Model predictive control for discrete systems with state and input delays Текст научной статьи по специальности «Математика»

2011

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА Управление, вычислительная техника и информатика

№ 1(14)

УПРАВЛЕНИЕ ДИНАМИЧЕСКИМИ СИСТЕМАМИ

УДК 519.2

M.Y. Kiseleva, V.I. Smagin

MODEL PREDICTIVE CONTROL FOR DISCRETE SYSTEMS WITH STATE AND INPUT DELAYS

The paper deals with Model Predictive Control synthesis based on the system output tracking with input and state delays. Input and state constraints are taken into account in MPC problem solving. A prediction is carried out on the base of object states estimation that is obtained by the Kalman filter. The criteria function is assumed to be convex quadratic.

Keywords: Model Predictive Control, discrete systems, state delay, input delay,

Kalman filter.

One of the modern formalized approaches to the system control synthesis based on mathematical methods of optimization is control methods using predictive models -Model Predictive Control (MPC).

his approach began to develop in the early 1960s. It was developed for equipment and process control in petrochemical and energy industries for which the application of traditional synthesis methods was extremely complicated according to mathematical

model’s complication. During the last years application was considerably extended cov-

ering technologic fields [1], economic system control [2], inventory control [3] and investment portfolio control [4].

The results of this paper extend the results of the paper [5].

1. Problem Statement

Suppose the object is described by the following state-space system of linear-difference equations

r ____ ______

xt+1 = Axt +x Ax - + But-h+wt, xk = xk, (k = -r,°X u = u, ( i = -h,-1) (1)

i=1

Vt = Hxt + vt, (2)

yt = Gxt. (3)

Here xt є Rn (xt = xt, t = -r,...,-1,0, xt is considered to be given) is the object state, ut єRm is the control input (ut = ut, t = -h ,-h+1,...,-1. ut is given), yt єRp is the output (which is to be controlled), yt є Rl is the observation (measured output), r, h are the state and input delay values respectively. Further, the state noise wt and meas-

urement noise vt are assumed to be Gaussian distributed with zero mean and covariances W and V respectively, i.e.

M{wtwTk} = W8t k, M{vvT} = VSa ,

where St k is the Kronecker delta. This model is used to make predictions about plant

behavior over the prediction horizon, denoted by N, using information (measurements

of inputs and outputs) up to and including the current time t.

It is supposed the plant operates under the constrained conditions:

a1 < S\xt < a2, (4)

91 (xt-h) < S2Ut-h < (?2(xt-h) (5)

Here S1 and S2 are structural matrices that are composed of zeros and units, identifying constrained components of vectors xt and ut ; a1, a2, 91(xt), 92(xt) are given constant vectors and vector-functions.

The problem is to determine an acting strategy on the base of the observation according to which the output vector of the system yt will be close to the reference taking into account constraints on the state and input.

2. Prediction

With the Gaussian assumptions on state and measurement noise it is possible to make optimal (in the minimum variance sense) predictions of state and output using a Kalman filter, see e.g. [6].

Let xij and y^j be estimates of the state and output at time i given information up

to and including time j where j < i. Then

r _____

xt+i|t = Axt\t-i +X4xMt-i-i + But-h + Kt(Vt-^xt|t-i), V-i = xk, k = ~r,0,

i=1

yt+i|t = Gxt+i|t,

Kt = APtHT (HPtHT + V)-1 ,

Pt+1 = W + APtAT - APtHT (HPtHT + V)-1 HPtAT, P0 = PXo, (6)

where Pxo is the given initial value of the variance matrix. Equation (6) for Pt is known

as the discrete-time Riccati-equation.

MPC usually requires estimates of the state and/or output over the entire prediction horizon N from time t + 1 until time t + N, and can only make these predictions based on information up to and including the current time t. Equations (6) can be used to obtain xt+1t , yt+1t. Optimal state/output estimates from time t + 2 to t + N can be obtained

as follows

r _____

xt+i+1|t=Axt+i|t+X Ajxt+<■- j|t - j-1+But-h+i|t, i=i>N, (7)

j=1

yt+i\t = G xt+i|t, i = 1>N. (8)

In the above the notation ut-h+j\t is used to distinguish the actual input at time t+i, namely ut-t+-i, from that used for prediction purposes, namely ut-h+i\t.

