International Journal of Coputing and Optiization Vol. 3, 2016, no. 1, 71-82 HIKARI Ltd, www.-hikari.co http://dx.doi.org/10.12988/ijco.2016.635 The Extended Balanced Truncation Algorith Cong Huu Nguyen Thai Nguyen Univerity Tan Thinh, Thai Nguyen City, Viet Na Copyright 2016 Cong Huu Nguyen. Thi article i ditributed under the Creative Coon Attribution Licene, which perit unretricted ue, ditribution, and reproduction in any ediu, provided the original work i properly cited. Abtract Model order reduction i a reearch direction which ha attracted ore interet to cientit in recent year. Actually there are any reearche on odel reduction for higher-order linear yte but thoe on odel reduction odel for untable higher-order linear yte are till liited and exited any diadvantage. Thi paper preent extended balanced truncation algorith for untable linear yte. At the ae tie, to coplete extended balanced truncation algorith for untable linear yte, the author give one definition and two new theore with adequate proof to deterine the upper bound forula of order reduction error thereby the algorith can autoatically reduce order of linear untable yte baed on that forula. The illutration how the correctne of the odel order algorith. Keyword: Model order reduction, extended balanced truncation algorith, order reduction error I. Introduction There are two baic approache to reduce order for untable linear yte a follow: The firt approach in [1] (indirectly order reduction (OR) ethod for untable yte) analye the untable yte into u of table part and untable part, and then, the OR algorith uch a balanced truncation [4] i applied to reduce order table part. The new OR yte i fored a the u of order reduction part of table yte and untable part. Under thi approach, the effectivene of order reduction depend ainly on order reduction algorith applied to the table yte. However, in thi approach the untable yte cannot be
72 Cong Huu Nguyen deleted in the OR yte o the OR yte alway ha higher order than the untable one, which ean thi approach ay not provide an enough good OR yte to in-out relation of the original yte in which untable part occupie ajority. But in reality, untable part often occupy a all hare in the original yte o thi approach can give good OR reult. In the econd approach (directly OR ethod for untable yte), the order of the untable yte i directly reduced under the extended balanced truncation algorith of Zhou [7], algorith of LQG [6], eleental analyi [2], applied balanced truncation algorith of Boe [5], and extended balanced truncation of Zilochian [3]. The advantage of thi approach i that order of OR yte doe not depend on order of the untable yte, which ean order of thi OR yte can be lower than the untable one. However, algorith following thee ethod have weaknee a follow: The extended balanced truncation algorith of Zhou [7] need to addre 4 Lyapunov equation o the coputational coplexity i high and it cannot be applied to all untable yte and cannot give upper bound forula of order error reduction (OER). The eleental analyi algorith [2] require experience of uer and cannot give upper bound forula of OER. The applied balanced truncation algorith of Boe [5] ha upper bound forula of OER but it only can be applied for dicrete yte. Study no. [3] ha propoed apping (tranaction axe ethod) baed on the value of the real part of untable pole having the larget real part value to witch untable yte into table one, thereafter applying balanced truncation algorith to reduce order. However, thi algorith ha not given upper bound forula of OER o it cannot perfor autoatic reducing baed on upper bound forula of OER. According to thee analye, the article will focu on copleting the tudy on the OR algorith for untable yte under the econd approach (directly OR untable yte ethod), naely the reearch on deterining upper bound forula of OER of extended balanced truncation algorith of Zilochian [3] o that the algorith can perfor autoatic reduce order baed on upper bound forula of OER. II. Extended balanced truncation algorith 2.1 Proble of order reduction odel A ultiple-input and output linear yte i given with continuou-tie contant paraeter decribed in pace tated in the following equation: x Ax Bu (2.1) y Cx In which, n, p, q nxn nxp qxn x u y, A, B, C The goal of the order reduction proble with odel decribed by (2.1) i to find odel decribed by yte of equation:
