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Proceedings of International Conference on Electrical, Electronics & Computer Science
Prof.Srikanta Patnaik Mentor IRNet India, [email protected]
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Rate Compatible Punctured Turbo-Coded Hybrid Arq For Ofdm System
RATE COMPATIBLE PUNCTURED TURBO-CODED HYBRID ARQ FOR OFDM SYSTEM
DIPALI P. DHAMOLE1& ACHALA M. DESHMUKH2
1,2Department of Electronics and Telecommunications, Sinhgad College of Engineering, Pune 411041, Maharashtra
Abstract-Now a day’s orthogonal frequency division multiplexing (OFDM) is under intense research for broadband wireless transmission because of its robustness against multipath fading. A major concern in data communication such as OFDM is to control transmission errors caused by the channel noise so that error free data can be delivered to the user. Rate compatible punctured turbo (RCPT) coded hybrid ARQ (RCPT HARQ) has been found to give improved throughput performance in a OFDM system. However the extent to which the RCPT HARQ improves the throughput performance of the OFDM system has not been fully understood. HARQ has been suggested as a means of providing high data rate and high throughput communication in next generation systems through diversity combining transmit attempts at the receiver. The combination of RCPT HARQ with OFDM provides significant bandwidth at close to capacity rates of channel. In this paper, we evaluate by computer simulations the throughput performance of RCPT code HARQ for OFDM system as compared to that of conventional OFDM.
Keywords: OFDM, hybrid ARQ, rate compatible punctured turbo codes, mobile communication, multipath fading
I. INTRODUCTION with each transmission would result in the highest throughput as unnecessary redundancy is avoided. Orthogonal frequency division multiplexing has been However in a practical system, [4] the number of considered as one of the strong standard candidates retransmissions allowed is limited to avoid for the next generation mobile radio communication unacceptable time delay before the successful systems. Multiplexing a symbol data serial stream transmission. When the number of retransmissions is into a large number of orthogonal sub channels makes limited, the residual bit error is produced. the OFDM [1] signals spectral bandwidth efficient. Recently, high speed and high quality packet data II. RCPT ENCODER/DECODER AND RCPT communication is gaining more importance. A major HARQ concern in wireless data communication is to control transmission errors caused by the channel noise so The RCPT encoder and decoder are included as a part that error free data can be delivered to the user. For of the transmission system model shown in fig.1. successful packet data communication, hybrid RCPT encoder consists of a turbo encoder, a puncture automatic repeat request (HARQ) is the most reliable and a buffer. Turbo encoder /decoder parameters are error control technique for the OFDM systems as in shown in Table 1. HARQ schemes, channel coding is used for the error correction which is not applied in the Simple ARQ Table 1. Turbo encoder/ decoder parameters schemes. In HARQ, the advantage of obtaining high Rate 1/3 reliability in ARQ system is coupled with advantage of forward error correction system to provide a good Component RSC Encoder throughput even in poor channel conditions. Turbo encoder codes, introduced in 1993 by Berrou et al., have been Interleaver S-random intensively studied as the error correction code for (S = K ½) mobile radio communications. Rate compatible Component Log-MAP punctured turbo (RCPT) coded HARQ (RCPT decoder Decoder HARQ) scheme was proposed in [2] and shown to Number of 8 achieve enhanced throughput performance over an iterations additive white Gaussian noise (AWGN) channel. In [3] it is shown that the throughput of the RCPT Turbo encoder considered in this paper is a rate 1/3 hybrid ARQ scheme outperforms other ARQ schemes encoder. Turbo encoded sequences are punctured by over fading and shadowing channels. the puncturer and the different sequences obtained are stored in the transmission buffer for possible In this paper, we evaluate the throughput performance retransmissions. With each retransmission request, a of RCPT coded HARQ for OFDM system over new sequence (previously unsent sequence) is additive white Gaussian noise channel. When the transmitted. Turbo decoder consists of a depuncturer, number of allowable retransmissions is unlimited, buffer and a turbo decoder. At the RCPT decoder, a transmitting minimum amount of redundancy bits newly received punctured sequence is combined with
