Abstract

Following the Volterra theorem, every nonlinear operator can be implemented with a sum of integrals applied over the input. The second-order Volterra operator that describes many useful systems can be related to a single integral that is a projection of an operation called the triple correlation. This operation may be easily implemented optically and thus be incorporated into fast real-time nonlinear control systems. We present a theoretical investigation of the relation existing between the Volterra operators and the triple correlation as well as an experimental demonstration that validates the theory.

© 2001 Optical Society of America

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References

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  1. I. Yau, S. K. Spurgeon, “Implementation of non-linear tracking control schemes for serial/parallel topology robot,” J. Syst. Control Eng. 211, 457–470 (1997).
  2. P. F. Pai, B. Wen, A. S. Naser, M. J. Schulz, “Structural vibration control using PZT patches and non-linear phenomena,” J. Sound Vib. 215, 273–296 (1998).
    [Crossref]
  3. W. J. Rugh, Nonlinear System Theory (Johns Hopkins U. Press, Baltimore, Md., 1981), pp. 1–52.
  4. A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
    [Crossref]
  5. M. Schetzen, Volterra and Wiener Theories of Non-Linear Systems (Wiley, New York, 1980), pp. 1–195.
  6. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 434–458.
  7. P. Nisenson, R. A. Sprague, “Real-time optical correlation,” Appl. Opt. 14, 2602–2606 (1975).
    [Crossref] [PubMed]
  8. D. Mendlovic, A. W. Lohmann, D. Mas, G. Shabtay, “Optoelectronic implementation of the triple correlation,” Opt. Lett. 22, 1018–1020 (1997).
    [Crossref] [PubMed]

1998 (1)

P. F. Pai, B. Wen, A. S. Naser, M. J. Schulz, “Structural vibration control using PZT patches and non-linear phenomena,” J. Sound Vib. 215, 273–296 (1998).
[Crossref]

1997 (2)

I. Yau, S. K. Spurgeon, “Implementation of non-linear tracking control schemes for serial/parallel topology robot,” J. Syst. Control Eng. 211, 457–470 (1997).

D. Mendlovic, A. W. Lohmann, D. Mas, G. Shabtay, “Optoelectronic implementation of the triple correlation,” Opt. Lett. 22, 1018–1020 (1997).
[Crossref] [PubMed]

1984 (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

1975 (1)

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 434–458.

Lohmann, A. W.

Mas, D.

Mendlovic, D.

Naser, A. S.

P. F. Pai, B. Wen, A. S. Naser, M. J. Schulz, “Structural vibration control using PZT patches and non-linear phenomena,” J. Sound Vib. 215, 273–296 (1998).
[Crossref]

Nisenson, P.

Pai, P. F.

P. F. Pai, B. Wen, A. S. Naser, M. J. Schulz, “Structural vibration control using PZT patches and non-linear phenomena,” J. Sound Vib. 215, 273–296 (1998).
[Crossref]

Rugh, W. J.

W. J. Rugh, Nonlinear System Theory (Johns Hopkins U. Press, Baltimore, Md., 1981), pp. 1–52.

Schetzen, M.

M. Schetzen, Volterra and Wiener Theories of Non-Linear Systems (Wiley, New York, 1980), pp. 1–195.

Schulz, M. J.

P. F. Pai, B. Wen, A. S. Naser, M. J. Schulz, “Structural vibration control using PZT patches and non-linear phenomena,” J. Sound Vib. 215, 273–296 (1998).
[Crossref]

Shabtay, G.

Sprague, R. A.

Spurgeon, S. K.

I. Yau, S. K. Spurgeon, “Implementation of non-linear tracking control schemes for serial/parallel topology robot,” J. Syst. Control Eng. 211, 457–470 (1997).

Wen, B.

P. F. Pai, B. Wen, A. S. Naser, M. J. Schulz, “Structural vibration control using PZT patches and non-linear phenomena,” J. Sound Vib. 215, 273–296 (1998).
[Crossref]

Wirnitzer, B.

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

Yau, I.

I. Yau, S. K. Spurgeon, “Implementation of non-linear tracking control schemes for serial/parallel topology robot,” J. Syst. Control Eng. 211, 457–470 (1997).

