Mukhina, O. O.

Smagin, Valery I.

English

Tomsk State University Repository

Филиал Кемеровского государственного университета  в г. Анжеро-Судженске Национальный исследовательский Томский государственный университет Кемеровский государственный университет Институт проблем управления им. В. А. Трапезникова РАН Институт вычислительных технологий СО РАН    ИНФОРМАЦИОННЫЕ ТЕХНОЛОГИИ  И МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ (ИТММ–2015) Материалы  XIV Международной  конференции   имени  А .  Ф .  Терпугова  18–22 ноября 2015 г. Часть 1       Издательство Томского университета  2015 177 ОПТИМИЗАЦИОННЫЕ МОДЕЛИ И ИССЛЕДОВАНИЕ ОПЕРАЦИЙ   DYNAMIC LOCALLY OPTIMAL CONTROL OF DISCRETE STATE DELAY SYSTEMS WITH RANDOM PARAMETERS О. О. Mukhina, V. I. Smagin National Research Tomsk State University, Tomsk, Russia  Control of the discrete systems with random parameters is considered in [1–4]. In this paper we propose for object with random parameters to realize synthesis of dynamic tracking system control on output based on optimization of the local criterion under indirect measurements taking into account state delay. To improve the quality control of objects practice of insertion in control law Lu-enberger observer [5] or dynamic feedback reduced dimensionality [6] is used. Control is defined as a function of the measured variables and the tracking sig-nal. Asymptotic behavior of the system, construct estimates for asymptotic tracking accuracy are researched. An example is given to illustrate the useful-ness of the proposed results. 1. Problem statement. Consider the following discrete-time system with time delay and random parameters: 1 1( 1) ( ( )) ( ) ( ( )) ( )r ri i i ii ix k A A k x k A A k x k h             +1( ( )) ( ) ( );ri iiB B k u k q k    ( ) ( ), ,1 ,2 , ,0; 0,1,2,x h h h k          ,                    (1) ).()()( kvkSxky                                             (2) In (1), (2) nRkx )(  is state vector; 0h  is positive integer time delay; mRku )(  is control input; lRky )(  is observations vector; riBBAAAA iii ,1,,,~,~,,   are constant matrices of appropriate dimensions; S is matrix of measurement channel; matrix B and S are of full rank; pairs of matric-es (A, B) and ( ),~ BA  are controllable; pairs of matrices ),( AS  and )~,( AS  are observable; ( )   is prescribed determinate function of initial conditions on the interval  1,,1,  hh , as this takes place 0(0) (0)x x    is random vector with characteristics:   00 xxM  ,   000 xPxxM  ; )(kq , v(k) are Gaussian ran-dom sequences of input disturbances and measurement errors with characteris-tics: ,0)}({  kq  ,0)}({  kv  ,0)}()({   jvkq  { ( ) ( )} ( ) kjq k q j Q k   , { ( ) ( )} ( ) kjv k v j V k    ( ,k j  is Kronecker delta), ,0)()(   kQkQ  178 0)()(   kVkV  are nonnegative definite matrices); ( )i k  is Gaussian random sequences (  ( ) 0iM k  ;  ( ) ( )i iM k j   jk , ). Local criterion described by:  ,)()())()1(())()1(()( kDukukzkwCkzkwMkI         (3) where )()( kHxkw   is controlled output of the system (H is matrix of the output of the system); 0,   DDCC  are weighting matrices; nRkz )(  is the reference input which described by the equation  )()()1( kqkFzkz z , ,2,1,0,)0( 0  kzz .              (4) In (4) )(kqz  is Gaussian random sequence with characteristics ,0)}({  kqz  ,0)}()({   jqkqz  ,0)}()({   jvkqz  ,{ ( ) ( )} ( ) ,z z z k jq k q j Q k    0z  are initial conditions ,}{,}{(000 0000 xzzPxzPzz    )}{0000 zxPzx   , F is ma-trix of the dynamics model of the reference input. It is required to construct a control of system (1), using observations (2) and minimizing the criterion (3). 