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 1 Rate Compatible Punctured Turbo-Coded Hybrid Arq For Ofdm System the previously received sequences stored in the adopted HARQ scheme. The punctured sequences are received buffer. The depuncturer inserts a channel of different length for different puncturing periods. value of 0 for those bits that are not yet received and The sequence to be transmitted is block interleaved 3 sequences, each equal to the information sequence by a channel interleaver and then transformed into length, are input to the turbo decoder where decoding PSK symbol sequence. The Nc data modulated is performed as if all 3 sequences are received. In this symbols are mapped onto Nc orthogonal subcarriers paper the type I HARQ is considered. For type I to obtain the OFDM signal waveform. This is done HARQ, the two parity bit sequences are punctured by applying the inverse fast Fourier transform (IFFT). with P = 2 and the punctured bit sequence is After the insertion of a guard interval (GI), the transmitted along with the systematic bit sequence. If OFDM signal is transmitted over Additive white the receiver detects errors in the decoded sequence, a Gaussion noise channel and received by multiple retransmission of that packet is requested. The antennas at the receiver. The OFDM signal received retransmitted packet uses the same puncturing matrix on each antenna is decomposed into the Nc as the previous packet. Instead of discarding the orthogonal subcarrier components by applying the erroneous packet, it is stored and combined with the fast Fourier transform (FFT) and each subcarrier retransmitted packet utilizing the packet combining or component coherently detected to be combined with time diversity (TD) combing effect. those from other antennas based on maximal ratio combining. The MRC combined soft decision sample III. TRANSMISSION SYSTEM MODEL [5] sequence is de-interleaved and input to the RCPT decoder which consists of a de-puncturer, a buffer The OFDM system model with RCPT coded HARQ and a turbo decoder. Error detection is performed by is shown in Fig 1. At the transmitter a CRC coded the CRC decoder which generates the ACK/NAK sequence of length K bits is input to the RCPT command and recovers the information in case of no encoder where it is coded, punctured and stored in the errors. buffer for possible retransmissions. The parity sequences are punctured before transmission based on
Fig. 1. OFDM with HARQ system model
IV. SIMULATION RESULTS N= 512 sub-channels, number of pilots P= N/8, total number of data sub-channels S=N-P and guard The turbo encoder/decoder parameters are as shown interval length GI=N/4. In our simulation, we assume in Table 1. The computer simulation conditions are channel length L=16, pilot position interval 8 and the summarized in Table 2. number of iterations in each evaluation are 500. A In this paper, the information sequence length K = rate 1/3 turbo encoder is assumed and a log-map 1024 bits is assumed. The turbo encoded sequence is decoding is carried out the receiver. The numbers of interleaved with a random interleaver. The data frame errors to count as a stop criterion are limited to modulation scheme considered is BPSK modulation 15. and the ideal channel estimation is assumed for data demodulation at receiver. We assume OFDM using
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 2 Rate Compatible Punctured Turbo-Coded Hybrid Arq For Ofdm System
Table 2. Simulation Conditions
Information sequence length K = 210 214 bits
Channel interleaver Block interleaver
Modulation / demodulation BPSK
