Abstract

Most broadband access networks such as passive optical networks (PONs) adopt the point-to-multipoint (P2MP) topology. One critical issue in such networks is the upstream resource management and allocation mechanism. Nonlinear predictor-based dynamic resource allocation (NLPDRA) schemes for improving the P2MP network upstream transmission efficiency have been investigated in an ad hoc manner. In this paper, we establish a general state space model to analyze the controllability and stability of the NLPDRA schemes from the P2MP network system’s point of view and propose controller design guidelines to maintain the system stability under different scenarios. Analytical results show that NLPDRA maintains the P2MP network system controllability even when the loaded network traffic changes drastically. We further prove that a P2MP network system with NLPDRA is stable by proper pole placements as the traffic changes. Finally, we provide guidelines to design a optimal compensator to achieve system accuracy.

© 2010 Optical Society of America

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References

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  1. A. Girard, FTTx PON Technology and Testing. Quebec City, Canada: EXFO Electro-Engineering Inc., 2005.
  2. IEEE Standard 802.3ah-2004.
  3. ITU-T, Gigabit-Capable Passive Optical Networks (GPON): General Characteristics, ITU-T G.984.1 Recommendation, 2003.
  4. ITU-T, Broadband Optical Access Systems Based on Passive Optical Networks (PON), ITU-T G.983.1 Recommendation, 2005.
  5. “MPCP—state of the art,” PDF presentation, 2002. Available: http://www.ieee802.org/3/efm/public/jan02/maislos_1_0102.pdf.
  6. H. Byun, J. Nho, J. Lim, “Dynamic bandwidth allocation algorithm in Ethernet passive optical networks,” Electron. Lett., vol. 39, no. 13, pp. 1001–1002, June 2003.
    [CrossRef]
  7. Y. Luo, N. Ansari, “Bandwidth allocation for multiservice access on EPONs,” IEEE Commun. Mag., vol. 43, no. 2, pp. S16–S21, Feb. 2005.
    [CrossRef]
  8. Y. Luo, N. Ansari, “Limited sharing with traffic prediction for dynamic bandwidth allocation and QoS provisioning over Ethernet passive optical networks,” J. Opt. Netw., vol. 4, no. 9, pp. 561–572, Sept. 2005.
    [CrossRef]
  9. S. Yin, Y. Luo, N. Ansari, T. Wang, “Controllability of non-linear predictor-based dynamic bandwidth allocation over EPONs,” in Proc. IEEE GLOBECOM 2007, Washington, DC, 2007, pp. 2199–2203.
  10. S. Yin, Y. Luo, N. Ansari, T. Wang, “Non-linear predictor-based dynamic bandwidth allocation over TDM-PONs: stability analysis and controller design,” in Proc. IEEE Int. Conf. on Communications (ICC 2008), Beijing, China, 2008, pp. 5186–5190.
  11. S. Haykin, Adaptive Filter Theory, 3rd ed.Prentice-Hall, 1996.
  12. S. Yin, Y. Luo, N. Ansari, T. Wang, “Bandwidth allocation over EPONs: a controllability perspective,” in Proc. IEEE GLOBCOM 2006, San Francisco, CA, 2006.
  13. Z. Bubnicki, Modern Control Theory. Berlin: Springer, 2005.
  14. C. T. Chen, Linear System Theory and Design, 3rd ed.Oxford U. Press, 1999.
  15. S. Yin, Y. Luo, N. Ansari, T. Wang, “Stability of predictor-based dynamic bandwidth allocation over EPONs,” IEEE Commun. Lett., vol. 11, no. 6, pp. 549–551, June 2007.
    [CrossRef]
  16. E. I. Jury, Inners and Stability of Dynamic Systems, 2nd ed. Malabar, FL: Krieger, 1982.
  17. J. Hellerstein, Y. Diao, S. Parekh, D. Tilbury, Feedback Control of Computing Systems. Hoboken, NJ: Wiley-Interscience, 2004.
    [CrossRef]
  18. C. V. Hollot, V. Misra, D. Towsley, W. Gong, “A control theoretic analysis of RED,” in Proc. of IEEE INFOCOM 2001, Anchorage, AK, 2001, vol. 3, pp 1510–1519.
  19. K. Y. Kim, B. S. Kim, Y. B. Choi, S. T. Ko, K. S. Kim, “Optimal rate based flow control for ABR services in ATM networks,” in Proc. of IEEE TENCON’99, Korea, 1999, vol. 1, pp. 773–776.

