Pareto-optimal Choice of Controller Dimension for Plasma Stabilization System

17th IFAC Workshop on Control Applications of Optimization 17th on Control Applications 17th IFAC IFAC Workshop Workshop onOctober Control 15-19, Applications of Optimization Optimization Yekaterinburg, Russia, 2018 of Available online at www.sciencedirect.com 17th IFAC Workshop on Control Applications of Optimization Yekaterinburg, Russia, October 15-19, 2018 17th IFAC Workshop on Control Applications of Optimization Yekaterinburg, Russia, October 15-19, 2018 17th IFAC Workshop on Control Applications of Optimization Yekaterinburg, Russia, Russia, October October 15-19, 15-19, 2018 2018 Yekaterinburg, Yekaterinburg, Russia, October 15-19, 2018

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IFAC PapersOnLine 51-32 (2018) 175–178

Pareto-optimal Choice of Controller Dimension Pareto-optimal Choice of Controller Dimension for Plasma Stabilization System Pareto-optimal Choice of Dimension Pareto-optimal Choice of Controller Controller Dimension for Plasma Stabilization System for Plasma Stabilization System for Plasma Stabilization System

D. Ovsyannikov*, S. Zavadskiy* D. S. D. Ovsyannikov*, Ovsyannikov*, S. Zavadskiy* Zavadskiy*  D. S.  D. Ovsyannikov*, Ovsyannikov*, S. Zavadskiy* Zavadskiy* D. Ovsyannikov*, S. Zavadskiy*  * Saint Petersburg State University, Saint  Petersburg, Russia ([email protected]) ** Saint Saint Petersburg Petersburg State State University, University, Saint Saint Petersburg, Petersburg, Russia Russia ([email protected]) ([email protected]) Saint Petersburg Petersburg State University, University, Saint Petersburg, Petersburg, Russia ([email protected]) ([email protected]) *** Saint Saint Petersburg State State University, Saint Saint Petersburg, Russia Russia ([email protected]) Abstract: The approach to synthesis and implementation of the controller of ITER plasma stabilization Abstract: The approachThe to synthesis synthesis andofimplementation implementation of the controller controller ofthe ITER plasma stabilization stabilization Abstract: approach to and the ITER plasma system is The considered. advantage the presented of approach is thatof controllers of various Abstract: The approachThe to synthesis synthesis andof implementation of the controller controller ofthe ITER plasma stabilization stabilization Abstract: The approach to and implementation of the of ITER plasma system is considered. advantage the presented approach is that controllers of various various system is considered. The advantage of the presented approach is that the controllers of Abstract: The approach to synthesis and implementation of the controller of ITER plasma stabilization dimensions are optimized. Software computes the controllers whose dimension and integral quality system is considered. considered. The Software advantagecomputes of the the presented presented approach is that that the controllers controllers of various various system is The advantage of approach is the of dimensions are optimized. the controllers whose dimension and integral quality dimensions are optimized. computes thealgorithm controllers dimension and integral quality system considered. The Software advantage of the the presented approach is that controllers of to various criterionisare Pareto-optimal. It is described to whose choose the the proper controller final dimensions are optimized. Software computes the controllers whose dimension and integral quality dimensions optimized. Software computes controllers dimension and integral criterion areare Pareto-optimal. It ofis isthe described thethe algorithm to whose choose the proper controller toquality final criterion are Pareto-optimal. It described the to choose the proper controller to final dimensions are optimized. Software computes thealgorithm controllers whose dimension and integral quality implementation when equations plasma stabilization system have a high dimension. The presented criterion are Pareto-optimal. It described the algorithm to choose proper controller to final criterion are It is described the to the proper controller to final implementation when equations ofis the plasma stabilization system have aaofthe high dimension. The presented presented implementation when equations plasma system have high dimension. The criterion are Pareto-optimal. Pareto-optimal. It of isthe described the algorithm algorithm to choose choose the proper controller to some final integral quality criterion evaluates the upper stabilization bound of the ensemble transient processes for implementation when equations of the plasma stabilization system have a high dimension. The presented implementation when equations of the plasma stabilization system have a high dimension. The presented integral quality criterion evaluates the upper bound of the ensemble of transient processes for some integral quality criterion evaluates the upper bound of the ensemble of transient processes for some implementation when equations of the plasma stabilization system