No. Of subcarreiers Nc = 256 OFDM Subcarrier spacing 1/Ts Guard interval Tg = 8/Ts Type Type I ARQ Max. No. Of tx. Ω Forward Additive white gaussion noise Propagation channel Reverse Ideal for a given range of signal-to noise ratio (SNR) while The throughput in bps/Hz is defined as the red line shows the bit error rate (BER) for turbo coded type I HARQ for the same range of given SNR. From graph (fig. 2) below it can be seen that the bit error rate of turbo coded HARQ decreases .....(1) with the increase in SNR however the BER for the conventional OFDM remains as it is for the entire Fig. 2 shows the bit error rate as a function of average range of SNR, hence we can get improved throughput bit energy- to- additive white Gaussian noise for turbo coded HARQ as compared to that of (AWGN) power spectrum density ratio. The conventional OFDM. Fig. 3 shows the bit error rate throughput in bps/Hz is defined as [6] the ratio of the as a function of given range of signal to noise ratio. number of information bits over the total number of Here the BER decrease is shown in between 0 to 1, transmitted bits, where the throughput loss due to GI where the BER changes rapidly for the lower SNR insertion is taken into consideration. In fig.2 the blue and we can get a very low BER even for higher SNR. line indicates the bit error rate of conventional OFDM
4.5
4
3.5
3
2.5
2 Bit Rate Error 1.5
1
0.5
0 0 5 10 15 20 25 30 SNR
Fig. 2. BER of conventional OFDM (+) and turbo coded HARQ (x)
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 3 Rate Compatible Punctured Turbo-Coded Hybrid Arq For Ofdm System
0.58
0.56
0.54
0.52
0.5
0.48
0.46
0.44 Bit Error Rate of Turbo code
0.42
0.4
0.38 0 5 10 15 20 25 30 SNR
Fig. 3. BER of turbo code
[2] Berrou, C., “Near optimum error correcting coding and V. CONCLUSIONS decoding: turbo codes.” IEEE Transactions on Communications vol. 44, No. 10, 1261-1271, 1996.
The throughput for the RCPT coded HARQ for an [3] Garg, D. Adachi, F. “Rate compatible punctured turbo-coded hybrid ARQ for OFDM in a frequency selective fading OFDM system was evaluated. It is shown that the bit channel.” IEEE, 2725-2729, 2008. error rate of turbo coded HARQ decreases with the increase in signal to noise ratio, and hence we can get [4] Lifeng, Y., Zhifang, L. “A hybrid automatic repeat request (HARQ) with turbo codes in OFDM system.” IEEE. a improved throughput for the OFDM with turbo coded HARQ as compared to that of conventional [5] Gacanin, H., Adachi, F. “Throughput of Type II HARQ- OFDM. It is shown that the throughput is almost OFDM/TDM using MMSE-FDE in a multipath channel.” Research Letters in Communications. 1-4, 2009. insensitive to the information sequence length. [6] Bahri, M.W. Boujemaa, H., Siala, M. “Performance comparison of Type I, II and III Hybrid ARQ schemes over REFERENCES AWGN channels.” IEEE International Conference on Industrial Technology, 1417-1421, 2004. [1] R. Prasad, “OFDM for wireless communications systems.” Ist [7] Sklar, B., Ray, P.K. “Digital communications: Fundamentals ed, Artech House, Inc. Boston London, 2004. and applications”. (2nd Edition) Pearson Publications, New Delhi, 2008.
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 4 Comparison Between Construction Methods of Unknown Input Reduced Order observer Using Projection Operator Approach And Generalized Matrix Inverse
COMPARISON BETWEEN CONSTRUCTION METHODS OF UNKNOWN INPUT REDUCED ORDER OBSERVER USING PROJECTION OPERATOR APPROACH AND GENERALIZED MATRIX INVERSE
AVIJIT BANERJEE1 & GOURHARI DAS2
1,2Dept. of Electrical Engineering, Jadavpur University(PG. Control System)Kolkata, India
Abstract—In this article a detailed comparison between two construction methods of unknown input reduced order observer is presented. The methods are namely - projection operator approach and generalized matrix inverse. An experimental study reveals that the order of the observer dynamics constructed using projection operator approach is dependent on the dimension of unknown input vector. Similarities and dissimilarities between the above mentioned methods are illustrated using an example and verified with experimental result.