Extended balanced truncation algorith 73 x A x B u y r r r r C x r r r (2.2) In which, r, p, q rxr rxp qxr xr ur yr, Ar, Br, Cr and r n, o that the odel decribed by (2.2) can be replaced by the odel decribed in (2.1) to be applied in analyi, deign and control yte. 2.2. Extended balanced truncation algorith The idea of the extended balanced truncation algorith of Zilochian [3] i carried out apping (the origin hift) to convert the original untable yte to table for, then reduce order for table yte according to the extended balanced truncation algorith obtaining the table yte. Finally, the algorith perfor backward projection (revere the origin helf) to convert the table yte to the untable for iilar to the original yte. The detailed content of the extended balanced truncation algorith of Zilochian [3] i decribed a follow: Algorith 2.2. The extended balanced truncation algorith of Zilochian [3] Input: The yte A, B, C i decribed a (2.1) (Untable yte). The tranfer function of yte (2.1) i given by ( ) : 1 G C I A B. Step 1: Identify the larget untable pole of yte (2.1). Set real( ), where arbitrarily all and 0. Step 2: Convert the yte A, B, C according to the following equation A A I, B C B, C. Step 3: Copute obervability Graian Q and controllability Graian P of yte A, B, C fro two Lyapunov equation: A P P A B B T T, A Q Q A C C T T. Step 4: Copute Choleky factorization of atrix upper triangle atrix. Step 5: Copute Choleky factorization of atrix upper triangle atrix. Step 6: Copute SVD of atrix R T R T U ΛV. o p Step 7: Copute noningular tranforation T T R V Λ. 1 1/ 2 p R i T P R pr p, where p T Q RoR o, where o R i
74 Cong Huu Nguyen Step 8: Copute Aˆ, Bˆ, Cˆ 1 1 T A T, T B, C T. Step 9: Chooe re-order r o that r << n. Step 10: A ˆ, ˆ, ˆ B C i partitioned a follow:: Aˆ ˆ ˆ ˆ 11 A 12 B ˆ 1 A,, ˆ ˆ ˆ B C 1 2, ˆ ˆ ˆ C C A21 A22 B2 x x x where ˆ r r, ˆ r p, ˆ q r A B C. 11 1 1 We obtain reduced table yte A ˆ ˆ ˆ 11, B1, C 1. Step 11: Convert reduced table yte A ˆ ˆ ˆ 11, B1, C 1 according to the following equation: Aˆ Aˆ I, 11 11 Bˆ Bˆ, 1 1 Cˆ ˆ 1 C1. Output: The reduced yte A ˆ, Bˆ, C ˆ 11 1 1 2.3. Copleting extended balanced truncation algorith The extended balanced truncation algorith ha not given upper bound forula of OER o it cannot perfor autoatic reduction baed on upper bound forula of OER. To coplete thi algorith, the author will identify and prove the correctne of the upper bound forula of OER of thi algorith. The tranfer function of yte (2.1) i given by ( ) 1 G C I A B Definition 1. The yte (2.1) i called - table if real( ( A)), with i a non-negative real nuber ( 0 ). The et of continuou - table yte i denoted by C. The H, -nor of G ( ) i denoted by: H, C G( ) : up ( G( )) real ( ( A)) ax ax up ( G( j)). Where ( ( )) ax G i the larget ingular value of G ( ). In the cae that 0 the yte (1) i called ayptotically table a ordinary eaning [1]. Matrix A in thi cae i Hurwitz atrix, i.e., real( ( A )) 0, and the H, -nor of G ( ) coincide with the H -nor of G ( ) a uual eaning G( ) G( ) up G j. H,0 H ax