have a high dimension. The presented arbitrary plasma disturbances. integral evaluates integral quality criterion evaluates the the upper upper bound bound of of the the ensemble ensemble of of transient transient processes processes for for some some arbitrary quality plasma criterion disturbances. arbitrary plasma disturbances. integral quality criterion evaluates the upper bound of the ensemble of transient processes for some arbitrary plasma disturbances. arbitrary plasma disturbances. © 2018, IFAC (International Federation of Automatic Methods Control) Hosting by Elsevier Ltd.Robust All rightsControl reserved.and Keywords: Optimization Methods; Numerical for Optimization; arbitrary plasma disturbances. Keywords: Optimization Methods; Methods; Numerical Numerical Methods Methods for for Optimization; Optimization; Robust Robust Control Control and and Keywords: Stabilization Optimization Keywords: Optimization Methods; Numerical Numerical Methods for for Optimization; Robust Robust Control and and Keywords: Optimization Stabilization Stabilization Keywords: Optimization Methods; Methods; Numerical Methods Methods for Optimization; Optimization; Robust Control Control and Stabilization Stabilization  Stabilization  measures 6 gaps between hot area of plasma and tokamak  1. INTRODUCTION measures gaps hot and tokamak measures 6 6plasma gaps between between hot area area of11plasma plasma andindicators. tokamak  chamber, current and otherof plasma 1. INTRODUCTION 1. INTRODUCTION measures 6 gaps between hot area of plasma and tokamak measures 6 gaps between hot area of plasma and tokamak chamber, plasma current and other 11 plasma indicators. chamber, plasma current and other 11 plasma indicators. measures 6 gaps between hot area of plasma and tokamak 1. INTRODUCTION These 18 values are treated as a control object output y. The 1. INTRODUCTION Problems of analysis1.and synthesis of stabilizing regulators of These chamber, plasma current and other 11 plasma indicators. INTRODUCTION chamber, plasmaare current and other 11object plasma indicators. 18 values treated as a control output y. The These 18 values are treated as a control object output y.That The Problems of analysis and synthesis of stabilizing regulators of chamber, plasma current and other 11 plasma indicators. chamber is divided into a large number of circuits. Problemsposition of analysis and synthesis ofinstabilizing regulators of ITER current, and plasma shape tokamak are of great These chamber 18 values valuesis are are treated asa aalarge control object output y.That The These 18 treated as control object output y. The Problems of analysis and synthesis of stabilizing regulators of ITER divided into number of circuits. Problems of analysis and synthesis of stabilizing regulators of ITER chamber is divided into a large number of circuits. That current, position and plasma shape in tokamak are of great These 18 values are treated as a control object output y. The high dimension ofinto control object. Linear systemsThat are current, position andand plasma shapeofinstabilizing tokamak are of great Problems of(McArdle analysis synthesis regulators of defines importance 1998), (Misenov 2000), (Ovsyannikov ITER chamber is divided a large number of circuits. ITER chamber is divided into a large number of circuits. That current, position and plasma shape in tokamak are of great defines high dimension of control object. Linear systems are current, position and plasma shape in tokamak are of great defines high dimension of control object. Linear systems are importance (McArdle 1998), (Misenov 2000), (Ovsyannikov ITER chamber is divided into a large number of circuits. That widely high used dimension in designofproblems of control systems for importance (McArdle 1998), (Misenov 2000), (Ovsyannikov current, position and plasma shape in tokamak are of great defines et al. 2006). Engineering implementation of the synthesized control object. object. Linear systems systems are are defines high dimension of control Linear systems importance (McArdle 1998), (Misenov 2000), (Ovsyannikov widely used in design problems of control for importance (McArdle 1998), (Misenov 2000), (Ovsyannikov widely used in design problems of control systems for et al. 2006). Engineering implementation of the synthesized defines high dimension of control object. Linear systems are complex objects. The synthesis of the regulator that stabilizes et al. 2006). Engineering implementation ofisthe synthesized importance (McArdle 1998), (Misenov 2000), (Ovsyannikov control laws in the stabilization system also complex widely used in design problems of control systems for widely used in design problems of control systems for et al. 2006). Engineering implementation of the synthesized complex objects. The synthesis of the regulator that stabilizes et al. 2006). Engineering implementation of the synthesized complex objects. The synthesis of the regulator that stabilizes control laws in the stabilization system is also complex widely used in design