2. Locally criterion optimization. Let the control law of the system (1) under observations (2) is defined as: 0 1 2 3 0( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )u k K k k K k y k K k y k h K k z k K k k           )()()()()()( 211 hkSxkKkvkKkSxkK )()()()( 32 kzkKhkvkK  ,  (5) where transfer coefficients are )(),(),(),( 3210 kKkKkKkK  to be determined and variable ( )k  is defined with by the reduced dimensionality equation [6]. Theorem 1. If for object (1), observations (2) and local criterion (3) ma-trices 0)()(1  riii CHBHBDCHBHBkC , ( ) ( )( ) ( ) ( )( )( , ) ( , )( ) ( )xx xx xz xzP k SP kP k S SP k S V kP kP k k h S SP k k h SP k SP k       ( , ) ( )( , ) ( )0( ) ( ) ( , )( , ) ( )x zx zxx zxxz zSP k h k P kSP k h k S P k SSP k h S V k h P k k h SSP k h k P k            (6) are positive definite for all ,2,1k , then optimal in the sense of minimum cri-teria (3) transfer coefficients for control (5) are determined by the formulas: dkcKkbKkaKkK  )()()()( *3*2*1*0 ;                          (7) 179 hkgKkfKkeKkK  )()()()( *2*1*0*1 ;                          (8) rkpKknKkmKkK  )()()()( *3*1*0*2 ;                           (9) ,)()()()( *2*1*0*3 kklKktKksKkK                           (10) where klsrpnmhgfedcba ,,,,,,,,,,,,,,  are variables depending on  )()(),( rzkzMrkPz  ;  ( , ) ( ) ( )P k r M k r    ; ( , )zP k r    ( , ) ( ) ( )zP r k M z k r    , which are defined by the system of difference ma-trix equations without delays with initial conditions: 0)0( zz PP  ; (0) (0) (0) 0z zP P P      and ),( rkPx  )()( rxkxM  ;   ),(),( krPrkP xzzx   )()( rxkzM  , which are defined by the system of difference matrix equations with time delays with initial and boundary conditions: ( , ) ( ) ( )xP t h j h t h j h        for 1,0,  hjt ; 0(0, ) ( )xP x     ; 0( ,0) ( )xP x    ; 0(0, ) ( )zxP z     ; 0( ,0) ( ) ;xzP z      ( ,0)xP     (0, ) 0xP    для 1,,2,1,  hhh ; 0)0( xx PP  ; 00)0( xzzx PP  ; ;)0(00zxxzPP   00)0( xzzx PP  ; (0) (0) 0x xP P   .  The proof of the theorem is performed similar paper [7]. 3. Asymptotic behavior. Asymptotic tracking accuracy for the object (1) is defined by calculating criterion estimation: },)1({lim 2zkxJk                                       (11) where . is Euclidean norm of vector, z is constant reference input. Theorem 2. Let in description of system (1), observations (2), criterion (3) and model of reference input (4) matrices DCVSQBBAAAA iii ,,,,,,,~,~,, , ri ,1  are constant; .0)(;  kqEF z  Then, if the condition (6) theorem 1 is satisfied, there exist steady-state solutions of difference equations for ),( jtPx , ),( jtPzx , ),( jtPxz , ( , )xP t j , ( , )xP t j  and condition is satisfied: 12121  ,                                                 (12) then for criterion (11) estimate is valid: 2 2 2 2 2 22 1 1 2 22 21 1[( ) ( ) ][( ) ( )]1 ( )G R g r trQJ             2 22 22 11 2 1 2 12 21 1( ) ( )2( ) ( ) ( )1 ( )G R g r trQ g r G R               1 2 1 1 221( )( ) ( )( )2( )1g r G Rg r           Qtr~ ,             (13) where 180 1s   ; 2 2s   ; 1 s   ; 2 2s   ; *1A BK S   ; *2 2A BK S   ; EBK  *33 ; 1 1 2 21 1( ) ( ) ; ( ) ( )r r* *i i i i i i i ii i(Aθ k Bθ k K S) (Aθ k Bθ k K S)         ; )(lim *1*1 kKK k  ; )(lim *2*2 kKK k  ; )(lim*3*3 kKK k  ; zr 31  ; zBKr *32  ; *31( )ri iiR B k K z  ; *0g BK  ; *01( )ri iiG B k K   ;  riii BVKKBBVKBKQ1*2*1*2*11~ ;   riii BVKKBBVKBKQQ1*1*1*1*1~  riii BVKKBBVKBK1*1*2*1*2 . 4. Illustrative example. Let the system and local criterion are described by the following matrices and vectors: ;0005,000;00005,0;11,0;003,000~;1025,0105,021  AABAA  ;001,003 A ;00003,0~;0000;01,00001654  AAAA;005,000~2 A  002,00 00~;00001,00~43 AA ;  00 00~~65 AA ; ;1010;1;025,00;0005,0;00654321  zFBBBBBB     ;01;10;02,00 002,0  HSQ 1; 0,2; 1.C D h    Parameters ( )i k  are modeled as Gaussian random variable with mathe-matical expectation is 0 and mean-square deviation is 1. To substantiate of the utility of introduction of the dynamic element ( )k  in control law we modeled four control systems (algorithms): algorithm 1 is con-trol built on the nominal values of the parameters; algorithm 2 is locally-optimal control calculated with regard to random parameters; algorithm 3 is control syn-thesized on the nominal values of the parameters and with the introduction of the dynamic element in the control law; algorithm 4 is locally-optimal control with the dynamic element, calculated taking into account the random parame-ters. In the table the values of quality criterion of the convergence of the state vector x(k) to the reference input z(k) for the four control systems is cited.    Average error Algorithm 1 2 3 4 e  2,325 2,103 1,806 1,638  181 The table shows that minimum average error deviation is reached at 4th algorithm with locally-optimal control with dynamic element, constructed on the transfer coefficients calculated with regard to the random parameters.  Conclusion. The problem of controlling the output of discrete systems with state delays and random parameters based on the synthesis of locally op-timal linear tracking control system of discrete systems with indirect observa-tions and introduction dynamic element in control law has been solved. The asymptotic behavior of the system has been analyzed. It is shown that introduc-tion of the dynamic element in system greatly improves system performance and reduces the sensitivity to disturbing influences. Also it is shown that the optimal dynamic control system, built on random parameters with constant transfer coef-ficients has the property of robustness.  References 1. Bar-Shalom Y., Sivan R. On the optimal control of discrete-time linear systems with random parameters // IEEE Trans. Automatic Control. 1969. Vol. 14, no. 1. P. 3–8. 2. Lee J. H., Cooley B. L. Optimal feedback control strategies for state-space systems with stochastic parameters // IEEE Trans. Automatic Control. 1998. Vol. 43. P. 1469–1474.  3. Dombrovsky V. V., Dombrovsky D. V., Lyashenko E. A. Model predictive control of systems with random dependent parameters under constraints and its application to the in-vestment portfolio optimization // Automation and Remote Control. 2006. Vol. 67, issue 12. P. 1927–1939. 4. Cannon M., Kouvaritakis B., Couchman P. Mean-variance receding horizon control for discrete time linear stochastic systems // Proc. of the 17th World Congress The Interna-tional Federation of Automatic Control. Seoul, Korea, 2008. P. 15321–15326. 5. Luenberger D. G. An introduction to observers // IEEE Trans. Automatic Control. 1971. Vol. 16, issue 6. P. 596–603. 6. Dombrovskii V. V. Synthesis of Reduced-order Dynamic Regulators Under H-in-finity-constraints // Automation and Remote Control. 1996. Vol. 57, issue 11. P. 1537–1543. 7. Mukhina O., Smagin V. Locally Optimal Control for Discrete Time Delay Systems with Interval Parameters / Dudin A., ed. // Communications in Computer and Information Science (CCIS 487). ITMM 2014. 2014. P. 301–311.    ИССЛЕДОВАНИЕ ЭФФЕКТИВНОСТИ ГЕНЕТИЧЕСКОГО АЛГОРИТМА БЕЗУСЛОВНОЙ ОПТИМИЗАЦИИ А. В. Дягель Сибирский федеральный университет, Красноярск, Россия  Оптимизационные задачи заключаются в нахождении экстремума за-данной целевой функции. Как правило, это сложная функция, зависящая от многих переменных (входных параметров). Требуется найти значения этих параметров, при которых целевая функция достигает своего экстремума. Одним из наиболее часто встречающихся на практике типов задач яв-ляются задачи безусловной оптимизации. В задачах такого типа требуется 

Dynamic locally optimal control of discrete state delay systems with random parameters

http://vital.lib.tsu.ru/vital/access/services/Download/vtls:000524300/SOURCE1

Dynamic locally optimal control of discrete state delay systems with random parameters

Abstract

Similar works

Full text

Available Versions

Tomsk State University Repository