Extended balanced truncation algorith 75 Set ( ) 1 G ˆ C ˆ I A ˆ B ˆ i the atrix tranfer function of yte 1 1 11 1 A ˆ ˆ ˆ 11, B1, C 1 and ( ) 1 reduced yte ˆ 11, ˆ ˆ 1, 1 follow: G ˆ C ˆ I A ˆ B ˆ i the atrix tranfer function of 1 1 11 1 A B C in Algorith 2.2, we have two theore a Theore 1: For any continuou yte G( ) repreented by (2.1), we conider the yte ( ) Then, the following thing hold: (i) G i ayptotically table, G with realization,,,, A B C A I B C. (ii) The H -nor of G ( ) i equal to the H, -nor of G ( ), G ( ) G ( ). H Proof of Theore 1: (i) Fro G( ), we have real( ( A )), 0 and real( ( A I )) 0. (ii) It hold that and thu H, G ( j) C ( ji A ) B 1 G H 1 C( ji A I) B 1 C(( j) I A) B G( j), up ( G ( j)) H, ax up ( G( j)) G ax. Theore 2: Let G( ) and G ˆ 1 ( ) be reduced order yte obtained a in Algorith 2.2. Then the following bound for the error yte hold: G( ) Gˆ ( ) 2, 1 r1 H, where,, 1 n are the Hankel ingular value of G ( ). Proof of Theore 2: Let E( ) G( ) Gˆ 1 1( ) Ce I Ae B e, we have n
76 Cong Huu Nguyen A 0 B A,, ˆ e e e 1. 0 ˆ B ˆ C C C A11 B1 Fro real( ( A)), real( ( A ˆ )) 11, we have E( ). Uing Theore 1 we obtain that E( ) E ( ) G ( ) G ˆ ( ), 1 H, H H 1 E ( ) Ce I Ae I B e. Since yte G ( ) and G ˆ 1 ( ) are ayptotically table and G ˆ 1 ( ) i reduced yte obtained by balanced truncation of G ( ), we get that where G ( ) ˆ G ( ) 2 n, 1 r1 H where,, 1 n are the Hankel ingular value of G ( ). III. The illutrative exaple 3.1. Reducing untable yte We have a high order untable yte a follow: ( ) S( ) H D( ) H( ) 1.26 110.4 3959 8.089.10 1.078.10 1.006.10 15 14 13 4 12 6 11 7 10 6.869.10 3.547.10 1.419.10 4.478.10 1.116.10 2.142.10 7 9 8 8 9 7 9 6 10 5 10 4 2.96.10 2.616.10 1.183.10 1.536.10 10 3 10 2 10 9 D( ) 0.000118 0.0202 1.051 22.89 222.5 165.6 1.99.10 (3.1) 15 14 13 12 11 10 4 9 2.433.10 1.533.10 5.942.10 1.438.10 2.042.10 1.401.10 5 8 6 7 6 6 7 5 7 4 7 3 6 2 8 2.108.10 1.49.10 Perforing reduced order odel untable yte (3.1) by extended balanced truncation algorith, the reult are hown in the following: Step 1: Identify the larget untable pole of yte (3.1). We obtained 10,313. Set real( ) 10,323 Step 2: Convert the untable yte S ( ) to table yte S ( ) according to the following equation
Extended balanced truncation algorith 77-181.5308-69.2774-23.5797-14.3273-1.3331 10.0562 7.6510 6.0266 2.9194 1.7667 1.2539 0.8603 0.5178 0.0000 0 128-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 64-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 160-10.3233 0 0 0 0 0 0 0 0 A A I 0 0 0 0 0 0 8-10.3233 0 0 0 0 0 0 0, 0 0 0 0 0 0 0 8-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.25-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.002-10.3233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.02-10.3233 8192 0 0 0 0 0 0 B B 0, 0 0 0 0 0 0 0 4 C C 1.0e 0.0108 0.0058 0.0020 0.0010-0.0008-0.0017-0.0011-0.0009-0.0004-0.0002-0.0002-0.0001-0.0001-0.0182-1.2074, 4 D D -1.0632e Step 3-10: Perforing reduced order odel yte follow tep 3 10, the reult are hown in the following table: Table 1. Reult of the order reduction of table yte S ( ) Order 6 5 4 3 Reduced order yte S ˆ ( ) 1 Error 1.063.10 1.092.10 4.207.10 8.124.10 8.488.10 4.612.10-1. 027. 10 186.3 4 1.008.10 5 3 6 2.026.10 1.724.10 6 5.301.10 5.283. 10 6 5 4 2 4 4 6 6 5 7 4 8 3 9 2 10 11 1.063.10 9.871.10 3.231.10 4.928.10 3.612.10 1.038.10 5 4 3 5 2 5 176.4 8335 1.201.10 5.352.10 5340 4 5 5 4 7 3 8 2 9 10 1.063.10 8.665.10 2.256.10 2.397.10 9.246.10 4 3 2 4 165.1 6470 4.765.10 475.8 4 4 5 3 7 2 8 8 1.063.10 3.28.10 3.174.10 7.74.10 3 2 114.5 399.5 3.983 4 3 5 2 6 6 S ( ) ˆ S1 ( ) H 3.484.10-7 3.5.10-7 0.0346 10.15 Step 11 : Convert reduced table yte S ( ) 1 to the reduced -table yte S ˆ ( ), the reult are preented in table 2: 1
