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DECENTRALIZED STABILIZATION OF A CLASS OF LARGE SCALE LINEAR INTERCONNECTED

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					INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING
 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME
                            & TECHNOLOGY (IJEET)

ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online)
                                                                               IJEET
Volume 4, Issue 3, May - June (2013), pp. 156-166
© IAEME: www.iaeme.com/ijeet.asp                                            ©IAEME
Journal Impact Factor (2013): 5.5028 (Calculated by GISI)
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      DECENTRALIZED STABILIZATION OF A CLASS OF LARGE SCALE
       LINEAR INTERCONNECTED SYSTEM BY OPTIMAL CONTROL

                                 Ranjana Kumari1, Ramanand Singh2
       1
           (Department of Electrical Engineering, Bhagalpur College of Engineering, P.O. Sabour,
                                      Bhagalpur-813210, Bihar, India)
             2
               (Department of Electrical Engineering (Retired Professor), Bhagalpur College of
                        Engineering, P.O. Sabour, Bhagalpur-813210, Bihar, India)



  ABSTRACT

          A major gap in literature in the aggregation procedure based on Algebraic Riccati
  equations when interaction terms in each subsystem of a linear interconnected system are
  aggregated with the co-efficient matrix has been removed by suggesting an alternative simple
  aggregation procedure. Optimal controls generated from the solution of the Algebraic Riccati
  equations for the resulting decoupled subsystem are the desired decentralized controls which
  guarantee the stability of the composite system with nearly optimal response and minimum
  cost of control energy. The procedure has been illustrated numerically.

  Keywords: Aggregation, decentralized, decoupled, optimal control.

 I.           INTRODUCTION

          Decentralized stabilization of large scale linear, bilinear, non-linear and stochastic
  interconnected systems etc. have been studied by various methods. The aim of the present
  work is to continue the further study of a computationally simple method in which the
  interaction terms of each subsystem is aggregated with the state matrix resulting in complete
  decoupling of the subsystems so that the decentralized stabilizing feedback control gain
  coefficients can be computed very easily. Two methods of aforesaid aggregation have been
  reported in the literature. The first method based on Liapunov function has been studied in [1]
  in which the basic methodology has been developed for linear interconnected system and the
  same has been extended for non-linear interconnected system in [2] and stochastic

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  interconnected system in [10]. But there are two drawbacks in the method. The method
  requires the solution of           linear algebraic equations for generating Liapunov function
  for the interaction free part of each subsystem of order . Secondly, if the interaction free
  parts are unstable, these have to be first stabilized by local controls. After a long gap, these
  two drawbacks were removed in [3] where the basic methodology for linear interconnected
  system was developed and the same was extended for stochastic bilinear interconnected
  system in [4] and stochastic linear interconnected system in [5]. This aggregation method is
  based on Algebraic Riccati equation for the interaction free part and requires the solution of
  only            non-linear algebraic equations for the interaction free part of the subsystem of
  order by using the simple method in [6]. The method was further improved in [7].
          However, after another long gap, the present authors have observed very recently that
  in the aggregation procedure of the above authors, there is a major gap affecting the results.
  This is illustrated as follows. In the aggregation procedure, the authors have considered the
  following inequalities:

         In [3]:
                                                                                           (1)
         In [7]:
                                                                                           (2)

  where the real symmetric positive definite matrices      in (1) and in (2) are obtained as the
  solution of the algebraic Riccati equation for the interaction free part of the    subsystem.
  results from the majorisation of the interaction terms and is a real positive definite diagonal
  matrix. It follows that is a real symmetric positive definite matrix whose all elements are
  positive. Hence the Eigen values of       are all real and positive. In (1),    is the minimum
  Eigen value of . In (2),          is the maximum Eigen value of . The inequalities (1) and
  (2) cannot be true since is not a diagonal matrix and its elements are all positive.
          Hence in order to remove the aforesaid major gap, the authors have suggested a
  different aggregation procedure which is simpler as well as is clear on comparison. The basic
  methodology has been developed for large scale linear interconnected system and can be
  extended for bilinear, non-linear and stochastic interconnected systems etc. The results have
  been illustrated through a numerical example.

II.      PROBLEM FORMULATION

         As in [1], Large Scale time-invariant systems are considered with linear
  interconnection:

                                                                                            (3)

                              In (3), in the interaction-free part of the   subsystem, is the
         state vector, the scalar control,        is        coefficient matrix and     is the
  driving vector. It is assumed that ( ,      ) is in companion form. In the interaction terms,
  is the        state vector,    are          constant real matrices.