2007 (1)

S. Yin, Y. Luo, N. Ansari, T. Wang, “Stability of predictor-based dynamic bandwidth allocation over EPONs,” IEEE Commun. Lett., vol. 11, no. 6, pp. 549–551, June 2007.
[CrossRef]

2005 (2)

2003 (1)

H. Byun, J. Nho, J. Lim, “Dynamic bandwidth allocation algorithm in Ethernet passive optical networks,” Electron. Lett., vol. 39, no. 13, pp. 1001–1002, June 2003.
[CrossRef]

Ansari, N.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Stability of predictor-based dynamic bandwidth allocation over EPONs,” IEEE Commun. Lett., vol. 11, no. 6, pp. 549–551, June 2007.
[CrossRef]

Y. Luo, N. Ansari, “Bandwidth allocation for multiservice access on EPONs,” IEEE Commun. Mag., vol. 43, no. 2, pp. S16–S21, Feb. 2005.
[CrossRef]

Y. Luo, N. Ansari, “Limited sharing with traffic prediction for dynamic bandwidth allocation and QoS provisioning over Ethernet passive optical networks,” J. Opt. Netw., vol. 4, no. 9, pp. 561–572, Sept. 2005.
[CrossRef]

S. Yin, Y. Luo, N. Ansari, T. Wang, “Bandwidth allocation over EPONs: a controllability perspective,” in Proc. IEEE GLOBCOM 2006, San Francisco, CA, 2006.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Controllability of non-linear predictor-based dynamic bandwidth allocation over EPONs,” in Proc. IEEE GLOBECOM 2007, Washington, DC, 2007, pp. 2199–2203.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Non-linear predictor-based dynamic bandwidth allocation over TDM-PONs: stability analysis and controller design,” in Proc. IEEE Int. Conf. on Communications (ICC 2008), Beijing, China, 2008, pp. 5186–5190.

Bubnicki, Z.

Z. Bubnicki, Modern Control Theory. Berlin: Springer, 2005.

Byun, H.

H. Byun, J. Nho, J. Lim, “Dynamic bandwidth allocation algorithm in Ethernet passive optical networks,” Electron. Lett., vol. 39, no. 13, pp. 1001–1002, June 2003.
[CrossRef]

Chen, C. T.

C. T. Chen, Linear System Theory and Design, 3rd ed.Oxford U. Press, 1999.

Choi, Y. B.

K. Y. Kim, B. S. Kim, Y. B. Choi, S. T. Ko, K. S. Kim, “Optimal rate based flow control for ABR services in ATM networks,” in Proc. of IEEE TENCON’99, Korea, 1999, vol. 1, pp. 773–776.

Diao, Y.

J. Hellerstein, Y. Diao, S. Parekh, D. Tilbury, Feedback Control of Computing Systems. Hoboken, NJ: Wiley-Interscience, 2004.
[CrossRef]

Girard, A.

A. Girard, FTTx PON Technology and Testing. Quebec City, Canada: EXFO Electro-Engineering Inc., 2005.

Gong, W.

C. V. Hollot, V. Misra, D. Towsley, W. Gong, “A control theoretic analysis of RED,” in Proc. of IEEE INFOCOM 2001, Anchorage, AK, 2001, vol. 3, pp 1510–1519.

Haykin, S.

S. Haykin, Adaptive Filter Theory, 3rd ed.Prentice-Hall, 1996.

Hellerstein, J.

J. Hellerstein, Y. Diao, S. Parekh, D. Tilbury, Feedback Control of Computing Systems. Hoboken, NJ: Wiley-Interscience, 2004.
[CrossRef]

Hollot, C. V.

C. V. Hollot, V. Misra, D. Towsley, W. Gong, “A control theoretic analysis of RED,” in Proc. of IEEE INFOCOM 2001, Anchorage, AK, 2001, vol. 3, pp 1510–1519.

Jury, E. I.

E. I. Jury, Inners and Stability of Dynamic Systems, 2nd ed. Malabar, FL: Krieger, 1982.

Kim, B. S.