problems of control systems for plasma shape in tokamak is done based on the linearization of control laws in the stabilization system is also complex et al. 2006). Engineering implementation of the synthesized (Ovsyannikov et the al. 2005). Controller equations should be complex objects. The synthesis of the regulator that stabilizes complex objects. The synthesis of the regulator that stabilizes control laws in stabilization system is also complex plasma shape in tokamak is done based on the linearization of control laws in inet the stabilization system is also alsoshould complex plasma shape in tokamak is done based on the linearization of (Ovsyannikov al. 2005). Controller equations be complex objects. The synthesis of the regulator that stabilizes differential equations that define plasma behavior (Zavadskiy (Ovsyannikov et al. 2005). Controller equations should be control laws the stabilization system is complex included in the etfirmware of the tokamakequations control system. The plasma shape in tokamak is done based on the linearization of plasma shape in tokamak is done based on the linearization of (Ovsyannikov al. 2005). Controller should be differential equations that define plasma behavior (Zavadskiy (Ovsyannikov et al. 2005). Controller equations should be differential equations that define plasma behavior (Zavadskiy included in the firmware of the tokamak control system. The plasma shape in tokamak is done based on the linearization of et al. 2010). There is a wide range of approaches for the included in the firmware of the tokamak control system. The (Ovsyannikov et al. 2005). Controller equations should be dimension of the controller equations affects the differential equations that define plasma behavior (Zavadskiy differential equations that define plasma behavior (Zavadskiy included in the firmware of the tokamak control system. The et al. 2010). There is a wide range of approaches for the included in the firmware of the tokamak control system. The et al. 2010). There is a wide range of approaches for the dimension of the controller equations affects the differential equations that define plasma behavior (Zavadskiy analytical regulator construction for complex systems dimension of the controller equations affects the included in the firmware of the tokamak control system. The computational Plasma equations has a wide range the of analytical et There aa wide of for et al. al. 2010). 2010).regulator There is is construction wide range rangefor of approaches approaches for the the dimension of of complexity. the controller controller affects complex systems dimension the equations affects the regulator construction for complex systems computational Plasma has aa wide range of et al. 2010). There is a 2017), wide range of approaches for the (Veremey and Knyazkin (Veremey 2017), (Balandin computational complexity. Plasma has wide range of analytical dimension of complexity. the instability controller equations affects instabilities. Any such must be worked out by the analytical regulator construction for complex systems analytical regulator construction for complex systems computational complexity. Plasma has a wide range of (Veremey and Knyazkin 2017), (Veremey 2017), (Balandin computational complexity. Plasma has a wide range of (Veremey and Knyazkin 2017), (Veremey 2017), (Balandin instabilities. Any such instability must be worked out by the analytical regulator construction for complex systems al. 2017),and (Ananyev 2018), (Talagaev 2017). Equations for instabilities.The Any such instability musthas be process worked out bymeet the computational complexity. Plasma a wide must range of et controller. dynamics of the transition (Veremey Knyazkin 2017), (Veremey 2017), (Balandin (Veremey and Knyazkin 2017), (Veremey 2017), (Balandin instabilities. Any such instability must be process worked out by the et (Ananyev 2018), (Talagaev 2017). Equations for instabilities. Any such instability must worked out by the et al. al. 2017), 2017), (Ananyev 2018), (Talagaev 2017). Equations for controller. The dynamics of transition must (Veremey and Knyazkin 2017), (Veremey 2017), (Balandin in circuits are linearized in the area of the equilibrium controller. The dynamics of the theaccuracy transition process must meet instabilities. Any such instability must be beand worked outsavings bymeet the current the requirements of control energy et al. al. 2017), 2017), (Ananyev 2018), (Talagaev (Talagaev 2017). Equations for et (Ananyev 2018), 2017). Equations for controller. The dynamics of the transition process must meet current in circuits are linearized in the area of the equilibrium controller. The dynamics of the transition process must meet current in circuits are linearized in the area of the equilibrium the requirements of control accuracy and energy savings et al. 2017), (Ananyev 2018), (Talagaev 2017). Equations for the requirements of control and energy savings controller. The dynamics of approach theaccuracy transition process meet (Veremey 2003). Presented