Decentralized stabilization of a class of large scale linear interconnected

111 views

Published on

Published in: Technology, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
111
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
1
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Decentralized stabilization of a class of large scale linear interconnected

  1. 1. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 156 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 INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOGY (IJEET) ISSN 0976 – 6545(Print) ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), pp. 156-166 © IAEME: www.iaeme.com/ijeet.asp Journal Impact Factor (2013): 5.5028 (Calculated by GISI) www.jifactor.com IJEET © I A E M E
  2. 2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 157 interconnected system in [10]. But there are two drawbacks in the method. The method requires the solution of ௡೔ ሺ௡೔ ାଵሻ ଶ 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 ሺ݊௜ െ 1ሻ 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]: ‫ܫ‬ ൑ ௌ೔ ఉ೔ ൌ ீ೔ షభ ோഢതതതሺீ೔ షభ ሻ೅ ఉ೔ (1) In [7]: ‫ܫ‬ ൒ ௌ೔ ఉ೔೘ೌೣ ൌ ீ೔ షభ ோ೔ሺீ೔ షభ ሻ೅ ఉ೔೘ೌೣ (2) where the real symmetric positive definite matrices ܴത௜ in (1) and ܴ௜ in (2) are obtained as the solution of the algebraic Riccati equation for the interaction free part of the ݅௧௛ subsystem. ‫ܩ‬௜ 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), ߚ௜ is the minimum Eigen value of ܵ௜. In (2), ߚ௜௠௔௫ 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: ‫ݔ‬పሶ ൌ ‫ܣ‬௜௜‫ݔ‬௜ ൅ ܾ௜‫ݑ‬௜ ൅ ∑ ‫ܣ‬௜௝‫ݔ‬௝ ே ௝ୀଵ,௝ஷ௜ , ݅ ൌ 1, 2, … , ܰ (3) In (3), in the interaction-free part of the ݅௧௛ subsystem, ‫ݔ‬௜ is the ݊௜ ܺ 1 state vector, ‫ݑ‬௜ the scalar control, ‫ܣ‬௜௜ is ݊௜ ܺ ݊௜ coefficient matrix and ܾ௜ is the ݊௜ ܺ 1 driving vector. It is assumed that (‫ܣ‬௜௜, ܾ௜ ) is in companion form. In the interaction terms, ‫ݔ‬௝ is the ݊௝ ܺ 1 state vector, ‫ܣ‬௜௝ are ݊௜ ܺ ݊௝ constant real matrices.
  3. 3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 158 The problem to be studied is the determination of the decentralized (1 x ni) state feedback control gain vector ࢑࢏ for generating the decentralized control: ‫ݑ‬௜ ൌ ࢑࢏‫ݔ‬௜, ݅ ൌ 1, 2, … , ܰ (4) for each of the ݅௧௛ 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 ‫ݑ‬௜ for the interaction free part of the ݅௧௛ subsystem of equation (3) which minimizes the quadratic performance criterion: ‫ܫ‬ ൌ ‫׬‬ ሺ‫ݔ‬௜ ்∞ ଴ ܳ௜‫ݔ‬௜ ൅ ߣ௜‫ݑ‬௜ ଶ ሻ ݀‫ݐ‬ (5) is given by: ‫ݑ‬௜ ൌ ݇௜‫ݔ‬௜, ݅ ൌ 1, 2, … , ܰ with: ݇௜ ൌ െ ଵ ఒ೔ ܾ௜ ் ܴ௜ (6) where ܴ௜ is an ݊௜ ܺ ݊௜ real symmetric positive definite matrix given as the solution of the Algebraic Riccati equation: ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ ൌ ‫ܣ‬௜௜ ் ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ ൅ ܳ௜ (7) In the equations (5), (6) and (7), ߣ௜ is a positive constant and ܳ௜ is an ݊௜ ܺ ݊௜ real