Keywords-Reduced order observer, Unknown Input Observer (UIO), Generalized Matrix Inverse, Das &Ghosal Observer(DGO), Projection Method Observer(PMO)
I. INTRODUCTION II. LIST OF NOTATIONS AND SYMBOLS All the symbols and notations used by two Over the past three decades many researchers have observer construction methods have been listed in paid attention to the problem of estimating the state of table I a dynamic system subjected to both known and TABLE I. unknown inputs. They have developed the reduced List of Notations and Symbols Serial and full order unknown input observers by different No Notations or Dimensio Purpose approaches[5-9]. In many application like fault Symbols n detection and isolation, cryptography, chaos 1. System matrix synchronization, unknown input observer[10-12] has 2. A Input matrix shown enormous importance and lots of work can be found in the literature. StefenHui and Stanislaw H.Zak 3. B Output coupling matrix [2, 3] developed full order and reduced order 4. C Unknown input coupling unknown input observer (UIO) using projection Matrix method (PMO) to state estimation. In [1] reduced 5. E Rank(C) Scalar order Das and Ghosal observer (DGO) [4], which uses 6. r Linearly independent generalized matrix inverse approach, is extended to column of such that construct reduced order UIO. In contrast to the well L Lg=(I-CgC) n n‐r known and well established Luenberger observer 7. Generalized Inverse of L theory, these two proposed methods does not pre Lmatrix 8. L Generalized Inverse of C n‐r n assume the observer structure and it is not necessary matrix for the system output matrix to be in specific form as 9. C Real matrix, is designed [I : 0] and the corresponding co-ordinate such that transformation is not required [13]. Both the M (I-MC) Ed=0 10. Arbitrary Design construction methods consider the system state to parameter matrix[2] for decompose into known and unknown components. H M PMO pre-assumes the unknown component as a skew 11. Arbitrary Design symmetric projection (not necessarily orthogonal) of parameter matrix for K H [1] n‐r p whole state along the subspace of available state 12. Projection matrix P=I-MC component.In contrast.DGO considers the unknown component is an orthogonal projection of known 13. p Co-ordinate transformation matrix component 14. Q State Vector In this paper first both of the methods are discussed briefly and then elaborated comparison is 15. x Estimated State Vector 1 done on the basis of observer structure and their 16. x Immeasurable State 1 performances. Numerical examples and simulation Vector used in [1] results are given for justifying the comparison. 17. h Observer State [1] 1 q n‐r 1
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 5 Comparison Between Construction Methods of Unknown Input Reduced Order observer Using Projection Operator Approach And Generalized Matrix Inverse
List of Notations and Symbols DGO). In general unknown input DGO dynamics is Serial given by the equation: No Notations or Dimensio Purpose Symbols n 18. Observer State [2] … (12) q (eqn. L23 AL of [1]) KCAL q L AL KCAL K q L AC KCAC y L B KCB u 19. Identity matrix ) The estimated states are given by, 1 … (13) (eqn. 19 20. Output Vector I of [1]) 21. y Control Input 1 Tox Lq made theC LKunknowny input ineffective in observer dynamics the observer gain K is designed by 22. u Observer Gain matrix 1 used in [1] following equation: 23. K Observer Gain matrix n‐r p used in [2] K L E … CE (14) (eqn. H I (22) L of[1])CE CE III. MATHEMATICAL PRELIMINARIES such that, . …(15) (eqn.(20) of[1]) For a system of equation … (1) where A is a B. Design of Reduced Order 0 Unknown Input given (m×n) matrix, y is a given (m×1) vector, x is an Projection Method Observer (PMO) unknown (n×1) vector. AlsoAx an y(n×m) matrix is The condition for existence of unknown input taken such that PMO is given by the following equation. ¾ …(2) Rank(C = Rank ( [pp. 432,434 of ¾ …(3) [2]) ¾ ...