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IDENTIFICATON-BASED APPROACH TO SOFT SENSORS DESIGN
IDENTIFICATON-BASED APPROACH TO SOFT SENSORS DESIGN Natalya Bachtadze, Vladimir V. Kulba, Vladimir A. Lototsky, Evgeny M.Maximov Institute of Control Sciences, 65, Profsoyuznaya, Moscow,117997 GSP, Russia Tel.: +(7)(095)3349201. E-mail: lotfone@ ipu.rssi.ru
Abstract. The definition of soft sensors as an algorithmic or software system, which is intended for real-time design of production processes mathematical models from both current and historical data about these processes is given. The virtual analyzers design methods and algorithms are presented. Keywords. Identification, soft sensors, virtual analysis, robust optimal control, robust stability, absorption condition.
1.
INTRODUCTION
In this paper, the problems of Soft Sensors (SS) design in an up-to-date technological environment i.e., in the united information space of the enterprise (company, corporation, etc.) is considered. This space, generated by the modern distributed information systems (IS) and large-scale enterprise technological data archives, forms the environment, where integration of main enterprise IS’s may be realized on much more advanced level compared to simple information support. Soft Sensors realize the processes of construction, adaptive adjustment, and real-time implementation of production processes models (with the purpose of direct computer control or decision making support). In the conditions of united information corporative medium, it becomes natural to do a simultaneous real-time adjustment of models for different levels of whole production chain. This chain includes both process control and management, transportation, inventory control, marketing, etc. (i.e., logistic cicle). The specifics of Soft Sensors in modern Manufacturing Execution Systems (MES) is determined by the fact that they realize monitoring, analysis, prediction of process parameters, and determination of control variables, basing both on empirical data (on-line measurements, archive information, knowledge base content, etc.) and on the models, used on the other control levels. For all that, the simulation results of different process fragments are not used as elements of more complicated model of higher level, but simply generate an input vector for this model. It means, that so called virtual measurements are used, which produce additional a priori information< mostly important while model adjustment.
Moreover, one can easily learn from real life experience that in different manufacturing systems the same sets of process data and similar algorithms may be successfully used to solve various control problems. So it follows, that modern SS realize the identification-based approach to control problems solution, but in a more generalized sense, comparing to traditional one. The actuality of such SS concept is determined by the necessity of new approach to integrated industrial MES design basing on modern information technologies. The new techniques, based on such approach will eliminate the artificial separation of tactic on-line process control problems and strategic manufacturing control problems (in particular, the logistics ones). The condition of high reliability is a principal one, which has to ensure the control strategy for many types of control plants (industrial processes, banking, stock exchange, etc.). In fact, high production rate of such processes results in great losses in case of failure. Frequently, lack of information about the control plant causes the necessity of robust control, which guarantees fulfillment of all technical requirements and attainment of high quality of finished product. The problem of stable control systems design under constrained uncertainty is a central one during projecting soft sensors (Bakhtadze, 2004). The constrained uncertainty of information about the control plant, which is determined by unaccounted plant dynamics, identification errors, diminishing of controller order for the sake of simplicity, etc., is interpreted as joint perturbations of plant and controller. The control strategy design providing the
best with respect to (w.r.t.) a given optimality criterion control quality under constrained uncertainty may be concerned as optimal control of certain class of plants. Polyak and Tsherbakov (2002) give detailed description of different uncertainty classes. Tsypkin (1991, 1992, 1996) introduced the concept of robust optimal and absolutely robust optimal linear discrete-time systems in constrained uncertainty environment and gave the design problem solution for the case, when plant disturbances can be restored for the finite number of steps. This is achieved by using the so called absorption principle and internal model technique. Later, Tsypkin and Vishnyakov (1996) suggested the design technique of systems, which are selectively invariant w.r.t. a class of disturbances with given finite spectrum. In what follows, the problem of robust optimal systems design for the class of disturbances, which are not lacunar (i.e., exactly predictable) is considered.
Q 0 () y (t ) Pu0 () u(t ) , where
Q 0 (), Pu0 () are fixed polynomials (so, it is assumed, that
P[0 () { 0) .
Then
Q()=Q 0() Q(), Pu()=Pu0() Pu() ; P[ () P[ () , where
G Q A G , G Pu A Gu , G P[ A G[
.