symmetric positive definite matrix. It follows that: ‫ݔ‬௜ ் ሺ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ൌ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ (8) Hence for the interconnected subsystems in (3) (proof in Appendix 1): ∑ ‫ݔ‬௜ ் ሺே ௜ୀଵ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ ∑ ሺ2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝ ே ௝ୀଵ,௝ஷ௜ ே ௜ୀଵ ே ௜ୀଵ ‫ݔ‬௝ሻ ൅ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ (9) Interaction terms in (9) can be bounded as (proof in Appendix 2): ∑ ∑ ሺ2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝ ே ௝ୀଵ,௝ஷ௜ ே ௜ୀଵ ‫ݔ‬௝ሻ ൑ ∑ ሺ‫ݔ‬௜ ் ‫ܯ‬௜‫ݔ‬௜ሻே ௜ୀଵ (10) where ‫ܯ‬௜ is an ݊௜ ܺ ݊௜ real positive definite diagonal matrix whose elements depend upon the elements of ܴ௜ and ‫ܣ‬௜௝. The term ‫ݔ‬௜ ் ‫ܯ‬௜‫ݔ‬௜ in the R.H.S. of inequality (10) can be bounded as (proof in Appendix 3): ‫ݔ‬௜ ் ‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜ ் ܴ௜‫ݔ‬௜ (11)
  4. 4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 159 where ݉௜ is a real and positive number given by: ݉௜ ൌ ఈ೔ ଶఉ೔ (12) ߙ௜ is the lowest diagonal element of ‫ܯ‬௜ and ߚ௜ is the highest diagonal element of ݊௜ ܺ ݊௜ real positive definite diagonal matrix ܵ௜ (different from the one in introduction) which is given by: ܵ௜ ൌ ‫ۏ‬ ‫ێ‬ ‫ێ‬ ‫ێ‬ ‫ۍ‬ ∑ ‫ݎ‬௜௪ଵ ௡೔ ௪ୀଵ ‫ڮ‬ 0 ‫ڭ‬ ∑ ‫ݎ‬௜௪ଶ ௡೔ ௪ୀଵ ‫ڰ‬ ‫ڭ‬ 0 ‫ڮ‬ ∑ ‫ݎ‬௜௪௡೔ ௡೔ ௪ୀଵ ‫ے‬ ‫ۑ‬ ‫ۑ‬ ‫ۑ‬ ‫ې‬ Where ‫,ݓ‬ ‫ݒ‬ ൌ 1, 2, . . . , ݊௜ and ‫ݎ‬௜௪௩ is the element in the ‫ݓ‬௧௛ row and ‫ݒ‬௧௛ column of ܴ௜. Using (10) and (11), the equation (9) is converted to the following inequality: ∑ ‫ݔ‬௜ ் ሺே ௜ୀଵ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ழ வ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ 2݉௜‫ݔ‬௜ ் ܴ௜‫ݔ‬௜ ே ௜ୀଵ ே ௜ୀଵ ൅ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ (13) The inequality (13) can be replaced by an equation by replacing ܴ௜ by another ݊௜ ܺ ݊௜ real symmetric positive definite matrix ࡾ࢏: ∑ ‫ݔ‬௜ ் ሺே ௜ୀଵ ଵ ఒ೔ ࡾ࢏ܾ௜ܾ௜ ் ࡾ࢏ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ 2݉௜‫ݔ‬௜ ் ࡾ࢏‫ݔ‬௜ ே ௜ୀଵ ே ௜ୀଵ ൅ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ (14) which is reduced to the following equation (Appendix 4): ଵ ఒ೔ ࡾ࢏ܾ௜ܾ௜ ் ࡾ࢏ ൌ ‫ܣ‬௠௜ ் ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௠௜ ൅ ܳ௜ (15) where: ‫ܣ‬௠௜ ‫؜‬ ‫ܣ‬௜௜ ൅ ݉௜‫ܫ‬ This is the Algebraic Riccati equation for the decoupled subsystems: ‫ݔ‬పሶ ൌ ‫ܣ‬௠௜‫ݔ‬௜ ൅ ܾ௜‫ݑ‬௜ , ݅ ൌ 1, 2, … , ܰ (16) If (16) is compared with (3), it is observed that the effects of interactions have been aggregated as ݉௜‫ܫ‬ into the coefficient matrix of the interaction free part of the ݅௧௛ 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 ‫ܣ‬௠௜ in equation (16) is the modified coefficient matrix of the ݅௧௛ subsystem incorporating the maximum possible interaction effects. Hence the decentralized stabilization of interconnected subsystems in (3) implies the stabilization of decoupled subsystems in (16).