(4) The observer dynamics can be expressed as: ¾ …(5) are satisfied where AAsuperscript A A T indicates … (16) A AA A transpose. The matrix is called the Moore- 0 PenroseGeneralized Matrix Inverse of A and is where, ... (16.a) unique for A. Now eqn. (1) is consistent if and only if ...(6) and if eqn. (1) is consistent then its ... (16.b) general solution is given by ...(7)where I is the identity matrix of proper … (16.c) (eqn. (13) of [2]) dimension and is an arbitrary (n×1) vector (Graybill and ...(16.d) 1969, Grayville 1975). Lemma1 used by DGO Whereas the observed state vector is represented as: For an (m×n) matrix C and an (n×k) matrix L, if the linear space spanned by columns of L is equal to … (17) the linear space spanned by the columns of ( , The contribution of unknown input in observer then is equal to . So by this Lemma, proposedg by dynamics has been0 eliminated by proper design of M g I C C Das and Ghosal, 1981: ( = ... (8). such that, LL g g C C L … (18) Lemma2 used by PMOI L Eqn.(18) provides the solution of M Ias, MC E 0 If be a projection and Rank ( … (19) , then there is an invertible matrix Q, having (Pp.433 of [2]) columnsP : same as eigenvector of such that Hence forth throughout the article DGO represents … (9) (Smith, 1984, unknown input reduced order Das and Ghosal pp. 156-158 and pp. 194-195) 0 observer similarly PMO represents unknown input 00 reduced order projection method observer. IV. BRIEF CONSTRUCTION METHODS V. NUMERICAL EXAMPLES A. Design of Reduced Order Unknown Input Das & Ghosal Observer(DGO) To study the performance of the DGO and the If an LTI system subjected to unknown input is PMO, we will consider the numerical example of the described in state space form as 5th order lateral axis model of an L-1011 fixed-wing … (10) (eqn.12 aircraft [2,3] at curies flight conditions neglecting of [1]). actuator dynamics (example.2 [2]). The following x The Ax Bu E reduced orderw unknown input DGO is numerical values [2] for the system matrices have governed by the following equations and conditions. been taken for the class of aircraft model. ... (11) (eqn. 21 of [1]; condition for existence of unknown input L E L E CE CE 0 International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 6 Comparison Between Construction Methods of Unknown Input Reduced Order observer Using Projection Operator Approach And Generalized Matrix Inverse
The unknown input reduced order PMOfor the above system has been realizedin [2] by following the equations (16&17) as bellow:
A. Matlab Simulation Responses Simulation has been carried out to analyze the performance of the DGO and the PMO. Their responses for the above numerical example are given in figure (4.1-4.5).
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 7 Comparison Between Construction Methods of Unknown Input Reduced Order observer Using Projection Operator Approach And Generalized Matrix Inverse
VI. COMPARATIVE STUDY TABLE II. A. Structure wise comparison Serial Unknown Input Reduced Order Projection No Method Observer (PMO) In unknown input PMO also the system state Se variable has been partitioned into two ria Unknown Input Reduced Order Das And Ghosal components where the unknown component is a l Observer(DGO) skew sympatric projection of the state and the 1. No decomposition of state vector is given by In unknown input DGO the system state variable has … (23) been decomposed into known and unknown Here the matrix (I-MC) is a projection which is components such that the unknown component is the not necessarily orthogonal to M. orthogonal projection of known component and The observer dynamic variable is considered as described as : . Where ...(22) 2. … (25) Here the matrix L is orthogonal to the matrix . And x C y Lh The observer state variable is represented as C where, The observer dynamic equation in presence of ... (24) unknown input for PMO can be written in the Or, q form: q h Ky 3. y q K I h ... 