, has no analyticalPlasma stabilization system design on the base of model predictive control 207representation and is given only algorithmically. Besides that, the specific character of theHerex

k

∈ E

ⁿ

is the actual state of the plant at time instantkor its estimation on the base ofmeasurement output.problem (27) is also defined by the complicated constraints imposed, which determines theLet desired object dynamics is presented by the given vector sequences{r

^x

_k

}^and{r

^u

_k

}^{, k =}admissible areas of eigenvalues displacement. It must be noted, that the dimension of the0,1,2,... . The linear mathematical model of the plant, describing its behavior in the neighbour-optimization problem (27) is defined only by the dimension of parameter vectorhand it doesnot depend on the prediction horizonPvalue.hood of the desired trajectory, can be obtained by performing the equations (21) linearization.As a result of this action, we get the linear system of difference equationsDefinition 1. We shall say that the controller (24) has afull structureif the degrees of polyno-mials in the numerators and denominators of the matrixW

(

q,h

)

components and the struc-x¯

k+1

=

Ax¯

k

+

Bu¯

k

+

Hϕ¯

_k

,ture of parameter vectorhare such that it is possible to assign any given roots for closed-loopy¯

_k

=

Cx¯

_k

, (23)system (23),(24) characteristic polynomial∆

(

z,h

)

by appropriate selection of parameter vectorwhere ¯x

_k

∈E

ⁿ

, ¯u

_k

∈E

^m

, ¯y

_k

∈E

^s

, ¯ϕ

_k

∈E

^l

are the vectors of the state, control input, measure-h.ments and external disturbances respectively. These vectors represent the deviations from theIn order to get another form of the presented definition, consider the equations of the closed-desired trajectory. Next we shall consider only such situations when all matrices in equationsloop system (23),(24). They can be represented in the normal form as follows(23) have constant elements. In the framework of proposed approach, the control input overthe prediction horizon is generated by the controller of the formx¯

_k+1

=

Ax¯

_k

+

Bu¯

_k

+

Hϕ¯

_k

,y¯

_k

=

C¯x

_k

,u¯

_k

=

W

(

q,h

)

y¯

_k

. (24)(28)ξ

_k+1

=

A

c

(

h

)

ξ

_k

+

B

c

(

h

)

y¯

k

,Hereqis the shift operator,W

(

q,h

)

is the controller transfer function with the fixed structureu¯

_k

=

C

c

(

h

)

ξ

_k

+

D

c

(

h

)

y¯

_k

,(that is the degrees of the polynomials in the numerator and denominator of all its componentswhereξ

_k

∈E

^ν

is a controller (24) state vector. After applying Z-transformation to the systemare given and fixed),h∈E

^r

is the vector of tuned parameters, which must be chosen on theof equations (28) with zero initial conditions, obtainstage of control design.The prediction model equations (22), closed by the feedback (24), can be presented as follows

(

E

n

z−A

)

x¯

=

Bu¯

+

Hϕ,¯x˜

_i+1

=

f

(

x˜

_i

, ˜u

_i

)

, i

=

k

+

j, j

=

0, 1, 2, ..., x˜

_k

=

x

_k

,

(

E

ν

z−A

c

(

h

))

ξ

=

B

c

(

h

)

C¯x,u¯

=

C

c

(

h

)

ξ

+

D

c

(

h

)

C¯x,u˜

i

=

r

^u

_i

+

W

(

q,h

)

C

(

x˜

i

−r

^x

_i

)

. (25)y¯

=

Cx¯,Let us assume that parameters vectorhis chosen and fixed. Then we can solve system ofor E

n

z−A−BD

c

(

h

)

C −BC

c

(

h

)

difference equations (25) with a given initial conditions for the instantsi

=

k,k

+

1, ...,k

+

P− Hx¯