specifies andmust fixes the position. current in circuits are linearized in the area of the equilibrium current in circuits are linearized in the area of the equilibrium the requirements of control accuracy and energy savings position. the requirements of control accuracy and energy savings position. (Veremey 2003). Presented approach specifies and fixes the current in circuits are linearized in the area of the equilibrium (Veremey 2003). Presented approach specifies and fixes the the requirements of controlwhich accuracy andinenergy savings set of plasma disturbances happen practice. This plasma state space position. (Veremey 2003). Presented specifies and the « l ,   drops » (Veremey 2003). Presented approach approach specifies and fixes fixesThis the position. set of disturbances which happen in position. equationspace set of plasma plasma disturbances which happen in practice. practice. This (Veremey 2003). Presented approach specifies and fixes the plasma disturbance set produces the ensemble of transient processes. plasmaBstate state spacef ( t ) «disturbance « ll ,,  x  A x  u G   drops drops »» set of plasma disturbances which happen in practice. This equation set of plasma disturbances which happen in practice. This equation plasma state space space disturbance set produces the ensemble of transient processes.  w (t»)  plasma state disturbance set produces the ensemble of transient processes. «disturbance l ,   drops  set of plasma disturbances which happen in practice. This x A x  u ff (( tt )) disturbance y  C x  FB ) Introduced integral quality criterion evaluates the upper »  f « l ,(t)  drops x  plasma Ax  Bstate u f ( tG Gspace equation   equation « l ,   drops disturbance set set produces the ensemble ensemble ofevaluates transient the processes. w ((tt»))   w disturbance produces the of transient processes. (xy yF E ) equation Introduced integral quality criterion upper C AEx x ,  Bu u Eff (( ttG G)), u  disturbance ff disturbance ((tt ))  y x   C xx  FB Introduced integral quality criterion evaluates the upper x  A ff (( tt ))    disturbance set produces the ensemble of transient processes. disturbance bound of the transient ensemble and evaluates gives estimations of  w ((tt ))  AExx ,yFB u Ef( tG), u  f (E t) ) ((xy x diagnostic signals Introduced integral quality criterion the upper upper    w yx  C CEx ,yF  Ef ( t ), u  E ) w t ( ) Introduced integral quality criterion evaluates the  ff ((tt ))   bound of the transient ensemble and gives estimations of w ((tt ))   CEx ,yF  Ef ( t ), u  E ) vector bound ofaccuracy the transient ensemble and for gives estimations of Introduced integral quality criterion evaluates the upper f (t )   w diagnostic ((yx control and energy costs the ensemble of  diagnostic signals signals x  E , y E , u E ) w t ( )   ( yvector E ) bound ofaccuracy the transient transient ensemble and for gives estimations of ( x E , y E , u E ) bound the ensemble and gives estimations vectorsignals control and costs the ensemble of diagnostic control of and energy energy costs the ensemble we of diagnostic signals bound ofaccuracy the transient ensemble and for gives estimations ( y  )) trajectories. By changing the dimension of the controller diagnostic vector ( yvector E Esignals control accuracy and energy costs for the ensemble of y control accuracy and costs for the ensemble of trajectories. By changing the the we vector (( y trajectories. By changing the dimension dimension of the controller controller we control accuracy and energy energy costs forof the ensemble of y  E E )) also change the integral criterion values. Practical advantage ( y  E )y y trajectories. By changing the dimension of the controller we trajectories. By changing the dimension of the controller we also change the integral criterion values. Practical advantage power system current and shape controller filters also the optimization integral criterion values. Practical advantage ysystem trajectories. By changing theapproach dimension the we of thechange proposed is of that wecontroller optimize the x  P and x  shape P y controller x  Afilters x  By y x  power A x  Bsystem u current system ysystem also change the integral criterion values. Practical advantage power system current and shape controller filters also change the integral criterion values. Practical advantage of the proposed optimization approach is that we optimize the y  C x u  P x   x  P x  P y x A  B y  x  A x  B u of the proposed optimization approach is that we optimize the u C also change the integral criterion for values. Practical advantage x  P and x  shape P y controller x  Afilters x  Bsystem y x  power A x  Bsystem u current whole ensemble of transients the controller of any power system current and shape controller filters system y x C CEfilters x, y Bsystem ux x P P Ex xx , u P E y , y  E ) (u (xy E , y E ) of the optimization approach is we the u C power current A u of the proposed proposed optimization approach is that that we optimize optimize the E whole ensemble of