Equation (7) can be expanded in terms of the initial state Xt+1|t and future control actions ut-h+i\t as follows

*t+,it = A-1 x+i|t + §A-k-1 iAA+k-jt-j-1 + § A-k-1But-h+klt , i = \N. (9)

k=1 j=1 k=1

Now in terms of predicting the output, equation (8) can be expanded in terms of the above expression for xt+^t, which results in a series of equations that provide optimal

output predictions. The key point to note is that each output prediction is a function of the initial state xt+1|t and future inputs ut-h+i^t only:

yt+i|t = GAi-1XM]t + G§ Ai-k-1 £ AjXt+k-^t-j-1 + Ai-k-1But-h+k]t, i = W. (10)

k=1 j=1 k=1

This series of prediction equations can be stated in an equivalent manner using ma-

trix vector notation. Denote

xt+1|t " yt+1|t" Ut -h+1|t

Tt = Xt+N |t , Y = _yt+N|t _ , Ut-h = Ut -h+N|t _

¥ = 1 1 , A = 1 A2 Ag i , % = Xt+1-i|t - i

An-1 GAn -1 Xt+N-i|t -

" 0 0 0 0' 0 0 0 0"

A 0 0 0 GA 0 0 0

m 0 = i aa Ai 0 0 , A 0 = — GA AA G 0 0

i A N i a" CO - N A A 0 _ gan-2a gan-3a GA 0 _

" 0 0 0 0' - 0 0 0 0'

B 0 0 0 GB 0 0 0

P = AB B 0 0 , ® = GAB GB 0 0 . (1

B <N - ‘ ^ A 1 An-3 B B 0 _ GAn-2 B GAn-3 B ••• GB 0 _

Here En is the «-by-« identity matrix. Then the predictive model (9)-(10) can be expressed as

Tt = ^^ +§T0 + PUt-h ,

i=1

Y =AXt+1,t +§A0X° +®Ut-h . (12)

i=1

3. Synthesis of Model Predictive Control

In order to solve the posed problem the following criterion is used as the criteria function

yt+k\t

+ u

t-h+k\t t-h+k-

(13)

where C > 0, D > 0 - weighing matrices.

In the case when the reference trajectory yt+k is unknown for k > 0 it seems reasonable to assume that yt+k = yt, i.e. the same reference point is held throughout the prediction horizon.

The summation terms in (13) can be expanded to offer a quadratic objective function

in terms of xt+it and Ut. Let

yt+1

yt+N

Then using (12) there is the following expression

1 i||yt+k\t -7t|| = ^IIY - Yt||C =

2 k=1 C 2

= 2Uj-h®TC®Ut-h + Uj-h[®TCAXt+i\t +®TCaOJ^O -®TCYt] + ] , (14)

2 i=1

where c1 is a constant term that does not depend on Ut-h or xt+j\t and C is given by

C =

In a similar manner

N

ut-h+ k\t ut-h+k-1\t|

C 0 : 0‘

0 C : 0

0 0 : C

2

D 1 UT 2 t h.DUt -,

\tDut -h + c2

(15)

where c2 is a constant term that does not depend on ut-h+k (k = 1, N) and D is given by

" 2D -D 0 i 0

-D 2D -D i 0

D =

-D 2D -D

0 -D 2D

Combining the above the criteria function can be expressed as

J (t) = 2 Uj-hFUt-h + Uj-hf + C3.

Here c3 is the combination of previous constant terms c1 and c2 and may be safely ignored. The terms F and f are given by

xt+1|t h ut Q

F = ®TC® + D , f = r IA0 X0 - 0

i=1 [ Y _ [ 0 _

r = [®TCA ®TC -®TC].

In the absence of constraints an analytical solution of the posed problem can be ob-d - = 0 using vector derivative formulas, see e.g. [7]:

tained from the condition

dUt

t-h

dJ

d

dUt

t-h

dU,

t-h

2 Uj-hFUt-h

T

+ UJ-hf + c

1 d(trFUf hUT h) d(UT hf) 1r T

= - dU + dU = T [ F t-h + FUt_h ] + f = 0. (17)

2 dUt-h dUt-h 2

As matrix F is symmetric the equation (17) can be expressed as follows

FU-h + f = 0.

So, the criteria function can be expressed as

U*-h = - (®TC® + D)-1(®TCA+nt - ®TC7t)-

Du

t-h

and the optimal predictive control has the form:

u*- h+ljt = (En 0 •

0)U

t-h .

Optimization of the constrained model (1)-(5) can be realized numerically using Matlab’s function named as «quadprog».

