MODEL PREDICTIVE CONTROL OF CONSTRAINED WITH NON LINEAR STOCHASTIC PARAMETERS SYSTEMS

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА

2011 Управление, вычислительная техника и информатика 3(16)

УПРАВЛЕНИЕ ДИНАМИЧЕСКИМИ СИСТЕМАМИ

УДК 519.865.5

V.V. Dombrovskii, T.U. Obyedko

MODEL PREDICTIVE CONTROL OF CONSTRAINEDWITH NON LINEAR STOCHASTIC PARAMETERS SYSTEMS

In this paper we consider the model predictive control problem of discrete-timesystems with non-linear random depended parameters for which only the first andsecond conditional distribution moments, the conditional autocorrelations and themutual cross-correlations are known. The open-loop feedback control strategy isderived subject to hard constraints on the control variables. The approach is ad-vantageous because the rich arsenal of methods of non-linear estimation or the re-sults of nonparametric estimation may be used directly for describing characteris-tics of random parameter sequences.

Keywords: discrete time control systems, model predictive control (MPC), non-linear stochastic parameters, multiplicative noise, constraints, computationalmethods, stochastic control.

Lately there has been a steadily growing need and interest in systems with stochasticparameters and/or multiplicative noise. The same systems have been gaining greater ac-ceptance in many engineering and finance applications.

Optimization techniques to various control and estimation problems for such sys-tems have been intensively studied in the literature.

In particular, a linear quadratic control for systems with random independent pa-rameters is studied in [1-3]. In [4-8] the authors consider stochastic optimal controlproblem of systems with dependent parameters which switch according to a Markovchain.

In above-mentioned papers there are no constraints on the state and control vari-ables. However, constraints arise naturally in many real world applications.

In recent years considerable interest has been focused on model predictive control(MPC), also known as receding horizon control (RHC), as an appropriate and effectivetechnique to solve the dynamic control problems having input and state/output con-straints. The basic concept of MPC is to solve an open-loop constrained optimizationproblem at each time instant and implement only the first control move of the solution.This procedure is repeated at the next time instant. Some of the recent works on thissubject can be found in literature [9-20].

MPC for constrained discrete-time linear systems with random independent pa-rameters is considered in [15, 16]. In [17, 18] MPC of linear with random dependent pa-rameters systems under constraints is examined, where parameters evolution is de-scribed by multidimensional stochastic difference equations. In [19] the MPC problemof discrete-time Markov jump with multiplicative noise linear systems subject to con-straints on the control variables is solved.

6 V.V. Dombrovskii, T.U. Obyedko

In this paper we consider MPC for constrained discrete-time with stochastic non-linear parameters systems. The main novelty of this paper is that no assumption is madeabout the form of the function describing parameters evolution. The first and secondconditional distribution moments, the conditional autocorrelations and the mutual cross-correlations are known only. The performance criterion is composed by a linear combi-nation of a quadratic and liner parts. The open-loop feedback control strategy is derivedsubject to hard constraints on the input variables. Predictive strategies computation in-cludes the decision of the sequence of quadratic programming tasks.

The approach is advantageous because the rich arsenal of methods of non-linear es-timation or if no model can be found that describes the underlying non-linear structureadequately, then the results of nonparametric estimation may be used directly for de-scribing characteristics of non-linear time series.

1. Problem formulation

We consider the following discrete-time with non-linear stochastic parameters sys-tem on the probabilistic space (Ω, F , P):

( 1) ( ) [η( 1), 1] ( )x k Ax k B k k u k+ = + + + , (1)where x(k) is the nx-dimensional vector of state, u(k) is the nu-dimensional vector ofcontrol; η(k) (k=0,1,2…) denotes a sequence of depended q-dimensional random vec-tors. A, B[η(k),k] are the matrices with appropriate dimensions, where B[η(k),k] linearlydepend on η(k).

Let F =( kF )k≥1 is the flow of sigma algebras defined on (Ω, F , P), where kF de-notes the sigma algebra generated by the (x(s), η(s)): s=0, 1, 2,…,k up to time instantk and it means the information (measurements) before the time k.

We assume that we know conditional moments for the process η(k) about kF :

η( ) / η( ),kM k i k i+ = +F (2)

η( ) η( ) η( ) η( ) ( ),

( 0,1,2,...), ( , 1, 2,...).

/Tk ijM k i k i k j k j k

k i j

⎡ ⎤ ⎡ ⎤+ − + + − + = Θ⎣ ⎦ ⎣ ⎦= =

F

(3)In that follows, we use notation: for any matrix ψ[η(k),k], dependent on η(k),

ψ( ) ψ[η( ), ] / kk E k k= F without indicating the explicit dependence of matrices onη(k). Also we use the standard notation, for square matrix A, A≥0 (A>0, respectively) todenote that the matrix A is positive semidefinite (positive definite), and tr(A) to repre-sent the trace of A.