Now one may rewrite the interval plant (1) equation as follows:
Pu0 ()u (t ) M (t ) ,
Q o () y (t ) where
M (t ) is a “generalized disturbance”,
M(t)=M in(t)+M ex(t), 2. PROBLEM STATEMENT
M in (t ) GPu ()u (t ) GQ() y (t )
Let linear discrete system be described by the equation
Q ( ) y (t )
Pu ( )u (t ) P[ ( )[ (t ) ,
(1)
Pr () r (t ) Py ( ) y (t ) ,
(2)
where y(t) is control plant (1) output, u(t) stands for control variable, r(t) – setting point, [ (t ) – constrained additive noise, denotes one step delay operator. The plant (1) is assumed to be stable and minimal phase.
(3)
K 0 ( )
Pu0 () PrQ () Q0 () RQ () Pu0 PyQ ()
E () Q ,
E( )=K
where
G()=Q()R()+Pu ()Py () . Plant uncertainty is interpreted as certain set of plants
Q A, Pu Au , Pv Av , i.e., as interval plant. to find the controller R (), Pr ( ), P[ ( ) , stabilizing plant (1) and
`
optimal in the sense, indicated below. To determine the optimality criterion one has to choose certain nominal plant in appropriate set of parameter space:
(4) Q
JQ
^
external
will be referred as optimal w.r.t. criterion H , Q=1, …, v, if its transfer function
G()y(t)=Pu()Pr()r(t)+R()P[()[(t),
is
-
Pu0 () PrQ ()r (t )
minimizes the functional
problem
ex
Clothed-loop nominal system
So, the closed-loop system equation will be as follows
The
and
(Q0 () RQ () Pu0 () PyQ ()) y (t )
and controller equation has the form
R ( )u (t )
parametric disturbance disturbance.
denotes the
0
e jZ ,where
( ) K
ref
( )
denote
the
discrepancy, K ref ( ) stands for transfer function of a certain reference system and symbol Q , denotes Q
the norm in Hardy space H , Q=1, …, v. In particular,
J2
Jf
ª 1 jS º jZ 2 E ( e ) d Z « » ³ ¬« 2Sj jS ¼»
sup E (e jZ ) ,
0dZ d 2S
1/ 2
,
where E stands for expectation operator.
is stable and minimal phase.
The interval plant control system
Introducing the notation
(Q ()R() + P u()Py())y(t) = =P0u()Pr()r(t) + R()M(t), characteristics of which coincide with (or are close
H 1 ()
0
0
Q
to) those of H -optimal nominal system, will be referred as robust optimal control system (Tsypkin, 1992). In what follows, the approach for such system design for the cases Q 2 and Q f under both white noise and Markovian disturbances is suggested. The approach is based on external disturbances model building using identification techniques.
3. DESIGN OF ROBUST OPTIMAL SYSTEM FOR WHITE NOISE DISTURBANCES In what follows, the equations of closed-loop control system for interval plant are given in the form, which is convenient for comparison with corresponding Q
equations of H -optimal nominal system. For this purpose, consider two alternately used in system (3) fixed controllers
{R i (), P (), P ()}, i 1,2 . Corresponding outputs are denoted as yi(t), i=1,2. The i r
i y
considerations will be restricted to the class of interval plants for which the transient processes when switching from one controller to another may be neglected. Let assume that these controllers can be chosen so that the following two conditions are met. A1. The closed systems with both mentioned controllers are stable and minimal phase for the plant (1) w.r.t. both disturbance and control.
A2. The outputs of respective closed-loop systems y1 (t ), y 2 (t ) are related by following equation
W2 ()
one obtains the equations of closed loop systems with controllers R1() , R2() as follows
y (t ) W1 ()r (t ) H1 () R1 ()M (t ) T () y (t ) W2 ()r (t )
(5)
H 2 () R2 (t )M (t ) ] (t ). Now taking into account the definition and properties of the functions H 1(),R1(),H 2(),R2() , one gets from (5):
(H 21 R21T H11 R11 ) y (W2 H 21 R21 W1 H11 R11 )r ]
(6)
where for simplicity the arguments and t are omitted. Eq. (6) is the equation of open loop stable minimal phase plant with output noise ]. Identification of this
plant provides the estimate T of polynomial T T( ) . Now let us construct the external disturbances model, using these two controllers and following relations
y(t)=W1()r(t)+H1()R1()M(t)
T ()y(t)=W2()t)+H 2()R2()M(t)
.
( T H 1 R1 H 2 R2 )M .