4. Economic system control modelling

Consider the economic system control intended for goods production, storage and delivery to consumers

qt+i = Mt +I An- +9t-h +^t, Qk = qk, k = -r ,0,

i=1

zt+i = zt + Brot-h - 9t-h + Ct, zo = zo, (18)

where qt e Rs, qit is the i-typed consumer’s goods amount at the moment t ( t = 1,T ,

i = 1,s ), zi,t is the i-typed goods amount in the producer’s store, o>i,t is the production of the i-typed goods, ^ is the delivery volume of the i-typed goods. Vector Gaussian sequences |t, Zt have the following characteristics: M{ |t } = 0, M{ Zt } = 0,

M{Çt^T } = £5tk, M{ZtZT} = H5tk , M{|tZT } = 0. The last vectors take into account

errors arisen from the model definition inaccuracy. Matrices A and B define the production and consumption dynamics. It is supposed the time delays r and h are integer.

The following constraints expressed as linear inequalities should be satisfied at each moment:

Zmin < Zt < Zmax, 0 < Q^h < ©max, 0 < <$t_h < Zt.

(19)

The variables ©t and are considered to be the controlling inputs. The problem is to

determine an optimal control strategy for goods production, storage and delivery on the base of the observation according to which the consumer’s goods amount qt will be close to given one taking into account constraints (19).

The model of the economic system (18) with constraints (19) can be transformed and expressed in terms of the model (1) with constraints (4) - (5) assuming n=2s. Let

~qt" , ut-h = >t-h" , A = ' A 0 " , Ai = ' A 0" , B = ' Es 0"

_ Zt _ .®t -h _ . 0 Es_ _ 0 0 _ Es B

X" , W = "S 0

.Zt. . 0 s

^2 =

ax = Zmm, a2 = Zmax, = [0 Es ], 9j( xt) = 0 ,

92( xt) = „Zi

Es 0

0 E„

The optimization problem is solved at each time interval. In order to solve the problem of criterion (16) minimization numerically in the Matlab code it is necessary to express constraints in terms of matrixes and vectors. Then the constraint on the production

ot -h+if < ramax for the expanded system is the following one

RUt-h < E“max.

The constraint on the delivery volume q>t-h+i,t < Zt-h+i,t-h is expressed as

R2Ut_h <R1(TXt-h+1M + XY°X!0) + R:PUt-2h . i=1

As ®t-h+i|t > 0 and 9t-h+i|t > 0, then

U-h > 0.

The constraints irt+i|t < Zmax , Zt+itt > Zmin can be expressed in the form:

RPUt-h < EZmax-R1(YX+1|t + £ Y°X°),

i=1

-RxPUt-h <-EZmin +R1(YXt+1|t +XY0^i°).

(20)

(21)

(22)

(23)

(24)

i=1

Matrices Rj, R2, E are assumed to be as follows

" 0 Es 0 0 0 0 " " Es 0 0 0 0 0 " " Es"

R = 0 0 0 Es 0 0 , R2 = 0 0 Es 0 • 0 0 , E = Es

_ 0 0 0 0 Es _ _ 0 0 0 Es 0 _ _ Es _

xt =

wt =

The simulation is based on the following initial data:

A 0,75 0 '

“[-0,25 0,9

roma

, A = i © © i B = " 0,3 0,1" , z ■ = 0,1" , z = 1,5 "

-0,1 0_ 0,2 0,8_ ’ mm .0,1. ’ max .1,5 _

" 0,8 " "0,2" "0" "1"

.0,7 _ , z0 = .0,2. , ?0 = 0_ , q = . 2_

, r =1, h =1, N=8, H = E4, W = 0,

V = diag{0,0005; 0,0005; 0,0005; 0,0005}.

The simulation results are shown in Figures 1 - 3 as the plots of processes.

qi

0,5

q2

.... 2

/ jr^~~ 1,5 q2 ! '

.q1( 1

! 0,5 / . .

5 10 15 20 25 t 0 5 10 15 20 25 t

Fig. 1. The consumer’s amount of the good.

Fig. 2. The goods amount in the producer’s store and the delivery volume of the goods.

Fig.3. The goods production.

Economic system modelling proved algorithm efficiency. It is shown the goal is achieved; state and input constraints are satisfied under time delay condition.

1

0

5. Conclusion

The Model Predictive Control of the system with state and input delays has been developed, guaranteeing constraints satisfaction and feasibility. The solution of the MPC synthesis problem is obtained. The extrapolator is offered to use in order to obtain predicted values of the system output.

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Киселева Марина Юрьевна Смагин Валерий Иванович Томский государственный университет

E-mail: [email protected]; [email protected] Поступила в редакцию 12 ноября 2010 г.