transients for the controller of any u(xxx  x  P P and x  shape P y controller x  A A xx  B y y C AE xx, u B Bsystem u , u E ) whole ensemble of transients for the controller of any of the proposed optimization approach is that we optimize the ((ux x ((xy x  AE xx  AE xx, u B E u , u E ) dimension. This makes it possible to construct Pareto-optimal  C xx,, y  P P xx ,, u x PE Ex u  P E E y ,, y y  E E )) x C Ex y B E Ey ,, y y  E E )) u  C (  y u whole ensemble of transients transients for the controller controller of any any u( x CE x, u  E , u  E ) whole ensemble of for the of dimension. This makes it possible to construct Pareto-optimal u C x (u x x P E Ex ,, u u E E ,, y y  E E )) (yx x  CE E x,, y y  E E ,, y y  E E )) dimension. This makes it possible to construct Pareto-optimal whole ensemble of transients for the controller of any ( ( E E ,, u u  E E ,, u u E E )) set of controllers and select the controller with proper (( xx  ( x  E , u  E , y  E ) ( x  E , y  E , y  E ) dimension. This makes makes it possible possible to construct construct Pareto-optimal ( x E , u E , u E ) dimension. This it to Pareto-optimal set of controllers and select the controller with proper set of controllers and select the controller with proper dimension. This makes it possible to construct Pareto-optimal dimension for final firmware (Ovsyannikov et al. 2007). The set of of controllers controllers and select select the controller controller with proper 1. Structure of plasma current and shape stabilization set and the proper dimension for (Ovsyannikov 2007). The dimension for final final firmware firmware (Ovsyannikov etforal. al.with 2007). The Fig. set of controllers and select the controller with proper algorithm includes an optimization cycle et 1. Fig. 1. Structure Structure of of plasma plasma current current and and shape shape stabilization stabilization dimension for final final firmware firmware (Ovsyannikov etforal. al.each 2007).Pareto The Fig. system dimension for (Ovsyannikov et 2007). The algorithm includes an optimization cycle each Pareto Fig. 1. 1. Structure Structure of of plasma plasma current current and and shape shape stabilization stabilization algorithm includes an optimization cycle etforal.each dimension for final firmware (Ovsyannikov 2007).Pareto The Fig. point. system system Fig. 1. Structure of plasma current and shape stabilization algorithm includes an optimization cycle for each Pareto algorithm includes point. system point. algorithm includes an an optimization optimization cycle cycle for for each each Pareto Pareto system system point. The structure of the control system is schematically point. point.2. MODEL OF ITER PLAZMA STABILIZATION The of the control is The structure structure of (Fig. the 1). control system is schematically schematically represented on the Plasmasystem disturbances is given as 2. PLAZMA The structure of the control system is schematically 2. MODEL MODEL OF OF ITER ITERSYSTEM PLAZMA STABILIZATION STABILIZATION The structure of the control system is schematically represented on the (Fig. 1). Plasma disturbances is represented on the (Fig. 1). Plasma disturbances is given given as as The structure of the control system is schematically 2. MODEL OF ITER PLAZMA STABILIZATION f(t) (Veremey 2003) 2. PLAZMA represented on the (Fig. 1). Plasma disturbances is given as SYSTEM 2. MODEL MODEL OF OF ITER ITERSYSTEM PLAZMA STABILIZATION STABILIZATION represented on the (Fig. 1). Plasma disturbances is given as f(t) (Veremey 2003) f(t) (Veremey 2003) represented on the (Fig. 1). Plasma disturbances is given as SYSTEM SYSTEM ITER tokamak has 11 SYSTEM control coils to provide proper f(t) (Veremey 2003) f(t) (Veremey 2003) f(t) (Veremey 2003) ITER tokamak has 11 control coils to provide proper ITER tokamak has 11 control coils to provide proper magnetic configuration plasma coils hold (Ovsyannikov et al. ITER tokamak has 11 11to control to provide provide proper proper ITER tokamak has control coils to magnetic configuration to plasma hold (Ovsyannikov et al. al. magnetic configuration plasma hold (Ovsyannikov et ITER tokamak has 11to control coils to provide proper 2006), (Zavadskiy et al. 2009). These 11 voltages are treated magnetic configuration to plasma hold (Ovsyannikov et al. al. magnetic configuration to plasma hold (Ovsyannikov et 2006), (Zavadskiy et al. 2009). These 11 voltages are treated 2006), (Zavadskiy et al. 2009). These 11 voltages are treated magnetic configuration to plasma hold (Ovsyannikov et al. as a control objectet input u. The ITER diagnostic 2006), (Zavadskiy al. 2009). 2009). These 11 voltages voltages are system treated 2006), (Zavadskiy et al. These 11 are treated as a control object input u. The ITER diagnostic system as a control objectet input u. The ITER diagnostic 2006), (Zavadskiy al. 2009). These 11 voltages are system treated as aa control control object input input u. The The ITER diagnostic diagnostic system as as a control object object input u. u. The ITER ITER diagnostic system system i i i