Denoting now
R1
B1 (1 D); R2
where
D
D ()
B2 (1 D), stands
for
“predicting
polynomial” of order Nin - 1 for disturbance
W2 -T()W1(), where Q0 () R1 () Pu0 Py1 ()
,
Q0 () R2 () Pu0 () Py2 ()
A3. There exist two controllers R1() , R2() , meeting conditions A1, A2 and so that unknown polynomial T() is external, and the function
W1 ()
1
(W1 T W2 )r
] (t ) is discrete white noise.
Pu0 ( ) Pr1 ()
Q0 () R1 () Pu0 () Py2 ()
From these equations one has
y 2 (t ) T () y1 (t ) ] (t ) , where
H 2 ()
1
such that
Q0 () R2 () Pu0 Py2 ()
can be exactly predicted by N in
its previous values, and B1, B2 are fixed polynomials, so that
,
Pu0 () Pr2 ()
M in (n)
M in (t) ,
M in (t) ,
D()M in (t - 1) .
Than accounting Tsypkin’s (1992) absorption condition
(1 D())M in (t )
w.r.t. unknown polynomials C() and Py, and second one – as equation w.r.t. Pr.
0 for t t N in ,
one gets
(W1 T W2 )r
Denote CR -B( 1-D)=Pu A in (12), where
(T H 1 R1 H 2 R2 ) P[ [ . (7)
A() is unknown polynomial, satisfying Eq. (12).
So, the closed loop control system for interval plant accounting the estimates obtained, may be written as follows
(Q () R() P Py ()) y (t ) 0
0 u
Pu0 () Pr (t )r (t ) R()M in (t )
0
(8)
Then one gets the following equations for Py, R:
Py
Q 0 A CPyQ ,
R
CR -Pu0 A.
(13)
R() P[ ()[ (t )
Similarly, one gets from (12)
or, equivalently
Pu0 Pr CR Pu0 A Pu0 Pr .
Q 0 () R() Pu0 Py ()) y (t ) Pu0 () Pr (t )r (t )
(9)
~ Pu0 C one gets the following
~ (14) A Pr C R . Substituting Py , R , Pr from (12 - 14) into
Pr
where
) = (W1Tˆ W2 )(TˆH 1 R1 H 2 R2 ) 1 . Now let us rewrite (8) accounting absorption condition (1 D ())M in (t ) 0 , where D() stands for mentioned above predicting polynomial for M in (t) , and taking into consideration the designation R()=B( )( 1-D()), where D() is assumed to be known and B() stands for a certain (currently unknown) auxiliary polynomial. So one gets (Q 0 ()R()+Pu0 ()Py ())y(t)= (10) Pu0 ()Pr ()r(t)+B()( 1-D ()) () So, the equation (8) of closed control system for interval plant accounting (10) may be transformed to the mode, convenient for comparison with respective H Q -optimal nominal system (4):
Pu0()Pr ()+B()(1-D())() y(t)= r(t). Q0()B()(1-D())+Pu0()Py () (11) Comparing right hand parts of Eqs. (11) and (4), one gets the following polynomial equations w.r.t. unknowns Pr , R, Py :
P ()Pr ()+B()(1-D()))()= 0 u
>
C
equation for determination of Pr
R()Min (t ) R()) ()r
C()Pu0()Pr(),
Denoting
@
Q 0() C()R ()-B(1-D())
(12)
Pu0()(C()Py()-Py ())=0 where C() is arbitrary fixed external polynomial. First equation in (12) may be considered as equation
controller equation, one gets
(CR Pu0 A)u
~ (A Pr C R )r
(Q 0 A CPy )y. So the closed system equation takes the form (Q0(CR Pu0 A)+Pu0(Q0 A CPy ))y=
~ (Pu0 A Pr CR )r+(CR Pu0 A)Min or equivalently ~ (Q0CR+Pu0CPy )y=(Pu0 A Pr C
R )r+(CR Pu0 A)Min or
(Q 0 R Pu0 Py )y=Pu0( Pu0
A) -1 )r+(R C
A )M in . C
For a minimal phase plant, one can in particular to set
Pu0
R 0 , so that A() { D().
4. ROBUST OPTIMAL DESIGN MARKOVIAN DISTURBANCES
UNDER
Let the external plant disturbances be a Markovian sequence of the form M ex (t ) S ()M ex (t 1) + [ (t). Suppose that S() is a known polynomial of order N 2 , and [ (t) is white noise. Let also
Dex ()
S () .
