Abstract

A novel modulation index stabilization technique for tracking the phase modulation index of integrated optic phase modulator (IOPM) is proposed to improve temperature performance of the resonant integrated optic gyro (RIOG). The influence mechanism of IOPM’s modulation index fluctuation on the RIOG, especially the angular velocity tracking loop of RIOG, is investigated. A Mach-Zehnder Interferometer (MZI) is ingeniously added into the conventional RIOG structure for detecting the modulation index fluctuation. For synchronously demodulating the output of RIOG and the gain of IOPM in real time, a novel six-state wave modulation scheme is also proposed. Moreover, considering the disturbance and nonlinearity, the system model of IOPM’s modulation index controller is established and designed to guarantee high speed and precision tracking. The experimental results demonstrate that the proposed modulation index stabilization technique can in real time demodulate and control the modulation index of IOPM. The gyro scale factor stability of RIOG resulting from the IOPM’s modulation index fluctuation is decreased to 189.26 ppm within −40°C to +60°C, which, to the best of our knowledge, is the first time stabilizing the modulation index of IOPM in RIOG at full temperature.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (2)

2017 (1)

2016 (2)

2015 (3)

2014 (1)

2013 (2)

2012 (1)

2011 (2)

2010 (1)

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

2008 (1)

D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008).
[Crossref]

2006 (1)

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

2000 (2)

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[Crossref]

K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18(1), 66–72 (2000).
[Crossref]

1997 (1)

M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
[Crossref]

1990 (2)

T. Fujiwara, T. Kawazoe, and H. Mori, “Temperature dependence of the half-wave voltage in Ti:LiNbO3 waveguide devices at 0.83μm,” Jpn. J. Appl. Phys. 29(Part 2, No. 12), L2229 (1990).
[Crossref]

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

1988 (1)

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

1985 (1)

H. C. Lefevre, P. Graindorge, H. J. Arditty, S. Vatoux, and M. Papuchon, “Double closed-loop hybrid fiber gyroscope using digital phase ramp,” Opt. Fiber Sensors PDS7, 1–5 (1985).
[Crossref]

1980 (1)

Abarkan, M.

Arditti, H. J.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Arditty, H. J.

H. C. Lefevre, P. Graindorge, H. J. Arditty, S. Vatoux, and M. Papuchon, “Double closed-loop hybrid fiber gyroscope using digital phase ramp,” Opt. Fiber Sensors PDS7, 1–5 (1985).
[Crossref]

Armenise, M. N.

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Campanella, C. E.

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Chen, Q. Y.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

Chrostowski, L.

Ciminelli, C.

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Cretu, E.

Dell’Olio, F.

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Demma, N. A.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

Ding, C.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Feng, L.

H. Li, L. Liu, Z. Lin, Q. Wang, X. Wang, and L. Feng, “Double closed-loop control of integrated optical resonance gyroscope with mean-square exponential stability,” Opt. Express 26(2), 1145–1160 (2018).
[Crossref] [PubMed]

Q. Wang, L. Feng, H. Li, X. Wang, Y. Jia, and D. Liu, “Enhanced differential detection technique for the resonator integrated optic gyro,” Opt. Lett. 43(12), 2941–2944 (2018).
[Crossref] [PubMed]

J. Wang, L. Feng, Q. Wang, H. Jiao, and X. Wang, “Suppression of backreflection error in resonator integrated optic gyro by the phase difference traversal method,” Opt. Lett. 41(7), 1586–1589 (2016).
[Crossref] [PubMed]

J. Wang, L. Feng, Q. Wang, X. Wang, and H. Jiao, “Reduction of angle random walk by in-phase triangular phase modulation technique for resonator integrated optic gyro,” Opt. Express 24(5), 5463–5468 (2016).
[Crossref] [PubMed]

Y. Zhi, L. Feng, J. Wang, and Y. Tang, “Reduction of backscattering noise in a resonator integrated optic gyro by double triangular phase modulation,” Appl. Opt. 54(1), 114–122 (2015).
[Crossref] [PubMed]

J. Wang, L. Feng, Y. Tang, and Y. Zhi, “Resonator integrated optic gyro employing trapezoidal phase modulation technique,” Opt. Lett. 40(2), 155–158 (2015).
[Crossref] [PubMed]

L. Feng, J. Wang, Y. Zhi, Y. Tang, Q. Wang, H. Li, and W. Wang, “Transmissive resonator optic gyro based on silica waveguide ring resonator,” Opt. Express 22(22), 27565–27575 (2014).
[Crossref] [PubMed]

L. Feng, M. Lei, H. Liu, Y. Zhi, and J. Wang, “Suppression of backreflection noise in a resonator integrated optic gyro by hybrid phase-modulation technology,” Appl. Opt. 52(8), 1668–1675 (2013).
[Crossref] [PubMed]

Fontana, M.

Fujiwara, T.