78 Cong Huu Nguyen Table 2. Reult of the order reduction of S ( ) Order Reduced order yte S ˆ 1 ( ) Error 4 6 5 5 6 4 7 3 7 2 7 7 6 1.063.10 4.337.10 2.689.10 5.225.10 2.36.10 1.503.10 + 1.135. 10 6 5 4 4 3 4 2 5 6 124.4 2063 3.711.10 1.5 57.10 1.65.10 4.41. 10 4 5 5 4 6 3 6 2 7 7 5 1.063.10 4.383.10 2.877.10 6.463.10 2.638.10 2.638.10 124.8 2117 3.62.10 1.45.10 1.027. 10 5 4 3 4 2 5 5 4 4 4 5 3 6 2 6 7 1.063.10 4.274.10 2.522.10 4.157.10 2.191.10 4 3 2 4 4 123.8 1998 3.757.10 2.789.10 3 4 3 2 5 6 1.063.10 1317 1.981.10 1.772.10 3 2 83.49 1644 6978 S( ) Sˆ 1( ) H, 3.484.10-7 3.5.10-7 0.0346 10.15 Step repone and bode plot of the original yte (15 th -order yte), reduced order yte reult are hown in Figure 1. -0.5 0 x 105 Step Repone -1 15th-order yte 6th-order yte 5th-order yte 4th-order yte 150 100 Bode Diagra 15th-order yte 6th-order yte 5th-order yte 4th-order yte Aplitude -1.5-2 Magnitude (db) 50-2.5-3 -3.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Tie (econd) 0 10-2 10-1 10 0 10 1 10 2 10 3 10 4 Frequency (rad/) (a) (b) Fig. 1. Step repone (a) and bode plot (b) of the original yte, reduced order yte Coent: Through the reult of the order reduction error and Figure 1, it can be een that: - The order reduction error under the tandard H, of 5 th, 6 th -order yte i all; the tep repone and bode plot of 5 th, 6 th -order yte copletely coincide with the one of the original yte. - The order reduction error inder the tandard H, of 4 th -order yte i uch ore than the order reduction error of the 5 th, 6 th -order yte; the tep repone of 4 th -order yte copletely coincide with the one of the original yte; bode plot of 4 th -order yte i deviation fro the bode plot of the original yte in the low frequency (le than 1.98 rad/). Therefore, depending on the order reduction error requet and the application range of the odel order reduction in pecific proble,we can chooe the corre-
Extended balanced truncation algorith 79 ponding odel order reduction for the original yte. Epecically, if we want the order reduction yte having the lowet poible order and ainly care about the fale tep repone copared to the all original yte, we can chooe the 4 th - order yte intead of the original yte. If we want the lowet order yte but all order reduction error, the fale tep repone and the fale bode plot veru the all original yte, we can chooe the 5 th -order yte intead of the original yte. The above reult how that the extended balanced truncation algorith can reduce order for high-order untable linear yte. 3.2. Reducing higher- order controller Deign of controller of the bicycle balanced yte under H utainable control algorith hown in detail in [8], H full order controller i deigned a follow: ( ) R( ) H D( ) (3.2) H( ) 2.23.10 4.67.10 0.266 22.96 1006 2.853.10 7 30 4 29 28 27 26 4 25 5.837.10 9.144. 10 1.139.10 1.158.10 9.776.10 5 24 6 23 8 22 9 21 9 20 10 19 11 18 12 17 6.949.10 4.199.10 2.172.10 9.663.10 3.71.10 12 16 13 15 1.231.10 3.53.10 8.74.10 1.862.10 3.398.10 14 14 14 13 14 12 15 11 15 10 5.276.10 6.903.10 7.511.10 6.676.10 4.721.10 15 9 15 8 15 7 15 6 15 5 15 4 14 3 14 2 13 2.556.10 9.953.10 2.482.10 2.977.10 0.00439 D( ) 4.971.10 2.032.10 2.663.10 1.221.10 9.72.10 14 30 10 29 7 28 4 27 3 26 0.3918 10.14 187.1 2612 2.862.10 2.523.10 25 24 23 22 4 21 5 20 6 19 7 18 7 17 1.82.10 1.088.10 5.428.10 2. 273.10 8.005.10 8 16 8 15 2.372.10 5.9.10 1.225.10 2.107.10 2.962.10 9 14 9 13 10 12 10 11 10 10 3.341.10 2.941.10 1.931.10 8.743.10 2.286.10 10 9 10 8 10 7 9 6 9 5 8 4 7 3 6 2 22 1.519.10 5.226.10 3.6.10 5.32.10 Converting controller (3.2) to inial realization and pole-zero cancellation, we obtain the following reult: R ( ) H D ( ) ( ) (3.3)