Obviously,
the
equation
(1 Dex ())M ex = [ (t ) 2004). Let M (t )
M in (t ) M ex (t )
holds
(Bakhtadze,
denote the generalized
disturbance and, denoting perturbation predicting polynomial as Din () , one can determine '
D () Din ( ) Dex ( ) Din ( ) Dex ( ) i.e., the generalized predicting polynomial. Then the equality
(1 D())M (t )
(1 Din ())[ (t )
may be referred as pseudo absorption condition. Let us denote
D() 1 Din () .
Let D ( ) be an external polynomial. Then the equation (10) of closed control system for interval plant takes the form
(Q 0 R Pu0 Py ) y Pu0 Pr r RM , where M M (t ) stands for generalized disturbance. Let R=B(1-D). Then (Q 0 R Pu0 Py )y=Pu0 Pr r B( 1-D) ˆ[ (M in M ex )=P Pr r B D
The remaining solution stages are quite similar to considered earlier case of uncorrelated disturbances. The similar approach can be used when external
[ (t )
s
¦K (t i) , i 1
where Ki are independent random variables with zero mean. In fact, as [ (t ) [ (t 1) K (t ) the problem reduces to previous one. 5. THE CASE OF LIMITED STATIONARY DISTURBANCES Such disturbances can be uniquely represented as a sum of uncorrelated singular and regular random series (Wold expansion, see Gikhman and Skorokhod, 1965):
M ex (t)=M (t)+M (t), 1 2 where M ex (t) is singular and M ex (t) - regular. ~ (t)=M (t ) M 1 (t) , and Denoting M in in ex ~ Din ()
(1 Din ( ))(1 Dex1 ( ))M~in (t )
~ (1 Din ())M~in (t )
M~in (t)
with predicting polynomial
disturbances is restricted by the condition of exact
~ Din () . For the
M ex2 (t) it is necessary and sufficient that
it was an output of causal filter with standard uncorrelated input sequence. So the regular component may be analyzed by techniques of section 3. Designed robust optimal controller is causal if respective closed control systems are robust stable. The criteria of robust stability are given by Polyak and Tsherbakov (2002).
6. CONCLUSION The possibility of effective control in uncertain environment conditions determines the major perspectives of robust algorithms use for soft sensors design. But its practical implementation meets a number of essential obstacles, mainly of subjective character, e.g., conservative technical conviction of potential users. Due to different evaluations, more than 90% really implemented controllers are simple PID ones, designed by elementary design techniques. The design of contemporary VA demands not only (maybe, to a minor extent) the modern theoretical techniques, but also deep analytic study of the problems to be solved, based both on engineer’s intuition and on adequate algorithmic foundation.
REFERENCES 1.
2.
Din () Dex1 () Din () Dex1 () ,
N in 1 for M in (t) which is assumed to be exactly predictable by its N in previous values. The class of
0.
So the singular component of external disturbance and perturbations compose “generalized disturbance”
2 ex
where Din () is predicting polynomial of order
0,
or, equivalently
.
0 u
1 ex
condition takes the form
regularity of
disturbances are martingale
M ex1 (t) by N ex previous values using the prediction polynomial of order N ex 1 . ~ (t) the absorption For generalized disturbance M in predictability of
3.
4.
Albertos P, G.C. Goodwin (2002). Virtual sensors for control applications // Annual Reviews in Control. Volume (issue) 26 (1). Bakhtadze N.N. (2004). Virtual analyzers: identification approach (a survey). Automation and Remote Control. No. 11. Pp. 3 - 18. Plenum Publ. New York. N.Y. (translated from Russian). Gikhman I.I. and Skorokhod A.V. (1965). Introfuction to the theory of random processes. Nauka Publishers, Moscow (in Russian). Polyak B.T. and Tsherbakov P.S. (2002). Robust Stability and Control. Nauka Publishers, Moscow (in Russian).
5.
6.
7.
8.
Tsypkin Ya.Z. (1991). Dynamic plants control under constrained uncertainty. Measurements, Control, and Automation. 3 – 4. Pp. 3 – 21 (in Russian). Tsypkin Ya.Z. (1992). Design of robust optimal control systems under constrained uncertainty. Automation and Remote Control. No. 9. Pp. 139 – 159. Plenum Publ. New York. N.Y. (translated from Russian). Tsypkin Ya.Z. (1996). Discrete stochastic systems with internal models. Problems of Informatics and Control. 1 – 2. Pp. 21 – 26 (in Russian). Tsypkin Ya.Z. and A.N.Vishnyakov (1992). Discrete control system design for nonminimal phase plant with delay. Differential Equations. No. 11. Pp. 2001 – 2006 (translated from Russian).
