Abstract

Optical systolic array processors constitute a powerful and general-purpose set of optical architectures with high computational rates. In this paper, Kalman filtering, a novel application for these architectures, is investigated. All required operations are detailed; their realization by optical and special-purpose analog electronics are specified; and the processing time of the system is quantified. The specific Kalman filter application chosen is for an air-to-air missile guidance controller. The architecture realized in this paper meets the design goal of a fully adaptive Kalman filter which processes a measurement every 1 msec. The vital issue of flow and pipelining of data and operations in a systolic array processor is addressed. The approach is sufficiently general and can be realized on an optical or digital systolic array processor.

© 1984 Optical Society of America

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References

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  1. H. J. Caulfield et al., “Optical Implementation of Systolic Array Processing,” Opt. Commun. 40, 86 (1981.
    [CrossRef]
  2. D. Casasent, “Acoustooptic Transducers in Iterative Optical Vector–Matrix Processors,” Appl. Opt. 21, 1859 (1982).
    [CrossRef] [PubMed]
  3. D. Casasent, J. Jackson, C. Neuman, “Frequency-Multiplexed and Pipelined Iterative Optical Systolic Array Processors,” Appl. Opt. 22, 115 (1983).
    [CrossRef] [PubMed]
  4. R. P. Bocker, H. J. Caulfield, K. Bromley, “Rapid Unbiased Bipolar Incoherent Calculator Cube,” Appl. Opt. 22, 804 (1983).
    [CrossRef] [PubMed]
  5. R. A. Athale, W. C. Collins, “Optical Matrix–Matrix Multiplier Based on Outer Product Decomposition,” Appl. Opt. 21, 2089 (1982).
    [CrossRef] [PubMed]
  6. M. Carlotto, D. Casasent, “Microprocessor-Based Fiber-Optic Iterative Optical Processor,” Appl. Opt. 21, 147 (1982).
    [CrossRef] [PubMed]
  7. D. Casasent, C. P. Neuman, M. Carlotto, “An Electro-Optical Processor for Optimal Control,” Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).
  8. C. P. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings Electro-Optical Systems Design Conference (Industrial & Scientific Conference Management, Chicago, 1981), pp. 311–320.
  9. J. Jackson, D. Casasent, “State Estimation Kalman Filter Using Optical Processing: Noise Statistics Known,” Appl. Opt. 23, 376 (1984).
    [CrossRef] [PubMed]
  10. R. H. Travassos, “Real-Time Implementation of Systolic Kalman Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 97 (1983).
  11. W. A. Roemer, P. S. Maybeck, “An Optically Implemented Multiple-Stage Kalman Filter Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 221 (1983).
  12. R. L. Barron et al., “A New Class of Guidance Laws for Air-To-Air Missiles,” in Proceedings, Third Meeting of the Coordinating Group on Modern Control Theory, Part 1 (Oct.1081), pp. 20–21.
  13. T. L. Riggs, P. L. Vergez, “Advanced Air-To-Air Missile Guidance Using Optimal Control and Estimation,” Report on Contrast AFATL/DLMA, AFATL-TR-81-56 (June1981).
  14. P. S. Maybeck, Stochastic Models, Estimation, and Control, Vol. 1, (Academic, New York, 1979).
  15. A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1974).
  16. A. E. Bryson, Y. C. Ho, Applied Optical Control (Blaisdell, Waltham, Mass., 1969).
  17. R. S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1962).
  18. R. E. Bellman, R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965).
  19. T. R. Blackburn, “Solution of the Algebraic Matrix Riccati Equation via Newton-Raphson Iteration,” AIAA J. 6, 951 (1968).
    [CrossRef]

1984 (1)

1983 (4)

R. H. Travassos, “Real-Time Implementation of Systolic Kalman Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 97 (1983).

W. A. Roemer, P. S. Maybeck, “An Optically Implemented Multiple-Stage Kalman Filter Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 221 (1983).

D. Casasent, J. Jackson, C. Neuman, “Frequency-Multiplexed and Pipelined Iterative Optical Systolic Array Processors,” Appl. Opt. 22, 115 (1983).
[CrossRef] [PubMed]

R. P. Bocker, H. J. Caulfield, K. Bromley, “Rapid Unbiased Bipolar Incoherent Calculator Cube,” Appl. Opt. 22, 804 (1983).
[CrossRef] [PubMed]

1982 (3)

1981 (2)

H. J. Caulfield et al., “Optical Implementation of Systolic Array Processing,” Opt. Commun. 40, 86 (1981.
[CrossRef]

D. Casasent, C. P. Neuman, M. Carlotto, “An Electro-Optical Processor for Optimal Control,” Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

1968 (1)

T. R. Blackburn, “Solution of the Algebraic Matrix Riccati Equation via Newton-Raphson Iteration,” AIAA J. 6, 951 (1968).
[CrossRef]

Athale, R. A.