K. Y. Kim, B. S. Kim, Y. B. Choi, S. T. Ko, K. S. Kim, “Optimal rate based flow control for ABR services in ATM networks,” in Proc. of IEEE TENCON’99, Korea, 1999, vol. 1, pp. 773–776.

Kim, K. S.

K. Y. Kim, B. S. Kim, Y. B. Choi, S. T. Ko, K. S. Kim, “Optimal rate based flow control for ABR services in ATM networks,” in Proc. of IEEE TENCON’99, Korea, 1999, vol. 1, pp. 773–776.

Kim, K. Y.

K. Y. Kim, B. S. Kim, Y. B. Choi, S. T. Ko, K. S. Kim, “Optimal rate based flow control for ABR services in ATM networks,” in Proc. of IEEE TENCON’99, Korea, 1999, vol. 1, pp. 773–776.

Ko, S. T.

K. Y. Kim, B. S. Kim, Y. B. Choi, S. T. Ko, K. S. Kim, “Optimal rate based flow control for ABR services in ATM networks,” in Proc. of IEEE TENCON’99, Korea, 1999, vol. 1, pp. 773–776.

Lim, J.

H. Byun, J. Nho, J. Lim, “Dynamic bandwidth allocation algorithm in Ethernet passive optical networks,” Electron. Lett., vol. 39, no. 13, pp. 1001–1002, June 2003.
[CrossRef]

Luo, Y.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Stability of predictor-based dynamic bandwidth allocation over EPONs,” IEEE Commun. Lett., vol. 11, no. 6, pp. 549–551, June 2007.
[CrossRef]

Y. Luo, N. Ansari, “Bandwidth allocation for multiservice access on EPONs,” IEEE Commun. Mag., vol. 43, no. 2, pp. S16–S21, Feb. 2005.
[CrossRef]

Y. Luo, N. Ansari, “Limited sharing with traffic prediction for dynamic bandwidth allocation and QoS provisioning over Ethernet passive optical networks,” J. Opt. Netw., vol. 4, no. 9, pp. 561–572, Sept. 2005.
[CrossRef]

S. Yin, Y. Luo, N. Ansari, T. Wang, “Controllability of non-linear predictor-based dynamic bandwidth allocation over EPONs,” in Proc. IEEE GLOBECOM 2007, Washington, DC, 2007, pp. 2199–2203.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Non-linear predictor-based dynamic bandwidth allocation over TDM-PONs: stability analysis and controller design,” in Proc. IEEE Int. Conf. on Communications (ICC 2008), Beijing, China, 2008, pp. 5186–5190.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Bandwidth allocation over EPONs: a controllability perspective,” in Proc. IEEE GLOBCOM 2006, San Francisco, CA, 2006.

Misra, V.

C. V. Hollot, V. Misra, D. Towsley, W. Gong, “A control theoretic analysis of RED,” in Proc. of IEEE INFOCOM 2001, Anchorage, AK, 2001, vol. 3, pp 1510–1519.

Nho, J.

H. Byun, J. Nho, J. Lim, “Dynamic bandwidth allocation algorithm in Ethernet passive optical networks,” Electron. Lett., vol. 39, no. 13, pp. 1001–1002, June 2003.
[CrossRef]

Parekh, S.

J. Hellerstein, Y. Diao, S. Parekh, D. Tilbury, Feedback Control of Computing Systems. Hoboken, NJ: Wiley-Interscience, 2004.
[CrossRef]

Tilbury, D.

J. Hellerstein, Y. Diao, S. Parekh, D. Tilbury, Feedback Control of Computing Systems. Hoboken, NJ: Wiley-Interscience, 2004.
[CrossRef]

Towsley, D.

C. V. Hollot, V. Misra, D. Towsley, W. Gong, “A control theoretic analysis of RED,” in Proc. of IEEE INFOCOM 2001, Anchorage, AK, 2001, vol. 3, pp 1510–1519.

Wang, T.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Stability of predictor-based dynamic bandwidth allocation over EPONs,” IEEE Commun. Lett., vol. 11, no. 6, pp. 549–551, June 2007.
[CrossRef]

S. Yin, Y. Luo, N. Ansari, T. Wang, “Bandwidth allocation over EPONs: a controllability perspective,” in Proc. IEEE GLOBCOM 2006, San Francisco, CA, 2006.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Controllability of non-linear predictor-based dynamic bandwidth allocation over EPONs,” in Proc. IEEE GLOBECOM 2007, Washington, DC, 2007, pp. 2199–2203.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Non-linear predictor-based dynamic bandwidth allocation over TDM-PONs: stability analysis and controller design,” in Proc. IEEE Int. Conf. on Communications (ICC 2008), Beijing, China, 2008, pp. 5186–5190.