Abstract. The definition of soft sensors as an algorithmic or software system, which is intended for real-time design of production processes mathematical models from both current and historical data about these processes is given. The virtual analyzers design methods and algorithms are presented. Keywords. Identification, soft sensors, virtual analysis, robust optimal control, robust stability, absorption condition.
1.
INTRODUCTION
In this paper, the problems of Soft Sensors (SS) design in an up-to-date technological environment i.e., in the united information space of the enterprise (company, corporation, etc.) is considered. This space, generated by the modern distributed information systems (IS) and large-scale enterprise technological data archives, forms the environment, where integration of main enterprise IS’s may be realized on much more advanced level compared to simple information support. Soft Sensors realize the processes of construction, adaptive adjustment, and real-time implementation of production processes models (with the purpose of direct computer control or decision making support). In the conditions of united information corporative medium, it becomes natural to do a simultaneous real-time adjustment of models for different levels of whole production chain. This chain includes both process control and management, transportation, inventory control, marketing, etc. (i.e., logistic cicle). The specifics of Soft Sensors in modern Manufacturing Execution Systems (MES) is determined by the fact that they realize monitoring, analysis, prediction of process parameters, and determination of control variables, basing both on empirical data (on-line measurements, archive information, knowledge base content, etc.) and on the models, used on the other control levels. For all that, the simulation results of different process fragments are not used as elements of more complicated model of higher level, but simply generate an input vector for this model. It means, that so called virtual measurements are used, which produce additional a priori information< mostly important while model adjustment.
Moreover, one can easily learn from real life experience that in different manufacturing systems the same sets of process data and similar algorithms may be successfully used to solve various control problems. So it follows, that modern SS realize the identification-based approach to control problems solution, but in a more generalized sense, comparing to traditional one. The actuality of such SS concept is determined by the necessity of new approach to integrated industrial MES design basing on modern information technologies. The new techniques, based on such approach will eliminate the artificial separation of tactic on-line process control problems and strategic manufacturing control problems (in particular, the logistics ones). The condition of high reliability is a principal one, which has to ensure the control strategy for many types of control plants (industrial processes, banking, stock exchange, etc.). In fact, high production rate of such processes results in great losses in case of failure. Frequently, lack of information about the control plant causes the necessity of robust control, which guarantees fulfillment of all technical requirements and attainment of high quality of finished product. The problem of stable control systems design under constrained uncertainty is a central one during projecting soft sensors (Bakhtadze, 2004). The constrained uncertainty of information about the control plant, which is determined by unaccounted plant dynamics, identification errors, diminishing of controller order for the sake of simplicity, etc., is interpreted as joint perturbations of plant and controller. The control strategy design providing the
best with respect to (w.r.t.) a given optimality criterion control quality under constrained uncertainty may be concerned as optimal control of certain class of plants. Polyak and Tsherbakov (2002) give detailed description of different uncertainty classes. Tsypkin (1991, 1992, 1996) introduced the concept of robust optimal and absolutely robust optimal linear discrete-time systems in constrained uncertainty environment and gave the design problem solution for the case, when plant disturbances can be restored for the finite number of steps. This is achieved by using the so called absorption principle and internal model technique. Later, Tsypkin and Vishnyakov (1996) suggested the design technique of systems, which are selectively invariant w.r.t. a class of disturbances with given finite spectrum. In what follows, the problem of robust optimal systems design for the class of disturbances, which are not lacunar (i.e., exactly predictable) is considered.
Q 0 () y (t ) Pu0 () u(t ) , where
Q 0 (), Pu0 () are fixed polynomials (so, it is assumed, that
P[0 () { 0) .
Then
Q()=Q 0() Q(), Pu()=Pu0() Pu() ; P[ () P[ () , where
G Q A G , G Pu A Gu , G P[ A G[
.
Now one may rewrite the interval plant (1) equation as follows:
Pu0 ()u (t ) M (t ) ,
Q o () y (t ) where
M (t ) is a “generalized disturbance”,
M(t)=M in(t)+M ex(t), 2. PROBLEM STATEMENT
M in (t ) GPu ()u (t ) GQ() y (t )
Let linear discrete system be described by the equation
Q ( ) y (t )
Pu ( )u (t ) P[ ( )[ (t ) ,
(1)
Pr () r (t ) Py ( ) y (t ) ,
(2)
where y(t) is control plant (1) output, u(t) stands for control variable, r(t) – setting point, [ (t ) – constrained additive noise, denotes one step delay operator. The plant (1) is assumed to be stable and minimal phase.
