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

Transcription

1 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 REDUCED ORDER CONTROLLER DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.4 REDUCED ORDER DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.5 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.6 NUMERICAL EXAMPLE 8.7 SUMMARY 57

2 CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 Introduction Thi chapter 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 controller pole can be eparated hence the pole-placement problem oberver deign are independent o each other. Thi chapter ha been organized a ollow. Section 8.2 decribe the reduced order model deign or linear dicrete-time control ytem. Section 8.3 decribe the reduced order controller deign or linear dicrete-time control ytem. Section 8.4 decribe the reduced order oberver deign or linear dicrete-time control ytem. Section 8.5 decribe the oberver baed reduced order controller deign or linear dicrete-time control ytem. Section 8.6 contain a numerical example illutrating the deign procedure. Section 8.7 contain a ummary o the main reult derived in thi chapter. 8.2 Reduced Order Model Deign or Linear Dicrete-Time Control Sytem Conider a large cale dicrete-time linear ytem modelled by the equation x( k 1) Ax Bu y Cx (8.1) n x R i the tate, m u R i the control or input 58 p y R i the ytem output. Aume that A, B C are real, contant matrice o dimenion nn, nm pn, repectively. Firt, aume that an identiication o the dominant (low) non-dominant (at) tate o the original linear ytem (8.1) were made uing the modal approach a decribed in Chapter 4.

3 Without lo o generality, it i aumed that x x, x (8.2) r x R repreent the dominant tate x nr R repreent the non-dominant tate o the ytem (8.1). The tate x correpond to the low mode o the ytem, while the tate to the at mode o the ytem. Then the linear ytem (8.1) become x correpond x ( k 1) A A x B u ( k ) x ( k 1) A A x B x y C C x (8.3) From (8.3), the plant equation can be written a x ( k 1) A x A x B u x ( k 1) A x A x B u y C x C x (8.4) Next, it i aumed that ytem matrix A i diagonalizable. In mot practical ituation, the matrix A will have ditinct eigenvalue hence thi aumption will be eaily atiied ince the eigenvector o a matrix correponding to dierent eigenvalue are linearly independent. Then there exit a modal matrix M coniting o n linearly independent eigenvector o A uch that M 1 AM, (8.5) i a diagonal matrix coniting o the n eigenvalue o A. A change o coordinate i introduced on the tate pace, which i given by Nx, N M 1 (8.6) 59

4 In the new coordinate (8.6), the linear ytem (8.3) become ( k 1) NBu y CM (8.7) Thu, ( k 1) NBu ( k 1) y CM (8.8) are r r ( n r) ( n r) diagonal matrice, repectively, coniting o the low at eigenvalue o A. Next, deine matrice,, by NB CM (8.9),, are r m, ( n r) m, pr p( n r) matrice, repectively. From (8.8) (8.9), it i een that the linear plant (8.3) ha the ollowing imple tructure in the new coordinate (8.6) ( k 1) u ( k 1) u y (8.10) Next, the ollowing two aumption are made: (H1) A k, ( k 1), i.e. take a contant value in the teady tate. (H2) The matrix I i invertible. Uing the aumption (H1), the econd equation in (8.10) become (or large value o k ) u (8.11) From Eq. (8.11) the aumption (H2), it ollow that 1 ( I ) u (8.12) 60

5 Subtituting (8.12) into (8.10), the reduced-order model o the linear ytem (8.1) in the coordinate can be written a ( k 1) u y k k I u k 1 ( ) (8.13) The reduced-order ytem model o the linear ytem (8.13) in the x coordinate obtained a ollow. i Set N 1 N N M, N N (8.14) N, N, N N are r r, r ( n r), ( n r) r ( n r) ( n r) matrice repectively. By the change o coordinate (8.6), it ollow that 1 M x Nx (8.15) Thu, x N N N x N N (8.16) From (8.13) (8.16), it ollow that 1 N x N x ( I ) u (8.17) i.e. N x k N x k I u k (8.18) 1 ( ) Next, the ollowing aumption i made. (H3) The matrix N i invertible. Uing (H3), the equation (8.18) become x k N N x k N I u k (8.19) ( ) 61