0 The observer dynamic equation in presence of (27) unknown input for DGO is given by: In PMO the effect Eof unknown input has been nullified from observer dynamics by proper selection of M so that, 4. q L AL KCAL …(26 q ) L AL KCAL K and the solution for such M is L AC KCAC y L B KCB u given by eqn. (19) To L Eeliminate KCE the w effect of unknown input from the 0 observer dynamics, for DGO the observer gain The existing condition for the PMO can be parameter K has been designed such that, stated as: g 1) Rank ( L Ed-KCEd=0 and the solution for such K is given by 5. m r n-m (n-r). …(29) eqn. (14) 2≤ 2 ≥ 2) The pairE Rank CEneeds to be The condition for existence of the DGO is as follows: completely observable. g g g 1) L Ed-L Ed(CEd) CEd =0…(28 ) Here the order of the observer , is . 6. 2) The pair has to be completely observable. Here it can be infer that , 7. Order of PMO (n-m2) ≥ Order of DGO (n-r) Here the order of the observer is . Special case: When r=n the order of the PMO varies from n r 8. Order of DGO (n-r) ≤ Order of PMO (n-m2) zero to n depending on the number of unknown input Here the observed state variable is expressed Special case: When r=n the order of the DGO always reduces to as, zero. 9. Here the observed state variable is given by, Or, 0 Or, x Lq C LK y Here the estimation0 error is given by q 10 … (31) (eqn. (20) of [2]) x L C LK y Here ̃ The error dynamics ̃ e(t) teds to zero for Here the observer error dynamics can be expressed 11 asymptotically stable as: … (30) For the PMO it has been observed that for some specific combination of C and observer poles The L errorAL dynamics KCAL e(t) teds to zero if may not be placed arbitrarily. 12 E is asymptotically stable. L AL TheKCAL gain K consists of two components. The arbitrariness of K depends on the arbitrary parameter I 0 Here the transformation Q-1PQ=P= n-m2 H. From the properties of generalized matrix inverse 13 00 we can infer that the observer poles cannot be placed is necessary to construct the observer. arbitrarily if the matrix is of full row rank. L can be directly obtainedCE from the independent column of and no additional transformation required.
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 8 Comparison Between Construction Methods of Unknown Input Reduced Order observer Using Projection Operator Approach And Generalized Matrix Inverse
B. Coefficient-wise comparison based on observer Performanc Comparative study equations e-wise Unknown Input Unknown Input Reduced Comparing the coefficients of observer dynamic comparison Reduced Order DGO Order PMO and figure 4) equations (eqns. (12) with eqn. (16)) and output equations (eqn. (13) and eqn. (14)) we get the following observations presented in table III TABLE III. Coefficient- Comparative study wise Unknown Input Unknown Input Reduced comparison Reduced Order DGO Order PMO g VII. CONCLUSION L AL A11 where Where, -1 A11 A12 g g A =Q AQ= LL =(In-C C) 1. A21 A22 In this paper a detailed and exhaustive I 0 -1 n-m2 comparative study between the unknown input Q PQ=P= 00 reduced order DGO and the PMO based on their C1 where structures, co-efficient of observer equations and 2. CAL C=CQ= C C 1 2 performances have been done. It has come out from
L 1where the study that the order of the DGO is always less or g g K=(L Ed)(CEd) + at best equal to that of the PMO. The unknown input 3. g H(I- CEd CEd) reduced order PMO and the DGO be of same order g only when the number of unknown input to a system L AL-KCAL A -L C 4. 11 1 1 exactly matches with the number of independent output. The construction method of the DGO is LgAL-KCAL K I 0 × n-m2 m2 relatively simpler and involves less complex + g g g L AC -KCAC + computation than that for the PMO. Finally a