(3)
K 0 ( )
Pu0 () PrQ () Q0 () RQ () Pu0 PyQ ()
E () Q ,
E( )=K
where
G()=Q()R()+Pu ()Py () . Plant uncertainty is interpreted as certain set of plants
Q A, Pu Au , Pv Av , i.e., as interval plant. to find the controller R (), Pr ( ), P[ ( ) , stabilizing plant (1) and
`
optimal in the sense, indicated below. To determine the optimality criterion one has to choose certain nominal plant in appropriate set of parameter space:
(4) Q
JQ
^
external
will be referred as optimal w.r.t. criterion H , Q=1, …, v, if its transfer function
G()y(t)=Pu()Pr()r(t)+R()P[()[(t),
is
-
Pu0 () PrQ ()r (t )
minimizes the functional
problem
ex
Clothed-loop nominal system
So, the closed-loop system equation will be as follows
The
and
(Q0 () RQ () Pu0 () PyQ ()) y (t )
and controller equation has the form
R ( )u (t )
parametric disturbance disturbance.
denotes the
0
e jZ ,where
( ) K
ref
( )
denote
the
discrepancy, K ref ( ) stands for transfer function of a certain reference system and symbol Q , denotes Q
the norm in Hardy space H , Q=1, …, v. In particular,
J2
Jf
ª 1 jS º jZ 2 E ( e ) d Z « » ³ ¬« 2Sj jS ¼»
sup E (e jZ ) ,
0dZ d 2S
1/ 2
,
where E stands for expectation operator.
is stable and minimal phase.
The interval plant control system
Introducing the notation
(Q ()R() + P u()Py())y(t) = =P0u()Pr()r(t) + R()M(t), characteristics of which coincide with (or are close
H 1 ()
0
0
Q
to) those of H -optimal nominal system, will be referred as robust optimal control system (Tsypkin, 1992). In what follows, the approach for such system design for the cases Q 2 and Q f under both white noise and Markovian disturbances is suggested. The approach is based on external disturbances model building using identification techniques.
3. DESIGN OF ROBUST OPTIMAL SYSTEM FOR WHITE NOISE DISTURBANCES In what follows, the equations of closed-loop control system for interval plant are given in the form, which is convenient for comparison with corresponding Q
equations of H -optimal nominal system. For this purpose, consider two alternately used in system (3) fixed controllers
{R i (), P (), P ()}, i 1,2 . Corresponding outputs are denoted as yi(t), i=1,2. The i r
i y
considerations will be restricted to the class of interval plants for which the transient processes when switching from one controller to another may be neglected. Let assume that these controllers can be chosen so that the following two conditions are met. A1. The closed systems with both mentioned controllers are stable and minimal phase for the plant (1) w.r.t. both disturbance and control.
A2. The outputs of respective closed-loop systems y1 (t ), y 2 (t ) are related by following equation
W2 ()
one obtains the equations of closed loop systems with controllers R1() , R2() as follows
y (t ) W1 ()r (t ) H1 () R1 ()M (t ) T () y (t ) W2 ()r (t )
(5)
H 2 () R2 (t )M (t ) ] (t ). Now taking into account the definition and properties of the functions H 1(),R1(),H 2(),R2() , one gets from (5):
(H 21 R21T H11 R11 ) y (W2 H 21 R21 W1 H11 R11 )r ]
(6)
where for simplicity the arguments and t are omitted. Eq. (6) is the equation of open loop stable minimal phase plant with output noise ]. Identification of this
plant provides the estimate T of polynomial T T( ) . Now let us construct the external disturbances model, using these two controllers and following relations
y(t)=W1()r(t)+H1()R1()M(t)
T ()y(t)=W2()t)+H 2()R2()M(t)
.
( T H 1 R1 H 2 R2 )M .
Denoting now
R1
B1 (1 D); R2
where
D
D ()
B2 (1 D), stands
for
“predicting
polynomial” of order Nin - 1 for disturbance
W2 -T()W1(), where Q0 () R1 () Pu0 Py1 ()
,
Q0 () R2 () Pu0 () Py2 ()
A3. There exist two controllers R1() , R2() , meeting conditions A1, A2 and so that unknown polynomial T() is external, and the function
W1 ()
1
(W1 T W2 )r
] (t ) is discrete white noise.