6 To impliy notation, deine the matrice R N N S N ( I ) (8.20) Uing (8.20), the equation (8.19) can be impliied a x R x S u (8.21) Subtituting (8.21) into (8.4), the reduced-order ytem model o the linear ytem (8.4) i obtained a x ( k 1) A x B u y C x D u (8.22) the matrice A, B, C D are deined a ollow: A A A R, B B A S C C C R, D C S (8.23) 8.3 Reduced Order Controller Deign or Linear Dicrete-Time Control Sytem In thi ection, the reduced order controller deign or a large cale dicrete-time linear ytem, whoe tate are ully obervable i decribed. Conider the linear control ytem given by x( k 1) Ax Bu (8.24) Under the aumption (H1)-(H3), the reduced order ytem model o the ytem (8.24) i obtained a x ( k 1) A x B u (8.25) A B are decribed by (8.23). Next, an ueul reult that precribe a imple method i derived or tabilizing the dominant tate x o the reduced-order linear plant (8.25). 62

7 Theorem 8.1 For the reduced-order ytem model (8.25), uppoe that the ytem pair ( A, B ) i completely controllable. Then there exit a tate eedback control law u F x (8.26) that tabilize the dominant tate x o the reduced order model (8.25) yielding any deired et o table eigenvalue or the cloed-loop ytem matrix A B F. Proo. Since the ytem pair ( A, B ) i completely controllable, it ollow by linear ytem theory that the pole o the eedback control ytem, which are the eigenvalue o the cloed-loop ytem matrix A B F can be arbitrarily placed by a tate eedback law o the orm (8.26). In particular, it ollow that the eigenvalue o the cloed-loop ytem matrix A B F can be placed in the table region given by : 1. Thu, it ollow that the tate eedback control law (8.26) tabilize the dominant tate x o the reduced order model (8.25) yielding any deired et o table eigenvalue or the cloed-loop ytem matrix A B F. Thi complete the proo. 8.4 Reduced Order Oberver Deign or Linear Dicrete-Time Control Sytem In thi ection, the reduced order oberver deign or a large cale linear dicrete-time control ytem i decribed. Conider the linear control ytem given by x( k 1) Ax Bu y Cx (8.27) Under the aumption (H1)-(H3), the reduced order ytem model o the ytem (8.27) i obtained a x ( k 1) A x B u y C x D u (8.28) A, B, C D are deined by (8.23). 63

8 Theorem 8.2 For the reduced order linear ytem (8.28), uppoe that the pair ( C, A ) i completely obervable. Then the ytem (8.28) ha a global exponential oberver deined by z ( k 1) A z B u K y C z D u (8.29) K i an output gain matrix that can be choen uch that the etimation error matrix E A K C (8.30) ha an arbitrarily aigned et o table eigenvalue. Proo. Deine the etimation error by e z x Then the error atiie the dierence equation e( k 1) E e, E A KC Then it ollow that k k e E e(0) ( A KC) e(0) or all k. Since the pair ( C, A ) i completely obervable, an oberver gain matrix K can be choen o that the oberver pole or the eigenvalue o the matrix E A KC can be arbitrarily placed in the table region : 1. Thi how that the etimation error e i globally exponentially table. 8.5 Oberver Baed Reduced Order Controller Deign or Linear Dicrete-Time Control Sytem In mot o the practical application, the dominant tate x o the reduced-order model (8.28) o the large cale linear ytem (8.27) may not be directly available or meaurement hence a tate eedback control law o the orm u F x cannot be implemented to tabilize the tate dynamic. To overcome thi practical diiculty, an important theorem i derived, 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 (8.27) alo demontrate that the oberver pole the cloed-loop controller pole can be eparated. Z 64