We impose the following inequality constraints on the control:

min max( ) ( ) ( ) ( ),u k S k u k u k≤ ≤ (4)where S(k) is the matrix with appropriate dimension.

The cost function of the RHC is defined as a function, composed by a linear combi-nation of a quadratic part and a linear part, which is to be minimized at every time k

1 31

( / ) ( ) ( , ) ( ) ( , ) ( ) /mT

ki

J k m k M x k i R k i x k i R k i x k i=

+ = + + − + +∑ F

1

2 40

( / ) ( , ) ( / ) ( , ) ( 1/ ) ,/mT

ki

M u k i k R k i u k i k R k i u k i k−

=+ + + − + −∑ F (5)

Model predictive control of constrained with non linear stochastic parameters systems 7

on trajectories of system (1) over the sequence of predictive controls u(k/k), …,u(k+m−1/k) dependent on the state x(k), under constraints (4); R1(k,i)≥0, R2(k,i)>0,R3(k,i) ≥0, R4(k,i) ≥0 are given symmetric weight matrices of corresponding dimensions;m is the prediction horizon.

Only the first control vector u(k/k) is actually used for control. Thereby we obtaincontrol u(k) as a function of state x(k), i.e. the feedback control. This optimization proc-ess is solved again at the next time instant k+1 to obtain control u(k+1). The synthesis ofpredictive control strategies leads to the sequence of quadratic programming problems.

2. Model predictive control strategies design

Consider the problem of minimizing the objective (5) with respect to the predictivecontrol variables u(k+i/k), subject to constraints (4).

Theorem. The set of predictive controls U(k)=[uT(k/k), …,uT(k+m-1/k)]T, such that itminimizes the objective (5) subject to (4), for each instant k is defined from the solvingof quadratic programming problem with criterion

( / ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( )T TY k m k x k G k F k U k U k H k U k⎡ ⎤+ = − +⎣ ⎦ (6)

under constraints

min max( ) ( ) ( ) ( ),U k S k U k U k≤ ≤ (7)

where ( ) diag( ( ),..., ( 1)),S k S k S k m= + −

min min min( ) [ ( ),..., ( 1)] ,T T TU k u k u k m= + − max max max( ) [ ( ),..., ( 1)] ,T T TU k u k u k m= + −

H(k), G(k), F(k) – are the block matrices

11 12 1

21 22 2

1 2

( ) ( ) ( )( ) ( ) ( )( ) ,

( ) ( ) ( )

m

m

m m mm

H k H k H kH k H k H kH k

H k H k H k

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(8)

[ ]1 2( ) ( ) ( ) ( ) ,mG k G k G k G k= (9)

[ ]1 2( ) ( ) ( ) ( ) ,mF k F k F k F k= (10)where the blocks satisfy the following recursive equations

2( ) ( , 1) ( ) ( ) ( )

( ) ( ) ( ) / ,

Ttt

Tk

H k R k t B k t Q m t B k tM B k t Q m t B k t

= − + + − + +

+ + − + F (11)

( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) / , ,

T T f ttf

T T f tk

H k B k t A Q m f B k f

M B k t A Q m f B k f t f

−

−

= + − + +

+ + − + <F (12)

( ) ( ), ,Ttf ftH k H k t f= > (13)

( ) ( ) ( ) ( ),t TtG k Q m t B k t= Α − + (14)

3 4( ) ( , ) ( ) ( , 1),m

j tt

j tF k R k j A B k t R k t−

== + + −∑ (15)


1( ) ( 1) ( , ), ( 1, ),TQ t A Q t A R k m t t m= − + − = (16)

1(0) ( , ).Q R k m= (17)The optimal control law is achieved by

( ) 0 0 ( ),u u un n nu k I U k⎡ ⎤= ⎣ ⎦ (18)

where unI is nu-dimensional identity matrix, 0

un is nu-dimensional zero matrix.In the case of unconstrained control the optimal control strategy for system (1) is

achieved by equation (18), where1 1( ) ( ) ( ) ( ) ( ) .

2T TU k H k G k x k F k− ⎡ ⎤= − −⎢ ⎥⎣ ⎦

(19)

In this case the optimal value of the criterion is achieved by equationopt 1

1

1

( / ) ( ) ( ) ( ,0) ( ) ( ) ( ) ( )1( ) ( ) ( ) ( ) ( ),4

T T

T

J k m k x k Q m R k G k H k G k x k

L k x k F k H k F k

−

−

⎡ ⎤+ = − − −⎣ ⎦

− − (20)

where 31

( ) ( , ) .m

i

iL k R k i A

== ∑

Proof. Denote

1 3

2 4

1 3

2 4

1

( 1) ( , 1) ( 1) ( , 1) ( 1)( ) ( , ) ( ) ( , ) ( )

( 2) ( , 2) ( 2) ( , 2) ( 2)( 1) ( , 1) ( 1) ( , 1) ( 1) ...

