OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN FOR THE STABILIZATION OF LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS

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1 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN FOR THE STABILIZATION OF LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS Sundarapandian Vaidyanathan 1 and Kavitha Madhavan 2 1 Reearch and Development Centre, Vel Tech Dr. RR & Dr. SR Technical Univerity Avadi, Chennai , Tamil Nadu, INDIA undarvtu@gmail.com 2 Department o Mathematic, Vel Tech Dr. RR & Dr. SR Technical Univerity Avadi, Chennai , Tamil Nadu, INDIA kavi78_m@yahoo.com ABSTRACT Thi paper dicue the deign o oberver-baed reduced order controller or the tabilization o large cale linear dicrete-time control ytem. Thi deign i carried out via deriving a reduced-order model or the given linear plant uing the dominant tate o the linear plant. Uing thi reduced-order linear model, uicient condition are derived or the deign o oberver-baed reduced order controller. A eparation principle ha been etablihed in thi paper which demontrate that the oberver pole and controller pole can be eparated and hence the pole-placement problem and oberver deign are independent o each other. KEYWORDS Dominant State, Reduced-Order Model, Stabilization, Oberver, Dicrete-Time Linear Sytem. 1. INTRODUCTION The reduced order controller deign and oberver deign are active reearch problem in the linear ytem literature and there ha been a igniicant attention paid in the literature on thee two problem during the pat our decade [1-10]. In the recent decade, there ha been a coniderable attention paid to the control problem o large cale linear ytem. Thi i mainly owing to the practical and technical iue like data acquiition, ening, computing acilitie, inormation traner network and the cot aociated with them, which arie rom the ue o ull order controller deign. Thu, there i a great demand in the indutry or the control o large cale linear ytem with the ue o reduced-order controller rather than ull-order controller. A recent approach or obtaining the reduced-order controller i via the reduced-order model o a linear plant preerving the dynamic a well a tatic propertie o the ytem and then deviing controller or the reduced-order model thu obtained [4-8]. Thi approach ha practical and DOI : /ijceit

2 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 technical beneit or the reduced-order controller deign or large-cale linear ytem with high dimenion and compleity. The oberver-baed reduced-order controller deign i motivated by the act that the dominant tate o the linear plant may not be available or meaurement and hence or implementing the pole placement law, only the reduced order eponential oberver can be ued in lieu o the dominant tate o the given dicrete-time linear ytem. In thi paper, we derive a reduced-order model or any linear dicrete-time control ytem and our approach i baed on the approach o uing the dominant tate o the given linear dicrete-time control ytem, i.e. we derive the reduced-order model or a given dicrete-time linear control ytem keeping only the dominant tate o the given dicrete-time linear control ytem. Uing the reduced-order model obtained, we characterize the eitence o a reduced-order eponential oberver that track the tate o the reduced-order model, i.e. the dominant tate o the original linear plant. We note that the reduced-order oberver deign detailed in thi paper i a dicretetime analogue o the reult o Aldeen and Trinh [8] or the oberver deign o the dominant tate o continuou-time linear control ytem. Uing the reduced-order model o the original linear plant, we alo characterize the eitence o a tabilizing eedback control law that ue only the dominant tate o the original plant. Alo, when the plant i tabilizable by a tate eedback control law, the ull inormation o the dominant tate may not be alway available. For thi reaon, we etablih a eparation principle o that the tate o the eponential oberver may be ued in lieu o the dominant tate o the original linear plant, which acilitate the implementation o the tabilizing eedback control law derived. The deign o oberver-baed reduced order controller or large cale dicrete-time linear ytem derived in thi paper ha important practical, indutrial application. Thi paper i organized a ollow. In Section 2, we decribe the contruction o the reducedorder plant o a given linear dicrete-time control ytem. In Section 3, we derive neceary and uicient condition or the eponential oberver deign or the reduced-order linear control plant. In Section 4, we derive neceary and uicient condition or the reduced-order plant to be tabilizable by a linear eedback control law and we alo preent a eparation principle or the oberver-baed reduced order controller deign. In Section 5, we decribe a numerical eample. In Section 6, we ummarize the main reult derived in thi paper. 2. REDUCED ORDER MODEL FOR DISCRETE-TIME LINEAR SYSTEMS In thi ection, we conider a large cale linear dicrete-time control ytem S 1 given by ( k + 1 = A + Bu y = C (1 n m p where R i the tate, u R i the control or input and y R i the ytem output. We aume that A, B and C are contant matrice with real entrie having dimenion n n, n m p n repectively. Firt, we uppoe that we have made an identiication o the dominant (low and nondominant (at tate o the original linear ytem (1 uing the modal approach a decribed in [9]. Without lo o generality, we may aume that 86