Pu0 ( ) Pr1 ()
Q0 () R1 () Pu0 () Py2 ()
From these equations one has
y 2 (t ) T () y1 (t ) ] (t ) , where
H 2 ()
1
such that
Q0 () R2 () Pu0 Py2 ()
can be exactly predicted by N in
its previous values, and B1, B2 are fixed polynomials, so that
,
Pu0 () Pr2 ()
M in (n)
M in (t) ,
M in (t) ,
D()M in (t - 1) .
Than accounting Tsypkin’s (1992) absorption condition
(1 D())M in (t )
w.r.t. unknown polynomials C() and Py, and second one – as equation w.r.t. Pr.
0 for t t N in ,
one gets
(W1 T W2 )r
Denote CR -B( 1-D)=Pu A in (12), where
(T H 1 R1 H 2 R2 ) P[ [ . (7)
A() is unknown polynomial, satisfying Eq. (12).
So, the closed loop control system for interval plant accounting the estimates obtained, may be written as follows
(Q () R() P Py ()) y (t ) 0
0 u
Pu0 () Pr (t )r (t ) R()M in (t )
0
(8)
Then one gets the following equations for Py, R:
Py
Q 0 A CPyQ ,
R
CR -Pu0 A.
(13)
R() P[ ()[ (t )
Similarly, one gets from (12)
or, equivalently
Pu0 Pr CR Pu0 A Pu0 Pr .
Q 0 () R() Pu0 Py ()) y (t ) Pu0 () Pr (t )r (t )
(9)
~ Pu0 C one gets the following
~ (14) A Pr C R . Substituting Py , R , Pr from (12 - 14) into
Pr
where
) = (W1Tˆ W2 )(TˆH 1 R1 H 2 R2 ) 1 . Now let us rewrite (8) accounting absorption condition (1 D ())M in (t ) 0 , where D() stands for mentioned above predicting polynomial for M in (t) , and taking into consideration the designation R()=B( )( 1-D()), where D() is assumed to be known and B() stands for a certain (currently unknown) auxiliary polynomial. So one gets (Q 0 ()R()+Pu0 ()Py ())y(t)= (10) Pu0 ()Pr ()r(t)+B()( 1-D ()) () So, the equation (8) of closed control system for interval plant accounting (10) may be transformed to the mode, convenient for comparison with respective H Q -optimal nominal system (4):
Pu0()Pr ()+B()(1-D())() y(t)= r(t). Q0()B()(1-D())+Pu0()Py () (11) Comparing right hand parts of Eqs. (11) and (4), one gets the following polynomial equations w.r.t. unknowns Pr , R, Py :
P ()Pr ()+B()(1-D()))()= 0 u
>
C
equation for determination of Pr
R()Min (t ) R()) ()r
C()Pu0()Pr(),
Denoting
@
Q 0() C()R ()-B(1-D())
(12)
Pu0()(C()Py()-Py ())=0 where C() is arbitrary fixed external polynomial. First equation in (12) may be considered as equation
controller equation, one gets
(CR Pu0 A)u
~ (A Pr C R )r
(Q 0 A CPy )y. So the closed system equation takes the form (Q0(CR Pu0 A)+Pu0(Q0 A CPy ))y=
~ (Pu0 A Pr CR )r+(CR Pu0 A)Min or equivalently ~ (Q0CR+Pu0CPy )y=(Pu0 A Pr C
R )r+(CR Pu0 A)Min or
(Q 0 R Pu0 Py )y=Pu0( Pu0
A) -1 )r+(R C
A )M in . C
For a minimal phase plant, one can in particular to set
Pu0
R 0 , so that A() { D().
4. ROBUST OPTIMAL DESIGN MARKOVIAN DISTURBANCES
UNDER
Let the external plant disturbances be a Markovian sequence of the form M ex (t ) S ()M ex (t 1) + [ (t). Suppose that S() is a known polynomial of order N 2 , and [ (t) is white noise. Let also
Dex ()
S () .