1

i

dropi

12 1

i

drop drop

st

p

st st st 67 st st st 67st 67st st st 67 st 67

modp p

st stst st st

12 12 1 2 2 2

drop drop drop

st

st st

st st st st st st

st st st

stmod p modp

mod mod mod

18

11

p mod 18 mod stmodp 18 mod p stmod p mod 18 18 st p st p 18 st p

67

11 11 11 11 11

18

st

18 18

st st

18 18 18

st st st

p

p

pp p

pp p

pp p pp p pp pp p p p p

p

p

pp p 11 pp pp p pp pp 11p p11 p p p p p p p11 p 11 p 11 p p

p p

11

p p p

11

c

1

c

2

c c

31 1 3 31 1

cc c k c cc kc

2 2

11 11

11 11

cc c cc c

11 11 11

11 11 11

c c c

k 31 cc 3 c 3 k kc k

f

f f 11 2 f 2 11 f 2 11 f 11 11 11

18

f f f f f f

18 18 18 18 18

2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 175Hosting by Elsevier Ltd. All rights reserved. Peer review© of International Federation of Automatic Copyright 2018 IFAC 175 Copyright ©under 2018 responsibility IFAC 175Control. 10.1016/j.ifacol.2018.11.376 Copyright © 2018 2018 IFAC IFAC 175 Copyright © 175 Copyright © 2018 IFAC 175