3 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 =, where R tate o the ytem. r repreent the dominant tate and Then the linear ytem (1 become ( k + 1 A A B u ( k ( k 1 = A A + B + y = C C From (2, we can write the plant equation a ( k + 1 = A + A + B u ( k + 1 = A + A + B u y = C + C n r R repreent the non-dominant (2 (3 Net, we hall aume that the matri A ha a et o n linearly independent eigenvector. In mot practical ituation, the matri A ha ditinct eigenvalue and thi condition i immediately atiied. Thu, it ollow that A i diagonalizable. Thu, we can ind a non-ingular (modal matri P coniting o n linearly independent eigenvector o A uch that P AP = Λ, where Λ i a diagonal matri coniting o the n eigenvalue o A. Now, we introduce a new et o coordinate on the tate pace given by ξ = P (4 In the new coordinate, the plant (1 become ξ k + = Λ ξ k + P Bu k Thu, we have ( 1 y = CPξ 87

4 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 ξ ( k + 1 Λ 0 ξ ξ ( k 1 = 0 ξ + + Λ ξ y = CP ξ P Bu k (5 where Λ and Λ are r r and ( n r ( n r diagonal matrice repectively, coniting o the dominant and non-dominant eigenvalue o A. Deine matrice Γ, Γ, Ψ and Ψ by Γ P B = Γ and CP = Ψ Ψ (6 where Γ, Γ, Ψ and Ψ are r m, ( n r m, p r and p ( n r matrice repectively. From (5 and (6, we ee that the plant (3 ha the ollowing imple orm in the new coordinate (4. ξ ( k + 1 = Λ ξ + Γ u ξ ( k + 1 = Λ ξ + Γ u y = Ψ ξ + Ψ ξ (7 Net, we make the ollowing aumption: (H1 A k, ξ ( k + 1 ξ, i.e. ξ take a contant value in the teady-tate. (H2 The matri I Λ i invertible. Then it ollow rom (7 that or large value o k, we have i.e. ξ Λ ξ + Γ u ξ k I Λ Γ u k (8 ( Subtituting (8 into (7, we obtain the reduced-order model o the linear plant (1 in the ξ coordinate a ξ ( k + 1 = Λ ξ + Γ u y k = Ψ ξ k + Ψ I Λ Γ u k ( (9 88

5 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 To obtain the reduced-order model o the linear plant (1 in the coordinate, we proceed a ollow. By the linear change o coordinate (4, it ollow that ξ = = Thu, we have P Q. ξ Q Q Q ξ = = Q Q (10 Uing (9 and (10, it ollow that ξ k Q ξ k Q ξ k I u k = + = ( Λ Γ or Q k = Q k + I Λ Γ u k (11 ξ ( Net, we aume the ollowing: (H3 The matri Q i invertible. Uing the aumption (H3, the equation (11 become k = Q Q k + Q I Λ Γ u k (12 ( To impliy the notation, we deine the matrice R = Q Q and 1 S = Q ( I Λ Γ (13 Uing (13, the equation (12 can be impliied a = R + Su (14 Subtituting (14 into (3, we obtain the reduced-order model S2 o the given linear ytem S1 a ( k + 1 = A + B u y = C + D u (15 where the matrice A, B, C and D are deined by 89

6 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 A = A + A R B = B + A S C = C + C R D = C S (16 3. REDUCED ORDER OBSERVER DESIGN In thi ection, we tate a new reult that precribe a imple procedure or etimating the dominant tate o the given linear control ytem S1 that atiie the aumption (H1-(H3 decribed in Section 2. Theorem 1. Let S 1 be the linear ytem decribed by ( k + 1 = A + Bu y = C (17 Under the aumption (H1-(H3, the reduced-order model S2 o the linear ytem S1 can be obtained (ee Section 2 a ( k + 1 = A + B u y = C + D u (18 where A, B, C and D are deined a in (16. To etimate the dominant tate by o the ytem S, conider the candidate oberver 1 S3 deined z ( k + 1 = A z + B u + K y C z D u (19 Let the etimation error be deined a e = z Then e 0 eponentially a k i and only i the matri K i uch that = i convergent. I ( C, A E A K C i obervable, then we can alway contruct an eponential oberver o the orm (19 having any deired peed o convergence. Proo. From Eq. (2, we have ( k + 1 = A + A + B u (20 Adding and ubtracting the term ( A K C R in the RHS o Eq. (20, we get 90