Yin, S.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Stability of predictor-based dynamic bandwidth allocation over EPONs,” IEEE Commun. Lett., vol. 11, no. 6, pp. 549–551, June 2007.
[CrossRef]

S. Yin, Y. Luo, N. Ansari, T. Wang, “Bandwidth allocation over EPONs: a controllability perspective,” in Proc. IEEE GLOBCOM 2006, San Francisco, CA, 2006.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Non-linear predictor-based dynamic bandwidth allocation over TDM-PONs: stability analysis and controller design,” in Proc. IEEE Int. Conf. on Communications (ICC 2008), Beijing, China, 2008, pp. 5186–5190.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Controllability of non-linear predictor-based dynamic bandwidth allocation over EPONs,” in Proc. IEEE GLOBECOM 2007, Washington, DC, 2007, pp. 2199–2203.

Electron. Lett. (1)

H. Byun, J. Nho, J. Lim, “Dynamic bandwidth allocation algorithm in Ethernet passive optical networks,” Electron. Lett., vol. 39, no. 13, pp. 1001–1002, June 2003.
[CrossRef]

IEEE Commun. Lett. (1)

S. Yin, Y. Luo, N. Ansari, T. Wang, “Stability of predictor-based dynamic bandwidth allocation over EPONs,” IEEE Commun. Lett., vol. 11, no. 6, pp. 549–551, June 2007.
[CrossRef]

IEEE Commun. Mag. (1)

Y. Luo, N. Ansari, “Bandwidth allocation for multiservice access on EPONs,” IEEE Commun. Mag., vol. 43, no. 2, pp. S16–S21, Feb. 2005.
[CrossRef]

J. Opt. Netw. (1)

Other (15)

S. Yin, Y. Luo, N. Ansari, T. Wang, “Controllability of non-linear predictor-based dynamic bandwidth allocation over EPONs,” in Proc. IEEE GLOBECOM 2007, Washington, DC, 2007, pp. 2199–2203.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Non-linear predictor-based dynamic bandwidth allocation over TDM-PONs: stability analysis and controller design,” in Proc. IEEE Int. Conf. on Communications (ICC 2008), Beijing, China, 2008, pp. 5186–5190.

S. Haykin, Adaptive Filter Theory, 3rd ed.Prentice-Hall, 1996.

S. Yin, Y. Luo, N. Ansari, T. Wang, “Bandwidth allocation over EPONs: a controllability perspective,” in Proc. IEEE GLOBCOM 2006, San Francisco, CA, 2006.

Z. Bubnicki, Modern Control Theory. Berlin: Springer, 2005.

C. T. Chen, Linear System Theory and Design, 3rd ed.Oxford U. Press, 1999.

E. I. Jury, Inners and Stability of Dynamic Systems, 2nd ed. Malabar, FL: Krieger, 1982.

J. Hellerstein, Y. Diao, S. Parekh, D. Tilbury, Feedback Control of Computing Systems. Hoboken, NJ: Wiley-Interscience, 2004.
[CrossRef]

C. V. Hollot, V. Misra, D. Towsley, W. Gong, “A control theoretic analysis of RED,” in Proc. of IEEE INFOCOM 2001, Anchorage, AK, 2001, vol. 3, pp 1510–1519.

K. Y. Kim, B. S. Kim, Y. B. Choi, S. T. Ko, K. S. Kim, “Optimal rate based flow control for ABR services in ATM networks,” in Proc. of IEEE TENCON’99, Korea, 1999, vol. 1, pp. 773–776.

A. Girard, FTTx PON Technology and Testing. Quebec City, Canada: EXFO Electro-Engineering Inc., 2005.

IEEE Standard 802.3ah-2004.

ITU-T, Gigabit-Capable Passive Optical Networks (GPON): General Characteristics, ITU-T G.984.1 Recommendation, 2003.

ITU-T, Broadband Optical Access Systems Based on Passive Optical Networks (PON), ITU-T G.983.1 Recommendation, 2005.