Obviously,
the
equation
(1 Dex ())M ex = [ (t ) 2004). Let M (t )
M in (t ) M ex (t )
holds
(Bakhtadze,
denote the generalized
disturbance and, denoting perturbation predicting polynomial as Din () , one can determine '
D () Din ( ) Dex ( ) Din ( ) Dex ( ) i.e., the generalized predicting polynomial. Then the equality
(1 D())M (t )
(1 Din ())[ (t )
may be referred as pseudo absorption condition. Let us denote
D() 1 Din () .
Let D ( ) be an external polynomial. Then the equation (10) of closed control system for interval plant takes the form
(Q 0 R Pu0 Py ) y Pu0 Pr r RM , where M M (t ) stands for generalized disturbance. Let R=B(1-D). Then (Q 0 R Pu0 Py )y=Pu0 Pr r B( 1-D) ˆ[ (M in M ex )=P Pr r B D
The remaining solution stages are quite similar to considered earlier case of uncorrelated disturbances. The similar approach can be used when external
[ (t )
s
¦K (t i) , i 1
where Ki are independent random variables with zero mean. In fact, as [ (t ) [ (t 1) K (t ) the problem reduces to previous one. 5. THE CASE OF LIMITED STATIONARY DISTURBANCES Such disturbances can be uniquely represented as a sum of uncorrelated singular and regular random series (Wold expansion, see Gikhman and Skorokhod, 1965):
M ex (t)=M (t)+M (t), 1 2 where M ex (t) is singular and M ex (t) - regular. ~ (t)=M (t ) M 1 (t) , and Denoting M in in ex ~ Din ()
(1 Din ( ))(1 Dex1 ( ))M~in (t )
~ (1 Din ())M~in (t )
M~in (t)
with predicting polynomial
disturbances is restricted by the condition of exact
~ Din () . For the
M ex2 (t) it is necessary and sufficient that
it was an output of causal filter with standard uncorrelated input sequence. So the regular component may be analyzed by techniques of section 3. Designed robust optimal controller is causal if respective closed control systems are robust stable. The criteria of robust stability are given by Polyak and Tsherbakov (2002).
6. CONCLUSION The possibility of effective control in uncertain environment conditions determines the major perspectives of robust algorithms use for soft sensors design. But its practical implementation meets a number of essential obstacles, mainly of subjective character, e.g., conservative technical conviction of potential users. Due to different evaluations, more than 90% really implemented controllers are simple PID ones, designed by elementary design techniques. The design of contemporary VA demands not only (maybe, to a minor extent) the modern theoretical techniques, but also deep analytic study of the problems to be solved, based both on engineer’s intuition and on adequate algorithmic foundation.
REFERENCES 1.
2.
Din () Dex1 () Din () Dex1 () ,
N in 1 for M in (t) which is assumed to be exactly predictable by its N in previous values. The class of
0.
So the singular component of external disturbance and perturbations compose “generalized disturbance”
2 ex
where Din () is predicting polynomial of order
0,
or, equivalently
.
0 u
1 ex
condition takes the form
regularity of
disturbances are martingale
M ex1 (t) by N ex previous values using the prediction polynomial of order N ex 1 . ~ (t) the absorption For generalized disturbance M in predictability of
3.
4.
Albertos P, G.C. Goodwin (2002). Virtual sensors for control applications // Annual Reviews in Control. Volume (issue) 26 (1). Bakhtadze N.N. (2004). Virtual analyzers: identification approach (a survey). Automation and Remote Control. No. 11. Pp. 3 - 18. Plenum Publ. New York. N.Y. (translated from Russian). Gikhman I.I. and Skorokhod A.V. (1965). Introfuction to the theory of random processes. Nauka Publishers, Moscow (in Russian). Polyak B.T. and Tsherbakov P.S. (2002). Robust Stability and Control. Nauka Publishers, Moscow (in Russian).
5.
6.
7.
8.
Tsypkin Ya.Z. (1991). Dynamic plants control under constrained uncertainty. Measurements, Control, and Automation. 3 – 4. Pp. 3 – 21 (in Russian). Tsypkin Ya.Z. (1992). Design of robust optimal control systems under constrained uncertainty. Automation and Remote Control. No. 9. Pp. 139 – 159. Plenum Publ. New York. N.Y. (translated from Russian). Tsypkin Ya.Z. (1996). Discrete stochastic systems with internal models. Problems of Informatics and Control. 1 – 2. Pp. 21 – 26 (in Russian). Tsypkin Ya.Z. and A.N.Vishnyakov (1992). Discrete control system design for nonminimal phase plant with delay. Differential Equations. No. 11. Pp. 2001 – 2006 (translated from Russian).