T. Fujiwara, T. Kawazoe, and H. Mori, “Temperature dependence of the half-wave voltage in Ti:LiNbO3 waveguide devices at 0.83μm,” Jpn. J. Appl. Phys. 29(Part 2, No. 12), L2229 (1990).
[Crossref]

Gaiffe, T.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Gille, S.

Graindorge, P.

H. C. Lefevre, P. Graindorge, H. J. Arditty, S. Vatoux, and M. Papuchon, “Double closed-loop hybrid fiber gyroscope using digital phase ramp,” Opt. Fiber Sensors PDS7, 1–5 (1985).
[Crossref]

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Guilbert, L.

Guillén-Torres, M. Á.

Haavisto, J.

Harumoto, M.

M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
[Crossref]

He, Z.

Hotate, K.

Jaeger, N.

Jia, Y.

Jiao, H.

Jin, Z.

Kawazoe, T.

T. Fujiwara, T. Kawazoe, and H. Mori, “Temperature dependence of the half-wave voltage in Ti:LiNbO3 waveguide devices at 0.83μm,” Jpn. J. Appl. Phys. 29(Part 2, No. 12), L2229 (1990).
[Crossref]

Kyuma, K.

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

Le Boudec, G.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Lefevre, H. C.

H. C. Lefevre, P. Graindorge, H. J. Arditty, S. Vatoux, and M. Papuchon, “Double closed-loop hybrid fiber gyroscope using digital phase ramp,” Opt. Fiber Sensors PDS7, 1–5 (1985).
[Crossref]

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Lefèvre, H. C.

H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

Lei, M.

Li, H.

Lin, Z.

Liu, D.

Liu, H.

Liu, L.

Ma, H.

Martin, P.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Maseeh, F.

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[Crossref]

Miesel, K. A.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

Monovoukas, C.

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[Crossref]

Mori, H.

T. Fujiwara, T. Kawazoe, and H. Mori, “Temperature dependence of the half-wave voltage in Ti:LiNbO3 waveguide devices at 0.83μm,” Jpn. J. Appl. Phys. 29(Part 2, No. 12), L2229 (1990).
[Crossref]

Morisse, J.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Pajer, G. A.

Papuchon, M.

H. C. Lefevre, P. Graindorge, H. J. Arditty, S. Vatoux, and M. Papuchon, “Double closed-loop hybrid fiber gyroscope using digital phase ramp,” Opt. Fiber Sensors PDS7, 1–5 (1985).
[Crossref]

Rouse, G. F.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

Salvestrini, J. P.

Sanders, G. A.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

Simonpiétri, P.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Strandjord, L. K.

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

Suzuki, K.

Swiecki, A. K.

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[Crossref]

Tai, S.

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

Takahashi, M.

M. Takahashi, S. Tai, and K. Kyuma, “Effect of reflections on the drift characteristics of a fiber-optic passive ring-resonator gyroscope,” J. Lightwave Technol. 8(5), 811–816 (1990).
[Crossref]

Takiguchi, K.

Tang, Y.

Taufflieb, E. M.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Vatoux, S.

H. C. Lefevre, P. Graindorge, H. J. Arditty, S. Vatoux, and M. Papuchon, “Double closed-loop hybrid fiber gyroscope using digital phase ramp,” Opt. Fiber Sensors PDS7, 1–5 (1985).
[Crossref]

Vivenot, P.

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

Wang, J.

Wang, L.

Wang, Q.

Wang, W.

Wang, X.

Ying, D.

D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008).
[Crossref]

Yu, X.

Zhang, J.

Zhang, X.

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Zhi, Y.

Adv. Opt. Photonics (1)

C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Photonic technologies for angular velocity sensing,” Adv. Opt. Photonics 2(3), 370–404 (2010).
[Crossref]

Appl. Opt. (2)

J. Lightwave Technol. (6)

Jpn. J. Appl. Phys. (1)

T. Fujiwara, T. Kawazoe, and H. Mori, “Temperature dependence of the half-wave voltage in Ti:LiNbO3 waveguide devices at 0.83μm,” Jpn. J. Appl. Phys. 29(Part 2, No. 12), L2229 (1990).
[Crossref]

Opt. Commun. (1)

D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281(4), 580–586 (2008).
[Crossref]

Opt. Eng. (1)

H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45(8), 080506 (2006).
[Crossref]

Opt. Express (5)

Opt. Fiber Sensors (1)

H. C. Lefevre, P. Graindorge, H. J. Arditty, S. Vatoux, and M. Papuchon, “Double closed-loop hybrid fiber gyroscope using digital phase ramp,” Opt. Fiber Sensors PDS7, 1–5 (1985).
[Crossref]

Opt. Laser Technol. (1)

M. Harumoto and K. Hotate, “Resonator fibre-optic gyro using digital serrodyne modulation – fundamental experiments and evaluation of the limitations,” Opt. Laser Technol. 29(2), xii (1997).
[Crossref]

Opt. Lett. (4)

Proc. SPIE (2)

G. A. Sanders, G. F. Rouse, L. K. Strandjord, N. A. Demma, K. A. Miesel, and Q. Y. Chen, “Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5-μm wavelength,” Proc. SPIE 985, 201–210 (1988).