“MPCP—state of the art,” PDF presentation, 2002. Available: http://www.ieee802.org/3/efm/public/jan02/maislos_1_0102.pdf.

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Figures (3)

Fig. 1
Fig. 1

Ethernet passive optical network.

Fig. 2
Fig. 2

Dynamic resource allocation in P2MP networks.

Fig. 3
Fig. 3

Controller design to meet transient performance objectives.

Equations (106)

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R i ( n + 1 ) = R i ( n ) + λ i ( n ) d i ( n + 1 ) .
d i ( n + 1 ) = min { G i ( n + 1 ) , R i ( n ) + λ i ( n ) } .
G i ( n + 1 ) = min { G i r ( n + 1 ) , G i max } .
G i r ( n + 1 ) = R i ( n ) + λ ̂ i ( n ) ,
λ ̂ i ( n + 1 ) = α i ( n ) λ i ( n ) ,
α i ( n + 1 ) = α i ( n ) + τ × e i ( n ) λ i ( n ) ,
e i ( n ) = λ i ( n ) λ ̂ i ( n ) .
α i ( n + 1 ) = α i ( n ) + τ τ × α i ( n ) × λ i ( n ) λ i ( n ) .
α i ( n + 1 ) = α i ( n ) .
x i ( n + 1 ) = A x i ( n ) + B u i ( n ) ,
d i ( n ) = min { G i r ( n ) , R i ( n 1 ) + λ i ( n 1 ) , G i max } .
G i r ( n + 1 ) = R i ( n ) + λ ̂ i ( n ) = R i ( n ) + α i ( n ) λ i ( n ) ,
R i ( n + 1 ) = R i ( n ) + λ i ( n ) G i r ( n + 1 ) .
R i ( n + 1 ) = λ i ( n ) α i ( n ) λ i ( n ) .
δ G i r ( n + 1 ) = δ R i ( n ) + λ i 0 δ α i ( n ) + α i 0 δ λ i ( n ) ,
δ R i ( n + 1 ) = δ λ i ( n ) λ i 0 δ α i ( n ) α i 0 δ λ i ( n ) ,
δ α i ( n + 1 ) = δ α i ( n ) τ λ i 0 λ i 0 δ α i ( n ) + τ α i 0 λ i 0 λ i 0 2 δ λ i ( n ) τ α i 0 λ i 0 δ λ i ( n ) ,
δ α i ( n + 1 ) = δ α i ( n ) .
δ x i ( n + 1 ) = A 1 δ x i ( n ) + B 1 δ u i ( n ) ,
A 1 = [ 0 1 0 λ i 0 0 0 0 λ i 0 0 0 1 τ λ i 0 λ i 0 0 0 1 0 ] , B 1 = [ 0 α i 0 1 α i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 ] .
U = [ B 1 A 1 B 1 ] = [ 0 α i 0 1 α i 0 1 α i 0 0 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 ] .
| U | = | 0 α i 0 1 α i 0 1 α i 0 0 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 | = τ 2 α i 0 2 ( λ i 0 α i 0 λ i 0 ) 2 λ i 0 4 ,
R i ( n + 1 ) = R i ( n ) + λ i ( n ) G i r ( n + 1 ) .
G i r ( n + 1 ) = α i ( n ) λ i ( n ) .
R i ( n + 1 ) = R i ( n ) + λ i ( n ) α i ( n ) λ i ( n ) .
δ G i r ( n + 1 ) = λ i 0 δ α i ( n ) + α i 0 δ λ i ( n ) ,
δ R i ( n + 1 ) = δ R i ( n ) + δ λ i ( n ) λ i 0 δ α i ( n ) α i 0 δ λ i ( n ) .
δ x i ( n + 1 ) = A 2 δ x i ( n ) + B 2 δ u i ( n ) ,
A 2 = [ 0 0 0 λ i 0 0 1 0 λ i 0 0 0 1 τ λ i 0 λ i 0 0 0 1 0 ] , B 2 = [ 0 α i 0 1 α i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 ] .
U = [ B 2 A 2 B 2 ] = [ 0 α i 0 0 0 1 α i 0 1 α i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 τ α i 0 λ i 0 λ 2 i 0 τ α i 0 λ i 0 ] .
| U | = | 0 α i 0 0 0 1 α i 0 1 α i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 τ α i 0 λ i 0 λ 2 i 0 τ α i 0 λ i 0 | = τ 2 α i 0 3 λ i 0 ( α i 0 λ i 0 λ i 0 ) λ i 0 4 .
G i r ( n + 1 ) = R i ( n ) + α i ( n ) λ i ( n ) ,
R i ( n + 1 ) = R i ( n ) G i max + λ i ( n ) .
δ G i r ( n + 1 ) = δ R i ( n ) + λ i 0 δ α i ( n ) + α i 0 δ λ i ( n ) ,
δ R i ( n + 1 ) = δ R i ( n ) + δ λ i ( n ) .
δ x i ( n + 1 ) = A 3 δ x i ( n ) + B 3 δ u i ( n ) ,
A 3 = [ 0 1 0 λ i 0 0 1 0 0 0 0 1 τ λ i 0 λ i 0 0 0 1 0 ] , B 3 = [ 0 α i 0 1 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 ] .
U = [ B 3 A 3 B 3 ] = [ 0 α i 0 1 0 1 0 1 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 ] .
| U | = | 0 α i 0 1 0 1 0 1 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 | = τ 2 α i 0 2 λ i 0 2 + τ 2 α i 0 3 λ i 0 λ i 0 3 .
u i ( n ) = K i x i ( n ) + F i r i ( n ) ,
x i ( n + 1 ) = ( A i B i K i ) x i ( n ) + B i F i r i ( n ) .
u i ( n ) = K i x i ( n ) ,
x i ( n + 1 ) = ( A i B i K i ) x i ( n ) .
δ x i ( n + 1 ) = A 1 δ x i ( n ) + B 1 δ u i ( n ) ,
A 1 = [ 0 1 0 λ i 0 0 0 0 λ i 0 0 0 1 τ λ i 0 λ i 0 0 0 1 0 ] , B 1 = [ 0 α i 0 1 α i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 ]
det ( z I A 1 ) = D ( z ) = z 4 z 3 + τ λ 0 λ 0 z 2 .
Rule 1 : a 0 2 a 4 2 a 0 a 3 + a 1 a 4 < 0 ,
Rule 2 : a 0 2 a 4 2 + a 0 a 3 a 1 a 4 < 0 ,
Rule 3 : a 0 3 + 2 a 0 a 2 a 4 + a 1 a 3 a 4 a 0 a 4 2 a 2 a 4 2 a 0 a 3 2 a 0 2 a 4 a 0 2 a 2 a 1 2 a 4 + a 4 3 + a 0 a 1 a 3 > 0 ,
Rule 4 : D ( 1 ) > 0 ,
Rule 5 : D ( 1 ) > 0.
δ x i ( n + 1 ) = A 2 δ x i ( n ) + B 2 δ u i ( n ) ,
A 2 = [ 0 0 0 λ i 0 0 1 0 λ i 0 0 0 1 τ λ i 0 λ i 0 0 0 1 0 ] , B 2 = [ 0 α i 0 1 α i 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 ] .
u i ( n ) = K 1 x i ( n ) ,
K 1 = [ k 11 k 12 k 13 k 14 k 21 k 22 k 23 k 24 ] .