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          The problem to be studied is the determination of the decentralized (1 x ni) state
   feedback control gain vector for generating the decentralized control:

                                                                                            (4)

   for each of the   subsystem of equation (3) such that the composite system is stabilized with
   optimal response and minimum cost of control energy.

III.       AGGREGATION-DECOMPOSITION AND DECOUPLED SUBSYSTEMS

          It is known that the optimal feedback control for the interaction free part of the
   subsystem of equation (3) which minimizes the quadratic performance criterion:
                                     ∞
                                                                                           (5)

   is given by:

   with:                                                                                    (6)

   where    is an         real symmetric positive definite matrix given as the solution of the
   Algebraic Riccati equation:

                                                                                                 (7)

   In the equations (5), (6) and (7), is a positive constant and   is an        real symmetric
   positive definite matrix. It follows that:

                                                                                                 (8)

   Hence for the interconnected subsystems in (3) (proof in Appendix 1):



                                                                                                 (9)

   Interaction terms in (9) can be bounded as (proof in Appendix 2):

                                                                                             (10)

   where     is an         real positive definite diagonal matrix whose elements depend upon the
   elements of     and    . The term            in the R.H.S. of inequality (10) can be bounded as
   (proof in Appendix 3):

                                                                                             (11)


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  where     is a real and positive number given by:
                                                                                            (12)

     is the lowest diagonal element of    and is the highest diagonal element of           real
  positive definite diagonal matrix (different from the one in introduction) which is given by:




  Where                      and       is the element in the         row and      column of      .
  Using (10) and (11), the equation (9) is converted to the following inequality:



                                                                                            (13)

  The inequality (13) can be replaced by an equation by replacing        by another           real
  symmetric positive definite matrix :



                                                                                            (14)

  which is reduced to the following equation (Appendix 4):

                                                                                            (15)
  where:

  This is the Algebraic Riccati equation for the decoupled subsystems:

                                                                                            (16)

          If (16) is compared with (3), it is observed that the effects of interactions have been
  aggregated as       into the coefficient matrix of the interaction free part of the   subsystem
  so that the N interconnected subsystems have been decomposed into the N decoupled
  subsystems of (16). The positive number               is, therefore, designated as interaction
  coefficient and the procedure is called aggregation-decomposition.

IV.        DECENTRALIZED STABILIZATION BY OPTIMAL FEEDBACK CONTROL

               in equation (16) is the modified coefficient matrix of the            subsystem
  incorporating the maximum possible interaction effects. Hence the decentralized stabilization
  of interconnected subsystems in (3) implies the stabilization of decoupled subsystems in (16).


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 It is noted that            in (16) is not in companion form. Hence on applying similarity
 transformation [8]:


     being           transformation matrix, subsystems (16) are transformed to:

                                                                                          (17)

 where            is in companion form. Referring to [9], the optimal control function , which
 minimizes the quadratic performance criteria so that the subsystems (17) and hence (16) are
 stabilized with optimal response and minimum cost of control energy, is given by:




                                                                                         (18)

 In equation (18),     is the solution of the Algebraic Riccati equation:

                                                                                         (19)

        Hence the decentralized stabilizing control gain vectors to generate the controls
 as per equation (4), which guarantees the stability of the interconnected subsystems (3), are
 optimal control gain vectors for the decoupled subsystems (16) and are given by (18).
 Response will be slightly deviated from the optimal and cost of control energy slightly higher
 than minimum due to majorization.

V.      NUMERICAL EXAMPLE

         The class of linear interconnected system consisting of three subsystems each of
 fourth order as in [3], [7] is considered corresponding to (3) as follows:



                                                                            +




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                                                                   +




                                                                   +




       With                       and       , on solving the Algebraic Riccati equations
corresponding to (7), values of         are obtained as:




       Solving the inequality (10)           are computed as:




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         are then computed. Hence, one gets:                                   and
                                                  . Then using (12),
                                            . Hence the three decoupled subsystems
corresponding to (16) are obtained. Then with the transformation matrices:




co-efficient matrices of the transformed decoupled subsystems corresponding to (17) are
obtained as:




       Hence             are computed by solving (19). Finally, corresponding to (18), the
desired decentralized stabilizing controller gain vectors are computed as:




