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 CBHerex

k

∈ E

ⁿ

is the actual state of the plant at time instantkor its estimation on the base ofrepresentation and is given only algorithmically. Besides that, the specific character of themeasurement 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

_k

^x

}^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

=

A¯x

k

+

Bu¯

k

+

Hϕ¯

_k

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

_k

=

C¯x

_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,¯

(

E

ν

z−A

c

(

h

))

ξ

=

B

c

(

h

)

C¯x,u¯

=

C

c

(

h

)

ξ

+

D

c

(

h

)

Cx,¯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¯