  5. 5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 160 It is noted that ሺ‫ܣ‬௠௜, ܾ௜ሻ in (16) is not in companion form. Hence on applying similarity transformation [8]: ‫ݔ‬పഥ ൌ ܲ௜‫ݔ‬௜ ܲ௜ being ݊௜ ܺ ݊௜ transformation matrix, subsystems (16) are transformed to: ‫ݔ‬ҧሶ௜ ൌ ‫ܣ‬ҧ௠௜‫ݔ‬ҧ௜ ൅ ܾ௜‫ݑ‬௜, ݅ ൌ 1, 2, … , ܰ (17) where ሺ‫ܣ‬ҧ௠௜, ܾ௜ሻ is in companion form. Referring to [9], the optimal control function ‫ݑ‬௜, 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: ‫ݑ‬௜ ൌ െ ଵ ఒ೔ ܾ௜ ் ܴത௜ ‫ݔ‬పഥ ൌ െ ଵ ఒ೔ ܾ௜ ் ܴത௜ ܲ௜‫ݔ‬௜ ֜ ‫ݑ‬௜ ൌ ࢑࢏‫ݔ‬௜, ‫݁ݎ݄݁ݓ‬ ࢑࢏ ൌ െ ଵ ఒ೔ ܾ௜ ் ܴത௜ ܲ௜, ݅ ൌ 1,2, … , ܰ (18) In equation (18), ܴത௜ is the solution of the Algebraic Riccati equation: ଵ ఒ೔ ܴത௜ܾ௜ܾ௜ ் ܴത௜ ൌ ‫ܣ‬ҧ௠௜ ் ܴത௜ ൅ ܴത௜‫ܣ‬ҧ௠௜ ൅ ܳ௜ (19) Hence the decentralized stabilizing control gain vectors ࢑࢏ to generate the controls ‫ݑ‬௜ 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: ‫ݔ‬ሶଵ ൌ ቎ 0 1 0 0 0 0 0 0 0 െ6 1 0 െ11 0 1 െ6 ቏ ‫ݔ‬ଵ ൅ ቎ 0 0 0 1 ቏ ‫ݑ‬ଵ ൅ ቎ 0 0 0 0 0 0 0 0 0 0.05 0 0 0 0 0 0 ቏ ‫ݔ‬ଶ + ቎ 0 0 0 0 0 0 0 0 0 0.04 0 0 0 0 0 0 ቏ ‫ݔ‬ଷ
  6. 6. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 161 ‫ݔ‬ሶଶ ൌ ቎ 0 1 0 0 0 0 0 0 0 0 1 0 െ2 0 1 െ3 ቏ ‫ݔ‬ଶ ൅ ቎ 0 0 0 1 ቏ ‫ݑ‬ଶ ൅ ቎ 0 0 0 0 0 0 0 0 0 0.02 0 0 0 0 0 0 ቏ ‫ݔ‬ଷ + ቎ 0 0 0 0 0 0 0 0 0 0.04 0 0 0 0 0 0 ቏ ‫ݔ‬ଵ ‫ݔ‬ሶଷ ൌ ቎ 0 1 0 0 0 0 0 0 0 6 1 0 െ1 0 1 െ4 ቏ ‫ݔ‬ଷ ൅ ቎ 0 0 0 1 ቏ ‫ݑ‬ଷ ൅ ቎ 0 0 0 0 0 0 0 0 0 0.03 0 0 0 0 0 0 ቏ ‫ݔ‬ଵ + ቎ 0 0 0 0 0 0 0 0 0 0.04 0 0 0 0 0 0 ቏ ‫ݔ‬ଶ With ܳଵ ൌ ܳଶ ൌ ܳଷ ൌ ‫ܫ‬ସ and ߣ ൌ ൅1, on solving the Algebraic Riccati equations corresponding to (7), values of ܴଵ, ܴଶ, ܴଷ are obtained as: ܴଵ ൌ ቎ 7.8202 12.0776 6.2574 1.0000 12.0776 6.2574 1.0000 22.1915 11.9341 1.8202 11.9341 7.7542 1.0776 1.8202 1.0776 0.2574 ቏ , ܴଶ ൌ ቎ 3.5170 5.6847 4.1676 1.0000 5.6847 4.1676 1.0000 15.8253 13.6576 3.5170 13.6576 14.1746 