C. Monovoukas, A. K. Swiecki, and F. Maseeh, “Integrated optical gyroscopes offering low cost, small size and vibration immunity,” Proc. SPIE 3936, 293–300 (2000).
[Crossref]

Other (4)

H. C. Lefevre, P. Martin, T. Gaiffe, P. Graindorge, G. Le Boudec, J. Morisse, P. Simonpiétri, E. M. Taufflieb, P. Vivenot, and H. J. Arditti, “Latest advances in fiber-optic gyroscope technology at Photonetics,” Fiber Optic and Laser Sensors XII2292, (1994).
[Crossref]

H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1987).

IEEE Standard Specification Format Guide and Test Procedure for Single-Axis Interferometric Fiber Optic Gyros, IEEE Standard 952, (1997).

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Figures (7)

Fig. 1
Fig. 1 The simplified block diagram of RIOG’s angular velocity tracking loop. The KFC refers the gain of forward channel which includes the gain of photodetectors and lock-in amplifier. KSW refers to the gain of sawtooth generator
Fig. 2
Fig. 2 The schematic diagram of RIOG with MIST. ISO, isolator; IOPM, integrated optic phase modulator; PD, photodetector; WRR, waveguide ring resonator; DEM, demodulator; C1, 99:1 optical coupler; C2, 99:1 optical coupler; C3, 50:50 optical coupler.
Fig. 3
Fig. 3 The simulation results of the demodulation outputs ΔI and ΔId at the different phase difference between the MZI’s two arms Δφ when (a) the modulation indexes are not more than 1.00 and (b) the modulation indexes are not less than 1.00.
Fig. 4
Fig. 4 The principle diagram of closed-loop tracking of IOPM’s modulation index.
Fig. 5
Fig. 5 The demodulation values ∆Pd and ∆P, and the demodulation result IDem when the IOPM’s modulation index is around (a) less than 1, and (b) more than 1.
Fig. 6
Fig. 6 The measurement and simulation results of the demodulation result IDem with different modulation indexes.
Fig. 7
Fig. 7 The measurement results of modulation indexes at full temperature (−40°C to + 60°C) experiment (a) without the MICCS and (b) with the MICCS.

Tables (1)

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Table 1 Key parameters of the feedback channel

Equations (17)

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Δ f Ω = D n e λ Ω,
G out = 2 V π Δ f Ω K SW V pp ,
ε r = V pp 2 V π .
K= D K SW ε r n e λ .
{ I 1 = α L k C I 0 [ 1+cos(Δφ) ] 0t< τ 2 /4 I 2 = α L k C I 0 [ 1+cos( π ε r /2 +Δφ ) ] τ 2 /4 t< τ 2 /2 I 3 = α L k C I 0 [ 1+cos( 3π ε r /2 +Δφ ) ] τ 2 /2 t< 3 τ 2 /4 I 4 = α L k C I 0 [ 1+cos( 2π ε r +Δφ ) ] 3 τ 2 /4 t< τ 2
Δ I 2π = I 1 I 4 = α L k C I 0 [ cos(Δφ)cos( 2π ε r +Δφ ) ] =2 α L k C I 0 sin( π ε r )sin( π ε r +Δφ )
Δ I d = I 2 I 3 = α L k C I 0 [ cos( π ε r /2 +Δφ )cos( 3π ε r /2 +Δφ ) ] =2 α L k C I 0 sin( π ε r /2 )sin( π ε r +Δφ )
I Dem = Δ I 2π Δ I d = 2 α L k C I 0 sin( π ε r )sin( π ε r +Δφ ) 2 α L k C I 0 sin( π ε r /2 )sin( π ε r +Δφ ) =2cos( π ε r /2 )
I Dem =2sin( π ε e /2 )
x(k+1)=Ax(k)Bkf( K c x(k))+Cw(k)
[ P+I K c T 0 A T G 2I 0 k B T G γ 2 C T G PG G T ]<0
K f = K d V refdac 2 V π 2 N dac =2.27× 10 5
K s = 1 K ¯ [ 1 Q1 i=1 Q ( K i K ¯ ) 2 ] 1/2
V(k+1)V(k) x T (k+1)Px(k+1) x T (k)Px(k)2 f T ( K c x(k))[ f T ( K c x(k)) K c x(k)] +[ x T (k)x(k) γ 2 w T (k)w(k)][ x T (k)x(k) γ 2 w T (k)w(k)] = η T (k)(Φ+ Ω T PΩ)η(k)[ x T (k)x(k) γ 2 w T (k)w(k)]
[ P+I K c T 0 A T 2I 0 k B T γ 2 C T P 1 ]<0
k=0 N1 { V(k+1)V(k) }=V(N)V(0) k=0 k1 η T (k)φη(k) k=0 k1 [ x T (k)x(k) γ 2 w T (k)w(k)]
k=0 k1 [ x T (k)x(k) γ 2 w T (k)w(k)] 0

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