( ) ( , ) (

Tk s

T

T

T

T

J x k s R k s x k s R k s x k su k s R k s u k s R k s u k s

x k s R k s x k s R k s x k su k s R k s u k s R k s u k s

x k m R k m x k

+ = + + + + + − + + + +

+ + + − + +

+ + + + + + − + + + +

+ + + + + + − + + + + +

+ + 3

2 4

) ( , ) ( )( 1) ( , 1) ( 1) ( , 1) ( 1).T

m R k m x k mu k m R k m u k m R k m u k m

+ − + +

+ + − − + − − − + −

It is easy to see that

1 3

2 4 1

( 1) ( , 1) ( 1) ( , 1) ( 1)( ) ( , ) ( ) ( , ) ( )

Tk s

Tk s

J x k s R k s x k s R k s x k su k s R k s u k s R k s u k s J

+

+ +

= + + + + + − + + + +

+ + + − + + (21)

and ( / ) / .k kJ k m k M J+ = F (22)Let us consider the following equation

1 1 3

2 4

( ) ( , ) ( ) ( , ) ( )( 1) ( , 1) ( 1) ( , 1) ( 1).

Tk m

TJ x k m R k m x k m R k m x k m

u k m R k m u k m R k m u k m+ − = + + − + +

+ + − − + − − − + − (23)

Using (1) in (23) for x(k+m), we obtain

1

2

3 3

4

( 1) (0) ( 1) 2 ( 1) (0)[η( ), ] ( 1) ( 1)[ [η( ), ] (0)

[η( ), ] ( , 1)] ( 1)( , ) ( 1) ( , ) [η( ), ] ( 1)

( , 1) ( 1)

T T T Tk m

T TJ x k m A Q Ax k m x k m A QB k m k m u k m u k m B k m k m Q

B k m k m R k m u k mR k m Ax k m R k m B k m k m u k m

R k m u k m

+ − = + − + − + + − ×

× + + + − + + − + + ×× + + + − + − −

− + − − + + + − −− − + − ,

where Q(0) are defined by (17).


We assume that for some q

1

1

1

2

( ) ( 1) ( )

2 ( ) ( ) ( 1) [η( 1), 1] ( )

( )[ [η( 1), 1] ( 1)

[η( 1), 1] ( , )] ( )

2 ( ) [η(

T Tk m q

qT T q i

iq

T T

i

T T

J x k m q A Q q Ax k m q

x k m q A Q i B k m i k m i u k m i

u k m i B k m i k m i Q i

B k m i k m i R k m i u k m i

u k m i B k

+ −

− +

=

=

= + − − + − +

+ + − − + − + + − + + − +

+ + − + − + + − + − ×

× + − + + − + + − + − +

+ + − +

∑

∑

1

1 1

3 41 1

1), 1] ( 1)

[η( 1), 1] ( )

( , 1) [η( 1), 1] ( , ) ( )

q qj i

i j i

q ii j

i j

m i k m i Q j A

B k m j k m j u k m j

R k m j A B k m i k m i R k m i u k m i

−−

= = +

−

= =

− + + − + − ×

× + − + + − + + − −

⎡ ⎤− − + + − + + − + + − + − −⎢ ⎥

⎢ ⎥⎣ ⎦

∑ ∑

∑ ∑

13

1( , 1) ( ),

qm i

iR k m i A x k m q− +

=− − + + −∑ (24)

where Q(i) are defined by (16) – (17).Prove that (24) holds for q+1. Indeed, from the (21) we have the result:

( 1) 1 3

2

4

( ) ( , ) ( ) ( , ) ( )

( ( 1)) ( , ( 1)) ( ( 1))( , ( 1)) ( ( 1)) .