7 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 ( k + 1 = ( A + A R K C + A ( A K C R + B u (21 Subtracting (21 rom (19 and impliying uing the deinition (16, we get e( k + 1 = ( A K C e ( A K C [ R Su] (22 A proved in Section 2, the aumption (H1-(H3 yield R + Su (23 Subtituting (23 into (22, we get which yield e( k + 1 = ( A K C e = Ee (24 k k e = ( A K C e(0 = E e(0 (25 From Eq. (25, it ollow that e 0a k i and only i E i convergent. I ( C, A i obervable, then it i well-known [10] that we can ind an oberver gain matri that will arbitrarily aign the eigenvalue o the error matri E = A KC. In particular, it ollow rom Eq. (25 that we can ind an eponential oberver o the orm (19 having any deired peed o convergence. 4. OBSERVER-BASED REDUCED ORDER CONTROLLER DESIGN In thi ection, we irt tate an important reult that precribe a imple procedure or tabilizing the dominant tate o the reduced-order linear plant derived in Section 2. Theorem 2. Suppoe that the aumption (H1-(H3 hold. Let S1 and S2 be deined a in Theorem 1. For the reduced-order model S, 2 the tate eedback control law u = F (26 K tabilize the dominant tate convergence. o the reduced-order model S2 having any deired peed o In practical application, the dominant tate o the reduced-order model S2 may not be directly available or meaurement and hence we cannot implement the tate eedback control law (26. To overcome thi practical diiculty, we derive an important theorem, uually called a the Separation Principle, which irt etablihe that the oberver-baed reduced-order controller indeed tabilize the dominant tate o the given linear control ytem S1 and alo demontrate that the oberver pole and the cloed-loop controller pole can be eparated. 91

8 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 Theorem 3 (Separation Principle. Suppoe that the aumption (H1-(H3 hold. Suppoe that there eit matrice F and K uch that A B F and A KC are both convergent matrice. By Theorem 1, we know that the ytem S3 deined by (19 i an eponential oberver or the dominant tate o the control ytem S. 1 Then the oberver pole and the cloed-loop controller pole are eparated and the control law u = F z (27 alo tabilize the dominant tate o the control ytem S. 1 Proo. Under the eedback control law (27, the oberver dynamic (19 o the ytem S 3 become z ( k + 1 = A B F KC K D F z + K C + C (28 By (14, we know that R + Su = R SF z (29 Subtituting (29 into (28 and impliying uing the deinition (16, we get z ( k + 1 = ( A B F K C z + K C (30 Subtituting the control law (27 into (3, we alo obtain ( k + 1 = A B F z (31 In matri repreentation, we can write equation (30 and (31 a ( k + 1 A B F k z ( k 1 = K z + C A B F KC (32 Since the etimation error e i deined by e = z, it i eay to ee rom equation (32 that the error e atiie the equation ( e( k + 1 = A K C e (33 Uing the (, e coordinate, the compoite ytem (32 can be impliied a ( k + 1 A B F B k k F M e ( k 1 = 0 A e = e + KC (34 where 92

9 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 M A B F B F =. 0 A KC (35 Since the matri M i block-triangular, it i immediate that ( ( eig( M = eig A B F U eig A K C (36 which etablihe the irt part o the Separation Principle namely that the oberver pole are eparated rom the cloed-loop controller pole. To how that the oberved-baed control law (27 indeed work, we note that the cloed-loop regulator matri A B F and A KC are both convergent matrice. From Eq. (36, it i immediate that M i alo a convergent matri. From Eq. (35, it i thu immediate that 0 and e 0a k or all (0 and e (0. 5. NUMERICAL EXAMPLE In thi ection, we conider a ourth-order linear dicrete-time control ytem decribed by ( k + 1 = A + B u y = C (37 where A =, B = and C = [ ]. (38 The eigenvalue o the matri A are λ = , λ = , λ = and λ 4 = ( From (39, we note that λ1, λ2are untable (low eigenvalue and λ3, λ4 are table (at eigenvalue o the ytem matri A. For thi linear ytem, the dominant and non-dominant tate are calculated. A imple calculation uing the procedure in [9] how that the irt two tate{, } are the dominant (low tate, while the lat two tate {, } are the non-dominant (at tate or the given ytem (37. Uing the procedure decribed in Section 2, the reduced-order linear model or the given linear ytem (37 can be obtained a 93