f

f

f f ff f f

ff ff f f 18 f f ff ff f 18 f

ff ff f f f f f

s s

s f s st s f st f st 18

f

18 ff ff f f f 18 18 f 18

s

f f f f f f f f f

st st 18 18 st 18 18 18

18

s s s s s s

18 18 18 18 18

IFAC CAO 2018 176 D. Ovsyannikov et al. / IFAC PapersOnLine 51-32 (2018) 175–178 Yekaterinburg, Russia, October 15-19, 2018

3. TRANSIENT ENSEMBLE MODELING

 w (t )  f (t )  f drop (t )   1 ,  w2 (t )  w1 (t )  d  e where



 t / t

d , dl , t , t l



Let’s specify disturbance set and let’s construct appropriate transient ensemble of the trajectories of system (2). Presented approach allows optimizing the transient ensemble of fullsize control object that is closed by regulator of dimension k. It is suggested to use an integral performance criterion as a functional that allows optimizing the transient process, perturbed at the initial point set and the set of external disturbances. The proposed approach is suitable for optimizing the structure and parameters of the stabilizing regulator as well as to control the beams of charged particles (Balabanov et al. 2018), (Golovkina 2017). Let x0 , f (t ) are

(1)

, w2 (t )  d l e t / tl  ,

are known real numbers which

depend on plasma mode (Zavadskiy 2014). The diagram (Fig. 1) also shows plasma state-space equations in deviation from equilibrium point, filter subsystem, current and shape controller, power subsystem, matrices of these subsystems are given constant matrices of ITER project. Let’s consider current and shape controller which has dimension k, and controller’s matrices are

 p1  P1    p  ( k 1)*k

 pk       p k *k 

an arbitrary disturbance that satisfy the following equation at the moment t  [0, T ] (Kirichenko 1978), (Ovsyannikov 1990)

( x0 , f (t ) )     (G1 , G2 (t ),  2 ),

,

  {( x0 , f (t )):

[ k k ]

(3)

t

x0 G1 x0   f * () G2 () f () d   2 }, *

 p k *k 1  p k *k 18    , P2       p   k *k  ( k 1)*18  p k *k  k *18 [ k 18]

t0

where T is the end of modeling, G1 , G2 () are the positively defined matrices and  is the real positive

 p k *k  k *181  p k *k  k *18 k       , P3    p   k *k  k *18( k 1)*11  p k *k  k *18 k *11 [11k ] where p 1 , …, p k*k  k*18 k*11 are the constant values of the regulator which should be found. The closed control object in the state space can be described by the following equations:

x  S x  G mod f(t), y  Cx, u  P3 x,

constant and symbol * is a matrix transposition. This disturbance ensemble  (3) includes practical plasma instabilities (1). Let us consider matrix D as solution of matrix differential equation (Zavadskiy and Kiktenko 2014)

D  SD  DS * ,

(4)

D (0)  G11 . Matrix D(t ) is modeling the ensemble, since for arbitrary ( x0 , f (t ) )   , it is performed (Zavadskiy 2014)

x(t, x0 , f (t )) {x(t ) : x(t )* D(t ) x(t )   2 }. Therefore the diagonal elements Di ,i (t ) of the matrix D(t )

(2)

are

x(0)  x 0 ,

the

maximum

for

xi (t )  Di ,i (t ) , 2

( x0 , f (t ) )   . The results of modeling the boundaries of the ensemble for Pareto point k=11 are presented on the figure (Fig. 2).