Tk m q

T

k m q

J x k m q R k m q x k m q R k m q x k m q

u k m q R k m q u k m qR k m q u k m q J

+ − +

+ −

= + − − + − − − + − +

+ + − + − + + − + −

− − + + − + + (25)

Applying (24) in (25) and using (1) for x(k+m-q) after some calculations, we obtain

( 1)

( 1)( 1) 1

1( 1)

1

2

( ( 1)) ( ) ( ( 1))

2 ( ( 1)) ( ) ( 1) [η( 1), 1] ( )

( )[ [η( 1), 1] ( 1) [η( 1), 1]

( , )] (

T Tk m q

qT T q i

iq

T T

i

J x k m q A Q q Ax k m q

x k m q A Q i B k m i k m i u k m i

u k m i B k m i k m i Q i B k m i k m i

R k m i u

+ − +

++ − +

=+

=

= + − + + − + +

+ + − + − + − + + − + + − +

+ + − + − + + − + − + − + + − + +

+ −

∑

∑1

1 1

( 1)

3 41 1

) 2 ( ) [η( 1), 1]

( 1) [η( 1), 1] ( )

( , 1) [η( 1), 1] ( , )

q qT T

i j i

j i

q ii j

i j

k m i u k m i B k m i k m i

Q j A B k m j k m j u k m j

R k m j A B k m i k m i R k m i

−

= = +

−

+−

= =

+ − + + − + − + + − + ×

× − + − + + − + + − −

⎡ ⎤− − + + − + + − + + − ×⎢ ⎥

⎢ ⎥⎣ ⎦

∑ ∑

∑ ∑

( 1)1

31

( ) ( , 1) ( ( 1)).q

m i

iu k m i R k m i A x k m q

+− +

=× + − − − + + − +∑ (26)

By the induction from (26) we have that (24) holds for each q=1,2,…m.


According to (22) и (24) we have that

( / ) ( ) ( 1) ( )T TJ k m k x k A Q m Ax k+ = − +

12 ( ) ( ) ( ) ( ) ( 1/ )

mT i T

ix k A Q m i B k i u k i k

=+ − + + − +∑

21

( 1/ ) ( ) ( ) ( ) ( , 1) ( 1/ )m TT

iu k i k B k i Q m i B k i R k i u k i k

=+ + − + − + + − + − +∑

1

( 1/ ) ( ) ( ) ( ) / ( 1/ )m

T Tk

iu k i k M B k i Q m i B k i u k i k

=+ + − + − + + − +∑ F

1

1 12 ( 1/ ) ( )( ) ( ) ( ) ( 1/ )

m m TT T j i

i j iu k i k B k i A Q m j B k i u k j k

−−

= = ++ + − + − + + − +∑ ∑

1

1 12 ( 1/ ) ( )( ) ( ) ( ) / ( 1/ )

m mT T T j i

ki j i

u k i k M B k i A Q m j B k i u k j k−

−

= = ++ + − + − + + − −∑ ∑ F

3 4 31 1

( , ) ( ) ( , 1) ( 1/ ) ( , ) ( ).m m m

j i i

i j i iR k j A B k i R k i u k i k R k i A x k−

= = =

⎡ ⎤− + + − + − −⎢ ⎥

⎢ ⎥⎣ ⎦∑ ∑ ∑ (27)

The (27) can be written on matrix form

( / ) ( ) ( 1) ( )( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ),

T T

T TJ k m k x k A Q m Ax k

L k x k x k G k F k U k U k H k U k+ = − −

⎡ ⎤− + − +⎣ ⎦ (28)

where the matrices H(k),G(k),F(k) take the forms (8)-(17), matrix L(k) takes the form

31

( ) ( , ) .m

i

iL k R k i A

== ∑

Thus we have that the problem of minimizing the criterion (28) subject to (4) isequivalent to the quadratic control problem with criterion (6) subject to (7).

Obviously, the optimal control law for system (1), such that it minimizes criterion(5), without constraint is achieved by equation (19). It is easy to show that in this casethe optimal value of the criterion (5) is achieved by equation (20).

Conclusion

In this paper the predictive control strategy for discrete-time linear systems withnon-linear random dependent parameters for which the first and the second conditionaldistribution moments are known is derived. The main novelty of this paper is that wedon’t assume the explicit form of model for describing the parameters evolution.

One can apply obtained results in a wide class of models with non-linear stochasticuncertainties. If no model can be found that describes the underlying non-linear struc-ture adequately, then the results of nonparametric estimation may be used directly fordescribing the characteristics of the multidimensional time series [21, 22].

In addition offered approach can be extended to the following cases:- when matrix A in (1) changes in time;- when system dynamics (1) contains additive noises depended on parameter η;


- when matrix A in (1) depends on stochastic independent parameters sequence non-correlated with η;

- when constraints (4) are defined by convex functions. In this case synthesis of pre-dictive control strategies leads to the sequence of convex optimization problems;

- when parameter η is non-observable. Then the cost functional (5) should be aver-aged over all possible states of process η.

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Домбровский Владимир ВалентиновичОбъедко Татьяна ЮрьевнаТомский государственный университетE-mail: [email protected]; [email protected] Поступила в редакцию 22 марта 2011 г.

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MODEL PREDICTIVE CONTROL OF CONSTRAINED WITH NON LINEAR STOCHASTIC PARAMETERS SYSTEMS