80 Cong Huu Nguyen H ( ) 4.485.10 4.231.10 1.912.10 5.51.10 1.138.10 6 27 8 26 10 25 11 24 13 23 1.794.10 2.245.10 2.29.10 1.939.10 1.381.10 14 22 15 22 16 20 17 19 18 18 18 17 19 16 20 15 20 14 21 13 8.359.10 4.33.10 1.929.10 7.413.10 2.462.10 7.066.10 1.75.10 3.732.10 6.814.10 1.058.10 21 12 22 11 22 10 22 9 23 8 1.385.10 1.508.10 1.341.10 9.487.10 5.138.10 23 7 23 6 23 5 22 4 22 3 22 2 21 20 2.001.10 4.99.10 5.987.10 D ( ) 2088 1.803.10 7.485.10 1.966.10 3.66.10 27 26 5 25 6 24 8 23 9 22 5.14.10 5.657.10 5.003.10 3.618.10 2.166.10 1.083.10 10 21 11 20 12 19 13 18 14 17 4.539.10 1.601.10 4.748.10 1.182.10 15 16 15 15 16 14 16 13 17 12 2.456.10 4.226.10 5.944.10 6.709.10 5.908.10 17 11 17 10 17 9 17 8 17 7 17 6 17 5 16 4 15 3 3.881.10 1.758.10 4.598.10 3.058.10 1.05 1.10 15 2 The controller R ( ) i an untable yte becaue the controller R ( ) ha 3 poitive pole which are 0; 0; 0.103. Perforing reduced order odel untable yte (3.3) by extended balanced truncation algorith, the reult are hown in the following table: Table 3. Reult of the order reduction of controller R ( ) Order Reduced order controller Rr ( ) 5 4 4.485.10 6.804.10 4.123.10 1.235.10 1.816.10 1.09.10 5 4 4 3 2 9 10 2009 1.833.10 1913 6.614.10 8.44.10 6 4 7 3 8 2 8 8 4.485.10 2.655.10 1.191.10 1.811.10 1.182.10 4 3 2 2000 205.6 0.1231 0.003463 6 5 7 4 8 3 9 2 9 9 Step repone and bode plot of the original controller (30 th -order controller), reduced order controller reult are hown Figure 2. Aplitude 0 x 106 Step Repone -0.5-1 -1.5-2 -2.5 30th-order controller 5th-order controller 4th-order controller Magnitude (db) 250 200 150 100 50 360 Bode Diagra 30th-order controller 5th-order controller 4th-order controller -3-3.5-4 Phae (deg) 180 0-4.5 0 0.002 0.004 0.006 0.008 0.01 Tie (econd) -180 10-2 10 0 10 2 10 4 Frequency (rad/) (a) (b) Fig. 2. Step repone (a) and bode plot (b) of the original controller, reduced order yte
Extended balanced truncation algorith 81 Reark of the reult: Step repone and bode plot of the 4 th, 5 th - order controller alot coincide with tep and bode plot of the original controller. Therefore, the 5 th, 4 th - order controller can be ued to replace the original controller. The above reult how that the higher-order controller (untable yte) can be reduced by extended balanced truncation algorith. IV. Concluion The article introduced the extended balanced truncation algorith. Beide, the author ha coe up with one definication and two new theore with the adequate proof to deterine the upper bound foula of the order reduction error in order to coplete the extended balanced truncation algorith for untable yte. The tiulation reult howed the correctne of the upper bound foula of the order reduction error and the introducted algorith. Reference [1] A. C Antoula, Approxiation of Large Scale Dynaical Syte, SIAM Society for Indutrial and Applied Matheatic, 2005. http://dx.doi.org/10.1137/1.9780898718713 [2] A. Varga, Coprie factor odel reduction baed on accuracy enhancing technique, Syte Analyi Modelling and Siulation, 11 (1993), 303-311. [3] A. Zilochian, Balanced Structure and Model Reduction of Untable Syte, IEEE Proceeding of Southeatcon 91, 2 (1991), 1198-1201. http://dx.doi.org/10.1109/econ.1991.147956 [4] B. C. Moore, Principal coponent analyi in linear yte: Controllability, obervability, and odel reduction, IEEE Tran. Auto. Contr., 26 (1981), 17-32. http://dx.doi.org/10.1109/tac.1981.1102568 [5] C. Boe, N. K. Nichol, A. Bune-Gertner, Model order reduction for dicrete untable control yte uing a balanced truncation approach, 2010. http://www.reading.ac.uk/web/files/ath/preprint_10_06_nichol.pdf [6] E. A. Jonckheere and L. M. Silveran, A New Set of Invariant for Linear Syte - Application to Reduced Order Copenator Deign, IEEE Tranaction on Autoatic Control, 28 (1983), no. 10, 953-964. http://dx.doi.org/10.1109/tac.1983.1103159 [7] K. Zhou, G. Saloon, E. Wu, Balanced realization and odel reduction ethod for untable yte, International Journal of Robut and Nonlinear Control, 9 (1999), no. 3, 183-198.
82 Cong Huu Nguyen http://dx.doi.org/10.1002/(ici)1099-1239(199903)9:3<183::aidrnc399>3.0.co;2-e [8] V. N. Kien, Reearching Model Order Reduction Algorith And Applying to Control Proble, Ph.D. Thei, Thai Nguyen Univerity of Technology, Viet Na, 2015. Received: March 24, 2016; Publihed: May 4, 2016