6545(Print), ISSN (Online) Volume 4, Issue 3, May - June (2013), IAEME & TECHNOLOGY (IJEET)

Transcription

1 INTERNATIONAL International Journal of JOURNAL Electrical Engineering OF ELECTRICAL and Technology (IJEET), ENGINEERING ISSN 976 & TECHNOLOGY (IJEET) ISSN (Print) ISSN (Online) Volume 4, Issue 3, May - June (213), pp IAEME: Journal Impact Factor (213): (Calculated by GISI) IJEET I A E M E DECENTRALIZED STABILIZATION OF A CLASS OF LARGE SCALE LINEAR INTERCONNECTED SYSTEM BY OPTIMAL CONTROL Ranjana Kumari 1, Ramanand Singh 2 1 (Department of Electrical Engineering, Bhagalpur College of Engineering, P.O. Sabour, Bhagalpur-81321, Bihar, India) 2 (Department of Electrical Engineering (Retired Professor), Bhagalpur College of Engineering, P.O. Sabour, Bhagalpur-81321, 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 156

2 interconnected system in [1]. 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 1 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]: In [7]: (1) (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:,, 1, 2,, (3) In (3), in the interaction-free part of the subsystem, is the 1 state vector, the scalar control, is coefficient matrix and is the 1 driving vector. It is assumed that (, ) is in companion form. In the interaction terms, is the 1 state vector, are constant real matrices. 157

3 The problem to be studied is the determination of the decentralized (1 x n i ) state feedback control gain vector for generating the decentralized control:, 1, 2,, (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:, 1, 2,, 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):,2 (9) Interaction terms in (9) can be bounded as (proof in Appendix 2):,2 (1) 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 (1) can be bounded as (proof in Appendix 3): 2 (11) 158

4 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, 1, 2,..., and is the element in the row and column of. Using (1) and (11), the equation (9) is converted to the following inequality: 2 (13) The inequality (13) can be replaced by an equation by replacing by another real symmetric positive definite matrix : which is reduced to the following equation (Appendix 4): 2 (14) where: (15) This is the Algebraic Riccati equation for the decoupled subsystems:, 1, 2,, (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). 159

5 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:, 1, 2,, (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:,, 1,2,, (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:

6 With and 1, on solving the Algebraic Riccati equations corresponding to (7), values of,, are obtained as: , , Solving the inequality (1),, are computed as: , ,

7 ,, are then computed. Hence, one gets:.232,.6,.7 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: , ,

8 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, 198, [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-1, No. 4, April 1981, [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, [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, [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 23, [6] R Singh, Optimal feedback control of Linear Time-Invariant Systems with Quadratic criterion, Institution of Engineers, Vol. 51, September 197, [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 2, Books [8] B C Kuo, Automatic Control Systems, (PHI, 6 th Edition, 1993), [9] D G Schultz and J L Melsa, State Functions and Linear Control Systems, (McGraw Hill Book Company Inc, 1967). Proceedings Papers [1] A K Mahalanabis and R Singh, On the stability of Interconnected Stochastic Systems, 8 th IFAC World Congress, Kyoto, Japan, 1981, No

9 APPENDIX 1 To derive the equation (9) Equation (8) is rewritten: Hence for the interaction free part of the subsystem in (3) 2 Thus for the linear interconnected subsystem in (3) 2 ),, (Since is a scalar) 2 2 ) 2 ), For the N linear interconnected subsystems (3), 2 To derive the inequality (1) It is noted that: 2 2 APPENDIX 2 If the multiplications are carried out in the RHS one gets the terms of the form: 2 r a x x, i, j 1,2,, N; w, v 1, 2,, ; w 1,2,,. which can be bounded as below : 2 r a x x. r a x x where r represents the, element of the matrix and a the w, w element of the matrix. Using this it is then possible to get the inequality: 2 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: 2,,,, 1 164,

10 1,, where 1 2, Hence: 2, 2, where is in general a real positive definite diagonal matrix. APPENDIX 3 In order to prove that 2, it is noted that: /2 = /2 /2 = /2 /2 Since is a real symmetric positive definite matrix:,, 1, 2,, /2 /2, (where ) = r where: r r It is noted that S i is a real positive definite diagonal matrix. where is the highest diagonal element of S i. Now referring to Appendix 2, since M i also is a real positive definite diagonal matrix,, where is the lowest diagonal element of 165

11 The equation (14) 2, where is a real and positive number APPENDIX 4 2 where which is the Algebraic Riccati equation for the decoupled subsystems:, 1, 2,, where 166