{ L 0 2 L 0 L 3 + L 1 1 < 0 L 0 2 + L 0 L 3 L 1 1 < 0 L 0 3 + 2 L 0 L 2 + L 1 L 3 L 0 L 2 L 0 L 3 2 L 0 2 L 0 2 L 2 L 1 2 + L 0 L 1 L 3 + 1 > 0 1 + L 3 + L 2 + L 1 + L 0 > 0 1 L 3 + L 2 L 1 + L 0 > 0 } ,
det ( z I A 2 ) = D ( z ) = z 4 2 z 3 + ( 1 + τ λ 0 λ 0 ) z 2 τ λ 0 λ 0 z .
det [ ( z I ) ( A 2 B 2 K 1 ) ] = D ( z ) = z 4 + L 3 z 3 + L 2 z 2 + L 1 z + L 0 .
L 0 = ( τ α 0 λ 0 λ 0 2 k 12 τ α 0 λ 0 k 22 ) [ ( λ 0 α 0 k 24 + k 14 ) α 0 k 21 + ( k 11 α 0 k 21 ) ( α 0 k 24 λ 0 ) ] α 0 k 22 ( k 11 α 0 k 21 ) ( τ λ 0 λ 0 τ λ 0 λ 0 k 24 + τ α 0 λ 0 λ 0 2 k 14 ) + ( τ α 0 λ 0 λ 0 2 k 11 τ α 0 λ 0 k 21 ) [ α 0 k 22 ( λ 0 α 0 k 24 + k 14 ) ( α 0 k 22 1 k 12 ) ( α 0 k 24 λ 0 ) ] ,
L 1 = ( τ α 0 λ 0 λ 0 2 k 11 τ α 0 λ 0 k 21 ) [ α 0 k 22 ( k 13 α 0 k 23 ) α 0 k 23 ( α 0 k 22 k 12 1 ) ] ( τ α 0 λ 0 λ 0 2 k 12 τ α 0 λ 0 k 22 ) × [ α 0 k 21 ( k 13 α 0 k 23 ) + α 0 k 23 ( k 11 α 0 k 21 ) ] + ( τ α 0 λ 0 λ 0 2 k 13 τ α 0 λ 0 k 23 1 ) [ α 0 k 21 ( α 0 k 22 k 12 1 ) α 0 k 22 ( k 11 α 0 k 21 ) ] + ( τ λ 0 λ 0 τ α 0 λ 0 k 22 + τ α 0 λ 0 λ 0 2 k 14 ) ( 2 α 0 k 22 k 12 ) ( λ 0 α 0 k 24 + k 14 ) × ( τ α 0 λ 0 λ 0 2 k 12 τ α 0 λ 0 k 22 ) ( τ α 0 λ 0 λ 0 2 k 11 τ α 0 λ 0 k 21 ) ( α 0 k 24 λ 0 ) ,
L 2 = α 0 k 21 ( α 0 k 22 k 12 1 ) α 0 k 22 ( k 11 α 0 k 21 ) + ( τ α 0 λ 0 λ 0 2 k 13 τ α 0 λ 0 k 23 1 ) × ( α 0 k 21 + α 0 k 22 k 12 1 ) α 0 k 23 ( τ α 0 λ 0 λ 0 2 k 11 τ α 0 λ 0 k 21 ) ( k 13 α 0 k 23 ) ( τ α 0 λ 0 λ 0 2 k 12 τ α 0 λ 0 k 22 ) + τ λ 0 λ 0 τ α 0 λ 0 k 24 + τ α 0 λ 0 λ 0 2 k 14 ,
L 3 = α 0 k 21 + α 0 k 22 k 12 + τ α 0 λ 0 λ 0 2 k 13 τ α 0 λ 0 k 23 2 ,
L 4 = 1.
{ L 0 2 L 0 L 3 + L 1 1 < 0 L 0 2 + L 0 L 3 L 1 1 < 0 L 0 3 + 2 L 0 L 2 + L 1 L 3 L 0 L 2 L 0 L 3 2 L 0 2 L 0 2 L 2 L 1 2 + L 0 L 1 L 3 + 1 > 0 1 + L 3 + L 2 + L 1 + L 0 > 0 1 L 3 + L 2 L 1 + L 0 > 0 } .
δ x i ( n + 1 ) = A 3 δ x i ( n ) + B 3 δ u i ( n ) ,
A 3 = [ 0 1 0 λ i 0 0 1 0 0 0 0 1 τ λ i 0 λ i 0 0 0 1 0 ] , B 3 = [ 0 α i 0 1 0 τ α i 0 λ i 0 λ i 0 2 τ α i 0 λ i 0 0 0 ] .
u i ( n ) = K 2 x i ( n ) ,
K 2 = [ p 11 p 12 p 13 p 14 p 21 p 22 p 23 p 24 ] .
{ M 0 2 M 0 M 3 + M 1 1 < 0 M 0 2 + M 0 M 3 M 1 1 < 0 M 0 3 + 2 M 0 M 2 + M 1 M 3 M 0 M 2 M 0 M 3 2 M 0 2 M 0 2 M 2 M 1 2 + M 0 M 1 M 3 + 1 > 0 1 + M 3 + M 2 + M 1 + M 0 > 0 1 M 3 + M 2 M 1 + M 0 > 0 } ,