3.6847 3.5170 3.6847 1.1676 ቏, ܴଷ ൌ ቎ 7.8202 12.0776 6.2574 1.0000 12.0776 6.2574 1.0000 94.1915 71.9341 13.8202 71.9341 57.7542 11.0776 13.8202 11.0776 2.2574 ቏ Solving the inequality (10) ‫ܯ‬ଵ , ‫ܯ‬ଶ , ‫ܯ‬ଷ are computed as: ‫ܯ‬ଵ ൌ ቎ 0.09 0 0 0 0 0 0 1.3833 0 0 0 0.0970 0 0 0 0.0232 ቏ , ‫ܯ‬ଶ ൌ ቎ 0.06 0 0 0 0 0 0 1.545 0 0 0 0.2211 0 0 0 0.0701 ቏, ‫ܯ‬ଷ ൌ ቎ 0.07 0 0 0 0 0 0 1.3210 0 0 0 0.7754 0 0 0 0.1580 ቏
  7. 7. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 162 ܵଵ, ܵଶ, ܵଷ are then computed. Hence, one gets: ߙଵ ൌ 0.0232, ߙଶ ൌ 0.06, ߙଷ ൌ 0.07 and ߚଵ ൌ 48.0233, ߚଶ ൌ 38.6846, ߚଷ ൌ 192.0233. Then using (12), ݉ଵ ൌ 2.4155‫ܧ‬ െ 4, ݉ଶ ൌ 7.755‫ܧ‬ െ 4, ݉ଷ ൌ 1.8227‫ܧ‬ െ 4 . Hence the three decoupled subsystems corresponding to (16) are obtained. Then with the transformation matrices: ܲଵ ൌ ቎ 1.0000 0 0 0 0.0002 0 0 1.0000 0.0005 0 0 1.0000 0.0007 0 0 1.0000 ቏ , ܲଶ ൌ ቎ 1.0000 0 0 0 0.0008 0 0 1.0000 0.0016 0 0 1.0000 0.0023 0 0 1.0000 ቏, ܲଷ ൌ ቎ 1.0000 0 0 0 0.0002 0 0 1.0000 0.0004 0 0 1.0000 0.0005 0 0 1.0000 ቏ co-efficient matrices of the transformed decoupled subsystems corresponding to (17) are obtained as: ‫ܣ‬ҧ௠ଵ ൌ ቎ 0 1.0000 0 0 0 0 0.0014 0 0 െ5.9947 1.0000 0 െ10.9957 0 1.0000 െ5.9990 ቏, ‫ܣ‬ҧ௠ଶ ൌ ቎ 0 1.0000 0 0 0 0 െ0.0000 0 0 0.0031 1.0000 0 െ1.9930 0 1.0000 െ2.9969 ቏, ‫ܣ‬ҧ௠ଷ ൌ ቎ 0 1.0000 0 0 0 0 െ0.0011 0 0 6.0004 1.0000 0 െ0.9978 0 1.0000 െ3.9993 ቏ Hence ܴതଵ, ܴതଶ, ܴതଷ are computed by solving (19). Finally, corresponding to (18), the desired decentralized stabilizing controller gain vectors are computed as: ࢑૚ ൌ ሾെ1.0019 െ 1.8236 െ 1.0795 െ 0.2577ሿ, ࢑૛ ൌ ሾെ1.0027 െ 3.5253 െ 3.6925 െ 1.1698ሿ, ࢑૜ ൌ ሾെ1.0014 െ 13.8247 െ 11.0806 െ 2.2580ሿ
  8. 8. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 163 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.