Barron, R. L.

R. L. Barron et al., “A New Class of Guidance Laws for Air-To-Air Missiles,” in Proceedings, Third Meeting of the Coordinating Group on Modern Control Theory, Part 1 (Oct.1081), pp. 20–21.

Baumbick, R.

C. P. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings Electro-Optical Systems Design Conference (Industrial & Scientific Conference Management, Chicago, 1981), pp. 311–320.

Bellman, R. E.

R. E. Bellman, R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965).

Blackburn, T. R.

T. R. Blackburn, “Solution of the Algebraic Matrix Riccati Equation via Newton-Raphson Iteration,” AIAA J. 6, 951 (1968).
[CrossRef]

Bocker, R. P.

Bromley, K.

Bryson, A. E.

A. E. Bryson, Y. C. Ho, Applied Optical Control (Blaisdell, Waltham, Mass., 1969).

Carlotto, M.

M. Carlotto, D. Casasent, “Microprocessor-Based Fiber-Optic Iterative Optical Processor,” Appl. Opt. 21, 147 (1982).
[CrossRef] [PubMed]

D. Casasent, C. P. Neuman, M. Carlotto, “An Electro-Optical Processor for Optimal Control,” Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

Casasent, D.

J. Jackson, D. Casasent, “State Estimation Kalman Filter Using Optical Processing: Noise Statistics Known,” Appl. Opt. 23, 376 (1984).
[CrossRef] [PubMed]

D. Casasent, J. Jackson, C. Neuman, “Frequency-Multiplexed and Pipelined Iterative Optical Systolic Array Processors,” Appl. Opt. 22, 115 (1983).
[CrossRef] [PubMed]

M. Carlotto, D. Casasent, “Microprocessor-Based Fiber-Optic Iterative Optical Processor,” Appl. Opt. 21, 147 (1982).
[CrossRef] [PubMed]

D. Casasent, “Acoustooptic Transducers in Iterative Optical Vector–Matrix Processors,” Appl. Opt. 21, 1859 (1982).
[CrossRef] [PubMed]

D. Casasent, C. P. Neuman, M. Carlotto, “An Electro-Optical Processor for Optimal Control,” Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

C. P. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings Electro-Optical Systems Design Conference (Industrial & Scientific Conference Management, Chicago, 1981), pp. 311–320.

Caulfield, H. J.

R. P. Bocker, H. J. Caulfield, K. Bromley, “Rapid Unbiased Bipolar Incoherent Calculator Cube,” Appl. Opt. 22, 804 (1983).
[CrossRef] [PubMed]

H. J. Caulfield et al., “Optical Implementation of Systolic Array Processing,” Opt. Commun. 40, 86 (1981.
[CrossRef]

Collins, W. C.

Gelb, A.

A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1974).

Ho, Y. C.

A. E. Bryson, Y. C. Ho, Applied Optical Control (Blaisdell, Waltham, Mass., 1969).

Jackson, J.

Kalaba, R. E.

R. E. Bellman, R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965).

Maybeck, P. S.

W. A. Roemer, P. S. Maybeck, “An Optically Implemented Multiple-Stage Kalman Filter Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 221 (1983).

P. S. Maybeck, Stochastic Models, Estimation, and Control, Vol. 1, (Academic, New York, 1979).

Neuman, C.

Neuman, C. P.

D. Casasent, C. P. Neuman, M. Carlotto, “An Electro-Optical Processor for Optimal Control,” Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

C. P. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings Electro-Optical Systems Design Conference (Industrial & Scientific Conference Management, Chicago, 1981), pp. 311–320.

Riggs, T. L.

T. L. Riggs, P. L. Vergez, “Advanced Air-To-Air Missile Guidance Using Optimal Control and Estimation,” Report on Contrast AFATL/DLMA, AFATL-TR-81-56 (June1981).

Roemer, W. A.

W. A. Roemer, P. S. Maybeck, “An Optically Implemented Multiple-Stage Kalman Filter Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 221 (1983).

Travassos, R. H.

R. H. Travassos, “Real-Time Implementation of Systolic Kalman Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 97 (1983).

Varga, R. S.

R. S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1962).

Vergez, P. L.

T. L. Riggs, P. L. Vergez, “Advanced Air-To-Air Missile Guidance Using Optimal Control and Estimation,” Report on Contrast AFATL/DLMA, AFATL-TR-81-56 (June1981).