M 0 = ( τ α 0 λ 0 λ 0 2 p 11 τ α 0 λ 0 p 21 ) [ p 14 ( α 0 p 22 1 ) ( α 0 p 24 λ 0 ) ( p 12 1 ) ] + ( τ α 0 λ 0 λ 0 2 p 12 τ α 0 λ 0 p 22 ) [ α 0 p 14 p 21 + p 11 ( α 0 p 24 λ 0 ) ] + ( τ α 0 λ 0 λ 0 2 p 14 τ α 0 λ 0 p 24 + τ λ 0 λ 0 ) [ α 0 p 21 ( p 12 1 ) p 11 ( α 0 p 22 1 ) ] ,
M 1 = p 13 ( α 0 p 22 1 ) ( α 0 p 24 λ 0 ) ( τ α 0 λ 0 λ 0 2 p 11 τ α 0 λ 0 p 21 ) × [ α 0 p 23 ( p 12 1 ) + α 0 p 24 λ 0 ] + ( τ α 0 λ 0 λ 0 2 p 12 τ α 0 λ 0 p 22 ) ( α 0 p 11 p 23 α 0 p 13 p 21 p 14 ) ,
M 2 = ( τ α 0 λ 0 λ 0 2 p 13 τ α 0 λ 0 p 23 1 ) ( α 0 p 21 + p 12 1 ) p 11 ( α 0 p 22 1 ) α 0 p 23 ( τ α 0 λ 0 λ 0 2 p 11 τ α 0 λ 0 p 21 ) p 13 ( τ α 0 λ 0 2 p 12 τ α 0 λ 0 p 22 ) + ( τ α 0 λ 0 λ 0 2 p 14 τ α 0 λ 0 p 24 + τ λ 0 λ 0 ) ,
M 3 = α 0 p 21 + p 12 + τ α 0 λ 0 λ 0 2 p 13 τ α 0 λ 0 p 23 2 ,
M 4 = 1.
X i ( n + 1 ) = A X i ( n ) + B U i ( n ) ,
Y i ( n ) = C X i ( n ) .
U i ( n ) = K i X i ( n ) + F i r ,
F i = [ K i 1 ] [ A i I B i C i 0 ] 1 [ 0 1 ] ,
[ A i I B i C i 0 ]
f 1 ( R i , α i , λ i ) = R i + α i λ i ,
f 2 ( λ i , α i , λ i ) = λ i α i λ i ,
f 3 ( α i , α i , λ i , λ i ) = α i + τ τ × α i × λ i λ i ,
f 4 ( α i ) = α i .
f 1 R i = 1 , f 1 α i = λ i 0 , f 1 λ i = α i 0 ,
f 2 λ i = 1 , f 2 α i = λ i 0 , f 2 λ i = α i 0 ,
f 3 α i = 1 , f 3 α i = τ λ i 0 λ i 0 , f 3 λ i = τ α i 0 λ i 0 λ i 0 2 ,
f 3 λ i = τ α i 0 λ i 0 ,
f 4 α i = 1 .
f 5 ( α i , λ i ) = α i λ i ,
f 6 ( G i r , α i , λ i , λ i ) = G i r + λ i + ( α i 1 ) λ i .
f 5 α i = λ i 0 , f 5 λ i = α i 0 ,
f 6 G i r = 1 , f 6 α i = λ i 0 , f 6 λ i = 1 ,
f 6 λ i = ( α i 0 1 ) .
f 7 ( R i , α i , λ i ) = R i + α i λ i ,
f 8 ( R i , λ i ) = R i G i max + λ i .
f 7 R i = 1 , f 7 α i = λ i 0 , f 7 λ i = α i 0 ,
f 8 R i = 1 , f 8 λ i = 1 .
U i ( n ) = K i ( X i ( n ) X i s s ) + U i s s .
U i ( n ) = K i X i ( n ) + [ K i 1 ] [ X i s s U i s s ] .
X i s s = A X i s s + B U i s s ,
Y i s s = C X i s s ,
Y i s s = r .
( A I ) X i s s + B U i s s = 0 C X i s s = r , i.e . , [ A I B C 0 ] [ X i s s U i s s ] = [ 0 r ] .
[ X i s s U i s s ] = [ A I B C 0 ] 1 [ 0 r ] ,
U i ( n ) = K i X i ( n ) + [ K i 1 ] [ A I B C 0 ] 1 [ 0 1 ] r .
F i = [ K i 1 ] [ A I B C 0 ] 1 [ 0 1 ] .