  9. 9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 164 APPENDIX 1 To derive the equation (9) Equation (8) is rewritten: ‫ݔ‬௜ ் ሺ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ൌ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ ൌ ‫ݔ‬௜ ் ‫ܣ‬௜௜ ் ܴ௜‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ ൌ ሺ‫ݔ‬௜ ் ‫ܣ‬௜௜ ் ܴ௜‫ݔ‬௜ሻ் ൅ ‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ (Since ‫ݔ‬௜ ் ‫ܣ‬௜௜ ் ܴ௜‫ݔ‬௜ is a scalar) ൌ ‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ Hence for the interaction free part of the ݅௧௛ subsystem in (3) ‫ݔ‬௜ ் ሺ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ൌ 2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ Thus for the ݅௧௛ linear interconnected subsystem in (3) ‫ݔ‬௜ ் ሺ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ൌ 2‫ݔ‬௜ ் ܴ௜ሺ‫ܣ‬௜௜‫ݔ‬௜ ൅ ∑ ‫ܣ‬௜௝ ே ௝ୀଵ,௝ஷ௜ ‫ݔ‬௝) ൅‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ ൌ 2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௜‫ݔ‬௜ ൅ 2‫ݔ‬௜ ் ܴ௜ሺ∑ ‫ܣ‬௜௝ ே ௝ୀଵ,௝ஷ௜ ‫ݔ‬௝) ൅‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ ֜ ‫ݔ‬௜ ் ሺ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ൌ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ 2‫ݔ‬௜ ் ܴ௜ሺ∑ ‫ܣ‬௜௝ ே ௝ୀଵ,௝ஷ௜ ‫ݔ‬௝) ൅‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ For the N linear interconnected subsystems (3) ∑ ‫ݔ‬௜ ் ሺ ଵ ఒ೔ ܴ௜ܾ௜ܾ௜ ் ܴ௜ሻ‫ݔ‬௜ ே ௜ୀଵ ൌ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ܴ௜ ൅ ܴ௜‫ܣ‬௜௜ሻ‫ݔ‬௜ ே ௜ୀଵ ൅ ∑ ∑ே ௝ୀଵ,௝ஷ௜ ே ௜ୀଵ ሺ2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൅ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ APPENDIX 2 To derive the inequality (10) It is noted that: 2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝‫ݔ‬௝ ൌ 2ሾ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ … ‫ݔ‬௜௡೔ ሿ ൦ ‫ݎ‬௜ଵଵ ‫ݎ‬௜ଶଵ … ‫ݎ‬௜௡೔ଵ ‫ݎ‬௜ଵଶ ‫ݎ‬௜ଶଶ … ‫ݎ‬௜௡೔ଶ ‫ڭ‬ ‫ݎ‬௜ଵ௡೔ ‫ݎ‬௜ଶ௡೔ … ‫ݎ‬௜௡೔௡೔ ൪ ‫ۏ‬ ‫ێ‬ ‫ێ‬ ‫ۍ‬ ܽ௜௝ଵଵ‫ݔ‬௝ଵ ൅ ܽ௜௝ଵଶ‫ݔ‬௝ଶ ൅ ‫ڮ‬ ൅ ܽ௜௝ଵ௡ೕ ‫ݔ‬௝௡ೕ ܽ௜௝ଶଵ‫ݔ‬௝ଵ ൅ ܽ௜௝ଶଶ‫ݔ‬௝ଶ ൅ ‫ڮ‬ ൅ ܽ௜௝ଶ௡ೕ ‫ݔ‬௝௡ೕ ‫ڭ‬ ܽ௜௝௡೔ଵ‫ݔ‬௝ଵ ൅ ܽ௜௝௡೔ଶ‫ݔ‬௝ଶ ൅ ‫ڮ‬ ൅ ܽ௜௝௡೔௡௝‫ݔ‬௝௡௝‫ے‬ ‫ۑ‬ ‫ۑ‬ ‫ې‬ If the multiplications are carried out in the RHS one gets the terms of the form: 2ሾ∑ ሺr୧୵୴a୧୨୵୵′ ሻሿx୧୴ x୨୵′ , i, j ൌ 1,2, … , N; w, v ൌ 1, 2, … … , ݊௜; w’ ൌ 1,2, … , ݊௝. ௡೔ ௪ୀଵ which can be bounded as below : 2ሾ∑ ሺr୧୵୴a୧୨୵୵′ ሻሿx୧୴ x୨୵′ . ௡೔ ௪ୀଵ ≤ ห∑ ௡೔ ௪ୀଵ ൫r୧୵୴a୧୨୵୵′ ൯หሺxଶ ୧୴ ൅ xଶ ୨୵′ሻ where r୧୵୴ represents the ‫,ݓ‬ ‫ݒ‬௧௛ element of the matrix ܴ௜ and a୧୨୵୵′ the w, w′୲୦ element of the matrix ‫ܣ‬௜௝. Using this it is then possible to get the inequality: 2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝‫ݔ‬௝ ൑ ‫ݔ‬௜ ் ‫′ܦ‬௜௜ ‫ݔ‬௜ ൅ ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ where ‫′ܦ‬௜௜ and ‫ܦ‬௜௝ are diagonal matrices whose elements are real non-negative numbers depending on the elements of the matrices ‫ܣ‬௜௝ and ܴ௜. It then follows that the following inequality must hold: ∑ ሺ2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே ௝ୀଵ,௝ஷ௜ ∑ே ௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௜ ் ‫′ܦ‬௜௜ ‫ݔ‬௜ ൅ ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ሻ ൌ ∑ே ௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௜ ் ‫′ܦ‬௜௜ ‫ݔ‬௜ሻ ൅ ∑ே ௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ሻ ൌ ሺܰ െ 1ሻሺ‫ݔ‬௜ ் ‫′ܦ‬௜௜ ‫ݔ‬௜ሻ ൅ ∑ே ௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ሻ
  10. 10. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 165 ൌ ‫ݔ‬௜ ் ሼሺܰ െ 1ሻ‫′ܦ‬௜௜ ሽ‫ݔ‬௜ ൅ ∑ே ௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ሻ ൌ ‫ݔ‬௜ ் ‫ܦ‬௜௜‫ݔ‬௜ ൅ ∑ே ௝ୀଵ,௝ஷ௜ ሺ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ሻ where ‫ܦ‬௜௜ ൌ ሺܰ െ 1ሻ‫′ܦ‬௜௜ ֜ ∑ ሺ2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே ௝ୀଵ,௝ஷ௜ ∑ே ௝ୀଵ ሺ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ሻ Hence: ∑ே ௜ୀଵ ∑ ሺ2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே ௝ୀଵ,௝ஷ௜ ∑ே ௜ୀଵ ∑ே ௝ୀଵ ሺ‫ݔ‬௝ ் ‫ܦ‬௜௝‫ݔ‬௝ሻ ൌ ∑ே ௜ୀଵ ∑ே ௝ୀଵ ሺ‫ݔ‬௜ ் ‫ܦ‬௝௜‫ݔ‬௜ሻ ൌ ∑ ‫ݔ‬௜ ்ே ௜ୀଵ ሺ∑ே ௝ୀଵ ‫ܦ‬௝௜ሻ‫ݔ‬௜ ֜ ∑ே ௜ୀଵ ∑ ሺ2‫ݔ‬௜ ் ܴ௜‫ܣ‬௜௝‫ݔ‬௝ሻ ൑ே ௝ୀଵ,௝ஷ௜ ∑ே ௜ୀଵ ሺ‫ݔ‬௜ ் ‫ܯ‬௜‫ݔ‬௜ሻ where ‫ܯ‬௜ ൌ ∑ே ௝ୀଵ ‫ܦ‬௝௜ is in general a real positive definite ݊௜ ܺ ݊௜ diagonal matrix. APPENDIX 3 In order to prove that ‫ݔ‬௜ ் ‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜ ் ܴ௜‫ݔ‬௜, it is noted that: ‫ݔ‬௜ ் ܴ௜‫ݔ‬௜ ൌ ൣ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ … ‫ݔ‬௜௡೔ ൧ ൦ ‫ݎ‬௜ଵଵ ‫ݎ‬௜ଶଵ … ‫ݎ‬௜௡೔ଵ ‫ݎ‬௜ଵଶ ‫ݎ‬௜ଶଶ … ‫ݎ‬௜௡೔ଶ ‫ڭ‬ ‫ݎ‬௜ଵ௡೔ ‫ݎ‬௜ଶ௡೔ … ‫ݎ‬௜௡೔௡೔ ൪ ൦ ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ ‫ڭ‬ ‫ݔ‬௜௡೔ ൪ ൌ ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ‫ݎ‬௜௪௩‫ݔ‬௜௪ ‫ݔ‬௜௩ ≤ ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ‫ݎ‬௜௪௩ሺ‫ݔ‬ଶ ௜௪൅ ‫ݔ‬ଶ ௜௩ ሻ/2 = ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ሺ‫ݎ‬௜௪௩‫ݔ‬ଶ ௜௪ሻ/2 ൅ ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ሺ ‫ݎ‬௜௪௩‫ݔ‬ଶ ௜௩ ሻ/2 = ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ሺ‫ݎ‬௜௩௪‫ݔ‬ଶ ௜௩ሻ/2 ൅ ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ሺ ‫ݎ‬௜௪௩‫ݔ‬ଶ ௜௩ ሻ/2 Since ܴ௜ is a real symmetric positive definite matrix: ‫ݎ‬௜௩௪ ൌ ‫ݎ‬௜௪௩, ‫,ݓ‬ ‫ݒ‬ ൌ 1, 2, … , ݊௜ ֜ ‫ݔ‬௜ ் ܴ௜‫ݔ‬௜ ൑ ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ሺ‫ݎ‬௜௪௩‫ݔ‬ଶ ௜௩ሻ/2 ൅ ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ሺ ‫ݎ‬௜௪௩‫ݔ‬ଶ ௜௩ ሻ/2 ൌ ∑௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ሺ‫ݎ‬௜௪௩‫ݔ‬ଶ ௜௩ሻ ൌ ∑ ሺ ௡೔ ௩ୀଵ ∑௡೔ ௪ୀଵ ‫ݎ‬௜௪௩ሻ‫ݔ‬ଶ ௜௩ ൌ ∑௡೔ ௩ୀଵ ‫݌‬௜௩‫ݔ‬ଶ ௜௩ , (where ‫݌‬௜௩ ൌ ∑௡೔ ௪ୀଵ ‫ݎ‬௜௪௩) ൌ ൣ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ … ‫ݔ‬௜௡೔ ൧ ൦ ‫݌‬௜ଵ 0 … 0 0 ‫݌‬௜ଶ 0 ‫ڭ‬ 0 ‫ڰ‬ … ‫ڭ‬ ‫݌‬௜௡௜ ൪ ൦ ‫ݔ‬௜ଵ ‫ݔ‬௜ଶ ‫ڭ‬ ‫ݔ‬௜௡೔ ൪ = ‫ݔ‬௜ ் ܵ௜‫ݔ‬௜ where: ܵ௜ ൌ ൦ ‫݌‬௜ଵ 0 … 0 0 ‫݌‬௜ଶ 0 ‫ڭ‬ 0 ‫ڰ‬ … ‫ڭ‬ ‫݌‬௜௡௜ ൪ ൌ ‫ۏ‬ ‫ێ‬ ‫ێ‬ ‫ۍ‬ ∑ r୧୵ଵ ୬୧ ୵ୀଵ ‫ڮ‬ 0 ‫ڭ‬ ‫ڭ‬ ∑ r୧୵ଶ ୬୧ ୵ୀଵ ‫ڰ‬ ‫ڭ‬ ‫ڭ‬ 0 … ∑ r୧୵୬୧ ୬୧ ୵ୀଵ ‫ے‬ ‫ۑ‬ ‫ۑ‬ ‫ې‬ ֜ ‫ݔ‬௜ ் ܵ௜‫ݔ‬௜ ൒ ‫ݔ‬௜ ் ܴ௜‫ݔ‬௜ It is noted that Si is a real positive definite ݊௜ ܺ ݊௜ diagonal matrix. ֜ ‫ܫ‬ ൒ ଵ ఉ೔ ܵ௜ where ߚ௜ is the highest diagonal element of Si. Now referring to Appendix 2, since Mi also is a real positive definite diagonal matrix, ‫ܯ‬௜ ൒ ߙ௜‫,ܫ‬ where ߙ௜ is the lowest diagonal element of ‫ܯ‬௜
  11. 11. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 3, May - June (2013), © IAEME 166 ֜ ‫ܯ‬௜ ൒ ఈ೔ ఉ೔ ܵ௜ ֜ ‫ܯ‬௜ ൒2݉௜ܵ௜, where ݉௜ ൌ ఈ೔ ଶఉ೔ is a real and positive number ֜ ‫ݔ‬௜ ் ‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜ ் ܵ௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜ ் ܴ௜‫ݔ‬௜ ֜ ‫ݔ‬௜ ் ‫ܯ‬௜‫ݔ‬௜ ൒ 2݉௜‫ݔ‬௜ ் ܴ௜‫ݔ‬௜ APPENDIX 4 The equation (14) ∑ ‫ݔ‬௜ ் ሺே ௜ୀଵ ଵ ఒ೔ ࡾ࢏ܾ௜ܾ௜ ் ࡾ࢏ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜ ் ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௜௜ሻ‫ݔ‬௜ ൅ ∑ 2݉௜‫ݔ‬௜ ் ࡾ࢏‫ݔ‬௜ ே ௜ୀଵ ே ௜ୀଵ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ ൌ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௜௜൅݉௜‫ܫ‬ሻ் ࡾ࢏‫ݔ‬௜ ே ௜ୀଵ ൅ ‫ݔ‬௜ ் ࡾ࢏ሺ‫ܣ‬௜௜൅݉௜‫ܫ‬ሻ‫ݔ‬௜ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ ൌ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௠௜ ் ࡾ࢏ ே ௜ୀଵ ൅ ࡾ࢏‫ܣ‬௠௜ሻ‫ݔ‬௜ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ where ‫ܣ‬௠௜ ‫؜‬ ‫ܣ‬௜௜൅݉௜‫ܫ‬ ֜ ∑ ‫ݔ‬௜ ் ሺே ௜ୀଵ ଵ ఒ೔ ࡾ࢏ܾ௜ܾ௜ ் ࡾ࢏ሻ‫ݔ‬௜ ൌ ∑ ‫ݔ‬௜ ் ሺ‫ܣ‬௠௜ ் ࡾ࢏ ே ௜ୀଵ ൅ ࡾ࢏‫ܣ‬௠௜ሻ‫ݔ‬௜ ൅ ∑ ሺ‫ݔ‬௜ ் ܳ௜‫ݔ‬௜ሻே ௜ୀଵ ֜ ଵ ఒ೔ ࡾ࢏ܾ௜ܾ௜ ் ࡾ࢏ ൌ ‫ܣ‬௠௜ ் ࡾ࢏ ൅ ࡾ࢏‫ܣ‬௠௜ ൅ ܳ௜ which is the Algebraic Riccati equation for the decoupled subsystems: ‫ݔ‬పሶ ൌ ‫ܣ‬௠௜‫ݔ‬௜ ൅ ܾ௜‫ݑ‬௜ , ݅ ൌ 1, 2, … , ܰ where ‫ܣ‬௠௜ ‫؜‬ ‫ܣ‬௜௜ ൅ ݉௜‫ܫ‬

×