10 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 ( k + 1 = A + B u y = C + D u (40 where A =, B = [ ] C = and D = The tep repone o the original plant and the reduced order plant are plotted in Figure 1, which validate the reduced-order model obtained or the given linear plant. Figure 1. Step Repone or the Original and Reduced Order Linear Sytem We note alo that the reduced-order linear ytem (40 i completely controllable and completely obervable. Thu, reduced-order oberver and oberver-baed controller or thi plant can be eaily contructed a detailed in Section 3 and CONCLUSIONS In thi paper, uing the dominant tate analyi o the given large-cale linear plant, we obtained the reduced-order model o the linear plant. Then we derived uicient condition or the deign o oberver-baed reduced-order controller. The oberver-baed reduced order controller are contructed by combining the reduced order controller or the original linear ytem which 94

We alo etablihed a eparation principle in thi paper which how that the pole placement problem and oberver problem are independent o each other. REFERENCES [1] Cumming, S.D.

11 International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT, Vol.1, No.5, December 2011 require the dominant tate o the original ytem and reduced order oberver or the original linear ytem which provide an eponential etimate o the dominant tate o the original linear ytem. We alo etablihed a eparation principle in thi paper which how that the pole placement problem and oberver problem are independent o each other. REFERENCES [1] Cumming, S.D. (1969 Deign o oberver or reduced dynamic, Electronic Letter, Vol. 5, pp [2] Fortman, T.E. & Williamon, D. (1971 Deign o low-order oberver or linear eedback control law, IEEE Tran. Automat. Control, Vol. 17, pp [3] Litz, L. & Roth, H. (1981 State decompoition or ingular perturbation order reduction a modal approach, Internat. J. Control, Vol. 34, pp [4] Latman, G. & Sinha, N. & Roza, P. (1984 On the election o tate to be retained in reducedorder model, IEE Proc. Part D, Vol. 131, pp [5] Anderon, B.D.O. & Liu, Y. (1989 Controller reduction: concept and approache, IEEE Tran. Automat. Control, Vol. 34, pp [6] Mutaa, D. & Glover, K. (1991 Controller reduction by H balanced truncation, IEEE Tran. Automat. Control, Vol. 36, pp [7] Aldeen, M. (1991 Interaction modelling approach to ditributed control with application to interconnected dynamical ytem, Internat. J. Control, Vol. 241, pp [8] Aldeen, M. & Trinh, H. (1994 Oberving a ubet o the tate o linear ytem, IEE Proc. Control Theory Appl., Vol. 141, pp [9] Sundarapandian, V. (2005 Ditributed control cheme or large cale interconnected dicrete-time linear ytem, Math. Computer Modelling, Vol. 5, pp [10] Ogata, K. (2010 Dicrete-Time Control Sytem, Prentice Hall, New Jerey. Author Dr. V. Sundarapandian obtained hi Doctor o Science degree in Electrical and Sytem Engineering rom Wahington Univerity, Saint Loui, USA under the guidance o Late Dr. Chritopher I. Byrne (Dean, School o Engineering and Applied Science in He i currently Proeor in the Reearch and Development Centre at Vel Tech Dr. RR & Dr. SR Technical Univerity, Chennai, Tamil Nadu, India. He ha publihed over 210 reereed international publication. He ha publihed over 100 paper in National Conerence and over 50 paper in International Conerence. He i the Editor-in-Chie o International Journal o Mathematic and Scientiic Computing, International Journal o Intrumentation and Control Sytem, International Journal o Control Sytem and Computer Modelling, etc. Hi reearch interet are Linear and Nonlinear Control Sytem, Chao Theory and Control, Sot Computing, Optimal Control, Proce Control, Operation Reearch, Mathematical Modelling, Scientiic Computing uing MATLAB etc. Mr. M. Kavitha received the B.Sc. degree in Mathematic rom Bharatidaan Univerity, Tiruchirappalli, Tamil Nadu in 1998, M.Sc. degree in Mathematic rom Bharatidaan Univerity, Tiruchirappalli, Tamil Nadu in 2000, P.G. Diploma in Computer Application rom Bharatidaan Univerity, Tiruchirappalli, Tamil Nadu, M.Phil in Mathematic rom Madurai Kamaraj Univerity, Madurai, Tamil Nadu in 2004 and currently puruing PhD degree in Mathematic rom Vel Tech Dr. RR & Dr. SR Technical Univerity, Chennai, India. She ha 9 year o teaching eperience and her thrut area o reearch are Dierence and Dierential Equation and Linear Control Sytem. 95

CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS

CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 INTRODUCTION 8.2 REDUCED ORDER MODEL DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.3