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VI.      CONCLUSION

          A computationally simpler aggregation procedure has been suggested to aggregate the
  interaction terms with the co-efficient matrix of each subsystem of a large scale linear
  interconnected system. Moreover, the major gap in the aggregation procedure in the
  concerned literature has been removed. The desired decentralized feedback co-efficients
  generated from the solution of the Algebraic Riccati equation for the resulting decoupled
  subsystems will guarantee the stability of the composite system with optimal response and
  minimum cost of control energy with slight deviation due to majorization of the interaction
  terms.
          The results of this paper can be easily extended for bilinear, non-linear and stochastic
  interconnected systems and even for time varying, uncertain and robust control
  interconnected systems and for the case with output feedback and pole-placement. The results
  can also be applied for improvement of dynamic and transient stability of multi-machine
  power systems etc. All the above cases will be reported in the literature by the present authors
  in due course. The procedure of the paper can be computerized and hence is applicable for
  higher order systems.

  REFERENCES

  Journal Papers
  [1] A K Mahalanabis and R Singh, On decentralized feedback stabilization of large-scale
       interconnected systems, International Journal of Control, Vol. 32, No. 1, 1980, 115-126.
  [2] A K Mahalanabis and R Singh, On the analysis and improvement of the transient
       stability of multi-machine power systems, IEEE Transactions on Power Apparatus and
       Systems, Vol. PAS-100, No. 4, April 1981, 1574-80.
  [3] K Patralekh and R Singh, Stabilization of a class of large scale linear system by
       suboptimal decentralized feedback control, Institution of Engineers, Vol. 78, September
       1997, 28-33.
  [4] K Patralekh and R Singh, Stabilization of a class of stochastic bilinear interconnected
       system by suboptimal decentralized feedback controls, Sadhana, Vol. 24, Part 3, June
       1999, 245-258.
  [5] K Patralekh and R Singh, Stabilization of a class of stochastic linear interconnected
       system by suboptimal decentralized feedback controls, Institution of Engineers, Vol. 84,
       July 2003, 33-37.
  [6] R Singh, Optimal feedback control of Linear Time-Invariant Systems with Quadratic
       criterion, Institution of Engineers, Vol. 51, September 1970, 52-55.
  [7] B C Jha, K Patralekh and R Singh, Decentralized stabilizing controllers for a class of
       large-scale linear systems, Sadhana, Vol. 25, Part 6, December 2000, 619-630.
  Books
  [8] B C Kuo, Automatic Control Systems, (PHI, 6th Edition, 1993), 222-225.
  [9] D G Schultz and J L Melsa, State Functions and Linear Control Systems, (McGraw Hill
       Book Company Inc, 1967).
  Proceedings Papers
  [10] A K Mahalanabis and R Singh, On the stability of Interconnected Stochastic Systems,
       8th IFAC World Congress, Kyoto, Japan, 1981, No. 248.



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                                         APPENDIX 1

To derive the equation (9)
Equation (8) is rewritten:



                                                                   (Since            is a scalar)

Hence for the interaction free part of the    subsystem in (3)


Thus for the     linear interconnected subsystem in (3)
                                                      )
                                                               )
                                                                            )
For the N linear interconnected subsystems (3)




                                         APPENDIX 2

To derive the inequality (10)
It is noted that:




If the multiplications are carried out in the RHS one gets the terms of the form:
                    ′          ′                                               ’
which can be bounded as below :
                       ′          ′ ≤                    ′               ′
                                                                                 ′
where       represents the           element of the matrix and         ′ the         element of
the matrix     . Using this it is then possible to get the inequality:
                         ′
where ′ and           are diagonal matrices whose elements are real non-negative numbers
depending on the elements of the matrices          and . It then follows that the following
inequality must hold:
                                                       ′
                                                    ′
                                                            ′

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                                                  ′

where                     ′

Hence:




where                     is in general a real positive definite              diagonal matrix.


                                            APPENDIX 3

    In order to prove that                              , it is noted that:




                ≤
                =
                =
Since     is a real symmetric positive definite matrix:




                                        ,    (where                       )




                   =


where:



It is noted that Si is a real positive definite         diagonal matrix.


where is the highest diagonal element of Si. Now referring to Appendix 2, since Mi also is
a real positive definite diagonal matrix,
                             , where is the lowest diagonal element of


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                      2     , where          is a real and positive number




                                      APPENDIX 4

The equation (14)




   where



   which is the Algebraic Riccati equation for the decoupled subsystems:
                                        where




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