IQn-m2 AM0m2 × practical numerical example has been considered (L- L(Ip-CM) 1011 lateral axis aircraft model) and both the observers have been applied for the system to I 0 × 5. n-m2 m2 understand their performances. L B KCB Q B REFERENCES In-m 6. Q 2 0 m2×n-m2 [1] Avijit Banerjee, Parijat Bhowmick and Prof. Gourhari Das, L “Construction of unknown input reduced order observer g g using generalized matrix inverse and Application to Missile C +LK M=(Ed(CEd) 7. g Autopilot”, International Journal of Engineering Research +H0(Ip- CEd CEd) and Development, Vol.4, issue.2, pp.15-18,2012 [2] Stefen Hui and Stanislaw H. Zak, “Observer design for systems with unknown inputs”, Int. J. Appl. Math. Comput. It can be found from table III referring table I that, Sci., 2005, Vol. 15, No. 4, 431–446 the complexity of each term for DGO is relatively less [3] Hui S. and ˙Z ak S. H. “Low-orderunknown input observers”, and involves less computation compare to the PMO. Proc. Amer. Contr. Conf., Portland, pp. 4192–4197. C. Performance-wise comparison [4] G. Das and T.K. Ghosal, “Reduced-order observer From the numerical example of the 5th order construction by generalized matrix inverse”, International system [2] consisting offour independent outputs Journal of Control, vol. 33, no. 2, pp. 371-378, 1981. given in section V, following observation have been [5] Shao-Kung Chang, Wen-Tong Youand Pau-Lo Hsu, summarize in the table IV. “General-Structured unknown input observers”,Proceedings TABLE IV. of the American Control Conference, Baltimore, Maryland, June 1994. Performanc Comparative study e-wise Unknown Input Unknown Input Reduced [6] Shao-Kung Chang, Wen-Tong Youand Pau-Lo Hsu, “Design comparison Reduced Order DGO Order PMO of general structured observers for linear systems with
st rd unknown inputs”, J. Franklin Inst. Vol. 334B, No. 2, pp. 213 1. 1 order observer 3 order observer 232, 1997. Single pole has to 3 poles have to place [7] Yuping Guan and Mehrdad Saif, “A novel approach to the 2. placed and placed at and placed at {-3,-4,-5} design of unknown input observers”, IEEE Transactions on (-5.1439} Automatic Control, Vol. 36, No. 5, May 1991. States X1 and X4 are directly available [8] P. Kudva, N. Viswanadharn, and A. Ramakrishna, State X1 and X4 in this in the output. Hence “Observers for linear systems with unknown inputs,’’ IEEE case does not match that it matches with the Trans. Automat.Contr., vol. AC-25, pp. 113-115, 1980 3. with the observed ones corresponding right from the beginning observed state right [9] Huan-Chan Ting, Jeang-Lin Chang, Yon-Ping Chen, (pp. 439 of [2]) from the initial “Unknown input observer design using descriptor system condition. (figure 1 approach”, SICE Annual conference, pp. 2207-2213, Taipei, Taiwan, August 2010
International Conference on Electrical, Electronics Engineering 9th December 2012, Bhopal, ISBN: 978-93-82208-47-1 9 Comparison Between Construction Methods of Unknown Input Reduced Order observer Using Projection Operator Approach And Generalized Matrix Inverse
[10] Jeang-Lin Chang, Tusi-Chou Wu, “Robust dissidence [14] Victor Lovass-Nagy, Richard J. Miller and David L. Powers, attenuation with unknown input observer and sliding mode “An Introduction to the Application of the Simplest Matrix- controller” Electrical Engineering (Archiv Fur Generalized Inverse in System Science”, IEEE Transactions Elektrotechnik) ,Vol.90, no 7, pp.493-502, 2008, doi: on Circuits and Systems, Vol. CAS-25, no. 9, pp. 766-771, 10.1007/s00202-008-0099-1. Sep. 1978 [11] Mou Chen, Changsheng Jiang, “Disturbance observer based [15] F.A. Graybill, “Introduction to Matrices with Applications in robust synchronization control of uncertain chaotic system, Statistics”, Belmont, CA: Wadsworth, 1969. Nonlinear Dynamics, Vol.70, no. 4, pp. 2421-2432, 2012. [16] A. Ben-Israel and T.N.E. Greville, “Generalized Inverses, [12] Maoyin Chen, Wang Min, “Unknown input observer based Theory and Applications”, New York: Wiley, 1974 chaotic secure communication”, Physics Letters A, 372 (2008) 1595-1600, October 2007. [17] J. O’Reilly, “Observers for Linear Systems”, London, Academic Press, 1983. [13] ParijatBhowmick and Dr.Gourhari Das, “A detail Comparative study between Reduced Order Luenbergerand [18] Elbert Hendrics, Ole Jannerup and Paul Hasse Sorensen, Reduced Order Das &Ghoshal Observer and their “Linear Systems Control – Deterministic and stochastic Application” Proceedings of the International Conference in Methods”, 1st Edition, Berlin, Springer Publishers, 2008. Computer, Electronics and Electronics Engineering, pp. 154 – 161, Mumbai, March 2012, doi:10.3850/978-981-07-1847-3 P0623.