9 Theorem 8.3 (Separation Principle) Suppoe that the aumption (H1)-(H3) hold or the original linear ytem (8.27). Then it ha a reduced order linear ytem given by (8.28), the matrice A, B, C D are deined by (8.23). Suppoe that there exit matrice F K uch that A B F A K C are both convergent matrice. By Theorem 8.2, the ytem deined by (8.29) i an exponential oberver or the dominant tate x o the original linear ytem (8.27). Then the oberver pole the cloed-loop controller pole are eparated the oberver-baed control law u F z (8.31) alo tabilize the dominant tate x o the large-cale control ytem (8.27). Proo. Under the eedback control law (8.31), the oberver dynamic (8.29) become ( 1) z k A B F KC K D F z k K Cx k C x k (8.32) By (8.21), x Rx Su Rx SF z (8.33) Subtituting (8.29) into (8.28) impliying uing the deinition (8.16), z ( k 1) ( A B F K C ) z K C x (8.34) Subtituting the control law (8.27) into (8.3), x ( k 1) A x B F z (8.35) In matrix repreentation, equation (8.34) (8.35) can be written a x ( k 1) A B F x k z ( k 1) K z C A B F KC (8.36) Since the etimation error e i deined by e z x, it i eay to ee rom Eq. (8.36) that the error atiie the equation ( 1) e k A K C e ( k ) (8.37) 65

10 Uing the ( x, e) coordinate, the compoite ytem (8.37) can be impliied a x ( k 1) A B F B x k x k F M e ( k 1) 0 A e e KC (8.38) M A B F B F 0 A KC. (8.39) Since the matrix M i block-triangular, it i immediate that eig( M ) eig A B F eig A K C (8.40) 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 (8.31) indeed work, it i noted that the cloed-loop regulator matrix A B F the oberver error matrix A K C are both convergent matrice. From Eq. (8.39), it i immediate that M i alo a convergent matrix. From Eq. (8.40), it i thu immediate that x 0 e 0 a k or all x (0) e (0). Thi complete the proo. 8.6 Numerical Example Conider a ourth-order linear dicrete-time control ytem decribed by x( k 1) A x B u y Cx (8.41) A, B C (8.42) The eigenvalue o the matrix A are , , (8.43)

11 From (8.43), it i noted that 1, 2 are untable (low) eigenvalue 3, 4 are table (at) eigenvalue o the ytem matrix A. For thi linear ytem, the dominant non-dominant tate are determined next. A imple calculation uing the procedure in Chapter 4 how that the irt two tate x x are the dominant (low) tate, while the lat two tate, 1, 2 (at) tate or the given ytem (8.41). x x are the non-dominant Uing the procedure decribed in Section 8.2, the reduced-order linear model or the given linear ytem (8.41) can be obtained a 3 4 x ( k 1) A x B u y C x D u (8.44) A , B C , , (8.45) D The impule repone o the original plant (8.41) the reduced order plant (8.44) are plotted in Figure 8.1. The tep repone o the original plant (8.41) the reduced order plant (8.44) are plotted in Figure

12 Figure 8.1 Impule Repone or the Original Reduced Order Linear Sytem Figure 8.2 Step Repone or the Original Reduced Order Linear Sytem 68

13 It i alo noted that the reduced-order linear ytem (8.44) i completely controllable completely obervable. Next, an exponential oberver i contructed or the reduced order ytem (8.44) with error dynamic having pole at {0.1,0.1}. Uing MATLAB, it i ound that the output gain matrix K (8.46) i uch that the error matrix E A KC ha eigenvalue {0.1,0.1}. Thu, a global exponential oberver or the linear ytem (8.44) i given by z ( k 1) A z B u K y C z D u (8.47) A, B, C, D I the etimation error i deined by K are deined by (8.45) (8.46). e z x, (8.48) then the error atiie the dierence equation e( k 1) Ee, E A KC (8.49) For imulation, take the initial value a z 2.1 (0) x (0) 4.8. The time-hitory o the etimation error correponding to thee initial condition i hown in Figure 8.3. A F , it i alo eay to ee that the cloed-loop eedback matrix Setting B F ha the eigenvalue {0.2,0.2}. Thu, by the reult in Section 8.6, it ollow that the oberver-baed tate eedback control law u F z alo tabilize the reduced-order linear ytem (8.44) a hown in Figure

14 Figure 8.3 Time-Hitory o the Etimation Error e1 e 2 Figure 8.4 Time-Hitory o the Oberver Baed Controller x 1 x 2 70

15 8.7 Summary In thi chapter, uing a modal approach an identiication o the dominant nondominant tate o large cale dicrete-time linear ytem, reduced order ytem model deign wa derived new reult or reduced order controller, reduced order oberver oberver-baed controller wa derived. A eparation principle in thi chapter or reduced order ytem model which how that the pole placement problem the oberver deign problem are independent o each other wa derived. A numerical example wa illutrated with a plot. 71