where

0 A  Af C S  0 P2 C f  0 0  where x  E

67 1811 k

0 0 P1 P3

BC p   0  , 0   A p 

is the state space vector, u  E

the control voltages vector,

yE

18

4. TRANSIENT OPTIMIZATION TOOL

11

It is suggested to use the following integral performance criterion that allows optimizing the set of transient processes perturbed by the sets of initial points and external disturbances:

is

is the diagnostic output * T   tr[ W1C D(t ) C ]    I ( pi )   2   dt  *  0   tr[ W2 P3 D(t ) P3 ]  

x0 is initial plasma shift from the equilibrium point, matrices A, B, C, A f , Ap , C f , C p , Gmod are given

signal vector,

 tr[ W3C D(T ) C * ]   min,

constant matrices of ITER (Ovsyannikov 2006).

176

(5)

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the increase of the dimension of the regulator causes additional computational load in real time control processes. We can say that these criteria are contradictory. Optimization software implements the iterative algorithm:

D i ,i (t ), 10 2  m 18 16

step 1: increase the controller dimension by one; step 2: initialize new added rows and columns by zeros; step 3: provide gradient optimization; step 4: transients ensemble modeling, compute the criteria; go to step 1. To find a balance between the dimension of the regulator and the integral criterion of the dynamics quality, we will take the only Pareto-optimal controllers. Software draws Pareto set of these two criterions – dimension via quality.

14 12 10 8 6 4 2 0 0

2

4

6

8

10

12

14

16

18

20

time (s)

Fig. 2. Modelling of transients ensemble for output y where tr[ A] is the trace of the matrix,

W1 , W2 , W3 are

pi is i-th parameter of the regulator, i  1,...,k * k  k *18  k *11. If regulator has symmetrical weight matrices,

dimension k then the gradient representation of the functional is *   S * 1 P2  D P G    2 2 T pi pi   I   2  2 tr   dt, (6) * pi 0    W P D P3 2 3   pi  

Fig. 3. Pareto-optimal set of controllers 6. CONCLUSIONS Given set of initial and external disturbances (3) gives the ensemble of transients (4). This disturbances set is considered constant. When setting up the structure, values and dimension of the closed-loop system’s controllers, it is possible to achieve the proper dynamics of the entire ensemble. The gradient optimization method is developed based on representations for the integral quality criterion (4)-(7). This integral criterion evaluates the upper bound of the entire ensemble of trajectories. Software iteratively increases the dimension of the controllers and conducts gradient optimization. In many cases, a larger dimension gives a better value of the integral criterion (5). These goals are contradictory. To find a balance between the dimension of the regulator and the integral criterion of the dynamics quality, software draws Pareto optimal set of these two goals: dimension via quality. Software users have the opportunities to select the appropriate controller in each particular case from Pareto-optimal set.

where matrix D is the solution of (4), matrix  is the solution of the following equation

  S  S *  (C *W C  P *W P ),  1 3 2 3 (T )  C *W3C.

(7)

Based on the analytical expressions (3)-(7) a gradient optimization method for the functional (5) with regulator of dimension k is implemented for C++ and MatLab environments (Zavadskiy and Sharovatova 2015). 5. PARETO-OPTIMAL DIMENSION VARIATION We do not change the input and output number of the controllers and we fix the disturbances set (3), but will vary the dimension k of the controller as well as dimension of matrices P1, P2 , P3 in controller representation (Fig. 1). For this purpose, it’s added zeros columns and rows to the matrices P1, P2 , P3 . New round of gradient optimization

REFERENCES

provides new value of integral criterion (6). Often the value of the criterion is better, but not always. On the other hand,

Ananyev, B. (2018). About control of guaranteed estimation. Cybernetics and Physics, volume 7, No. 1, pp. 18-25. 177

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