AIAA J. (1)

T. R. Blackburn, “Solution of the Algebraic Matrix Riccati Equation via Newton-Raphson Iteration,” AIAA J. 6, 951 (1968).
[CrossRef]

Appl. Opt. (6)

Opt. Commun. (1)

H. J. Caulfield et al., “Optical Implementation of Systolic Array Processing,” Opt. Commun. 40, 86 (1981.
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

D. Casasent, C. P. Neuman, M. Carlotto, “An Electro-Optical Processor for Optimal Control,” Proc. Soc. Photo-Opt. Instrum. Eng. 295, 176 (1981).

R. H. Travassos, “Real-Time Implementation of Systolic Kalman Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 97 (1983).

W. A. Roemer, P. S. Maybeck, “An Optically Implemented Multiple-Stage Kalman Filter Algorithm,” Proc. Soc. Photo-Opt. Instrum. Eng. 431, 221 (1983).

Other (8)

R. L. Barron et al., “A New Class of Guidance Laws for Air-To-Air Missiles,” in Proceedings, Third Meeting of the Coordinating Group on Modern Control Theory, Part 1 (Oct.1081), pp. 20–21.

T. L. Riggs, P. L. Vergez, “Advanced Air-To-Air Missile Guidance Using Optimal Control and Estimation,” Report on Contrast AFATL/DLMA, AFATL-TR-81-56 (June1981).

P. S. Maybeck, Stochastic Models, Estimation, and Control, Vol. 1, (Academic, New York, 1979).

A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge, Mass., 1974).

A. E. Bryson, Y. C. Ho, Applied Optical Control (Blaisdell, Waltham, Mass., 1969).

R. S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1962).

R. E. Bellman, R. E. Kalaba, Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965).

C. P. Neuman, D. Casasent, R. Baumbick, “An Electro-Optical Processor for the Optimal Control of F100 Aircraft Engines,” in Proceedings Electro-Optical Systems Design Conference (Industrial & Scientific Conference Management, Chicago, 1981), pp. 311–320.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of a frequency-multiplexed acoustooptic systolic array processor.3

Fig. 2
Fig. 2

Triple-nested discrete-time EKF algorithm.

Fig. 3
Fig. 3

Optical random-access AO storage/multiplier/summer for calculation of the Jacobian.

Fig. 4
Fig. 4

Optical systolic discrete-time EKF processor architecture.

Tables (2)

Tables Icon

Table I Component Requirements and Performance for the Linear Algebraic Operations Required in Our EKF Algorithm

Tables Icon

Table II Operations and Timing for Our EKF Processor (*Defines Critical Calculation Time Path)

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

x ˙ ( t ) = Fx ( t ) + u ( t ) + w ( t ) ,
z ( t ) = h [ x ( t ) ] + w s ( t ) .
H ( t ) h [ x ( t ) ] x ( t ) | x ( t ) = x ^ ( t ) .
x ^ ( t ) = Fx ^ ( t ) + K ( t ) { z ( t ) - h [ x ^ ( t ) ] } + u ( t ) ,
P ( t ) = FP ( t ) + P ( t ) F T + Q ( t ) - P ( t ) H T ( t ) R - 1 ( t ) H ( t ) P ( t ) ,
K ( t ) = P ( t ) H T ( t ) R - 1 ( t ) .
P ( t ) = E { [ x ^ ( t ) - x ( t ) ] [ x ^ ( t ) - x ( t ) ] T } .
K k = P k H k T R k - 1
x ^ k + 1 = { l + T F } x ^ k + T K k { z k - h ( x ^ k ) } + T u k ,
P k + 1 M k + 1 P k + 1 + { P k + 1 L T + L P k + 1 } + C k = 0.
M k + 1 = H k + 1 T R k + 1 - 1             H k + 1 , L = [ ( 1 / T ) 1 - F ] , C k = P k M k P - P k [ ( 1 / T ) l + F ] T - [ ( 1 / T ) l + F ] P k - 2 Q .
b k + 1 = ω ( Ab k - c ) + b k ,
b = A - 1 c .
G k = P k + 1 M k + 1 P k + 1 + { P k + 1 L T + L P k + 1 } + C k = 0
p n + 1 = p n - J [ p n ] - 1 g [ p n ] ,
J ( i , j ) = g i / p j p n for i , j = 1 , , 81.
J ( p n ) s n = - g [ p n ]
s n = p n + 1 - p n .
s r + 1 = s r + ω { g ( p n ) + J ( p n ) s r } .
J = A T l + l A T ,
A = M k + 1 P n + 1 + L ,
M P = ( m i j P ) i j .
T k = 42 T B + 30 μ sec + n [ 18 T B + 20 μ sec + r 455 T B ] ,
T k = 44.7 + n [ 26 + 159 r ] μ sec .

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