Appl. Opt. (1)

J. Sound Vib. (1)

P. F. Pai, B. Wen, A. S. Naser, M. J. Schulz, “Structural vibration control using PZT patches and non-linear phenomena,” J. Sound Vib. 215, 273–296 (1998).
[Crossref]

J. Syst. Control Eng. (1)

I. Yau, S. K. Spurgeon, “Implementation of non-linear tracking control schemes for serial/parallel topology robot,” J. Syst. Control Eng. 211, 457–470 (1997).

Opt. Lett. (1)

Proc. IEEE (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

Other (3)

M. Schetzen, Volterra and Wiener Theories of Non-Linear Systems (Wiley, New York, 1980), pp. 1–195.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 434–458.

W. J. Rugh, Nonlinear System Theory (Johns Hopkins U. Press, Baltimore, Md., 1981), pp. 1–52.

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Figures (6)

Fig. 1
Fig. 1

Joint transform correlator generating the auto correlation of f(x).

Fig. 2
Fig. 2

Model for controlling the velocity of an airplane under external forces. The coefficient of the linear part is taken as 1, and the coefficient of the nonlinear part is taken as 10.

Fig. 3
Fig. 3

Result of a pulse force inflicted on the object. The resulting velocity is highly nonlinear, and thus the use of linear techniques would not be sufficient to stabilize the object.

Fig. 4
Fig. 4

Model for controlling the angle of a pendulum under external forces. The coefficient of the linear part is taken as a=2 and b=4, and thus the coefficient of the nonlinear part is taken as 4/6.

Fig. 5
Fig. 5

Implementations of ATC by use of acousto-optic and electro-optic devices. (a) Acousto-optic modulators along the path of the modulated light source generate a time-dependent image in the output plane, and integration yields the ATC. (b) A computer generates the required spatial modulation so that in the Fourier domain an ATC is obtained.

Fig. 6
Fig. 6

Pendulum angle as a function of time, as a response to a pulse signal similar to the one given before.

Equations (44)

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H1(t)=-h1(τ)x(t-τ)dτ.
y(t)=-h1(τ1)x(t-τ1)dτ1+--h2(τ1, τ2)x(t-τ1)x(t-τ2)dτ1dτ2++-  -hn(τ1, τ2 , τn)x(t-τ1)×x(t-τ2)  x(t-τn)dτ1dτ2  dτn+
hn(τ1, τ2 , τn)=0foranyτj<0,j=1, 2, n.
H2(t)=--h2(τ1, τ2)x(t-τ1)x(t-τ2)dτ1dτ2.
y(t, τ)=-x1(η)x2(η+t)x3(η+τ)dη.
y(t, τ)=-x(η)x(η+t)x(η+τ)dη.
y˜(ν1, ν2)=x˜1(-ν1, -ν2)x˜2(ν1)x˜3(ν2).
H^2(t1, t2)=--h2(τ1, τ2)x(t1-τ1)×x(t2-τ2)dτ1dτ2.
H2(t)=H^2(t, t).
H˜2(ν1, ν2)=h˜2(-ν1, -ν2)x˜(ν1)x˜(ν2).
x˜3(ν)=x˜2(ν)=x˜(ν),
h˜2(ν1, ν2)=x˜1(ν1+ν2).
h˜2(ν1, ν-ν1)=h˜2(ν-ν2, ν2)=x˜1(ν),
h2(t, τ)=--h˜2(ν1, ν2)exp[j2π(ν1t+ν2τ)]dν1dν2=--x˜(ν1+ν2)×exp[j2π(ν1t+ν2τ)]dν1dν2.
h2(t, τ)=-exp[j2πν1t]×-x˜(ν1+ν2)exp(j2πν2τ)dν2dν1=-exp(j2πν1t)x(τ)exp(-j2πν1τ)dν1=x(τ)-exp[j2πν1(t-τ)]dν1=x (τ)δ(t-τ),
x(t)=-h2(t, τ)dτ,
g(s, θ)=Rf=--f(u, ν)δ(u cos θ+ν sin θ-s)dudν.
g(s, θ)=-f(u, s cos ecθ-u cot θ)du.
g(s, θ)=-f(s cos θ-η sin θ, s sin θ+η cos θ)dη.
hcut(t, τ)=hcut(t)=h2(t, t tan θ).
hparallel-cut(t)=h2(t, t tan θ-t0tan θ).
hdom(t)=-h2(t, t tan θ-t0tan θ)dt0,
hdom(t)=-tan θ-h2(t, μ)du.
x(t)=-h2(t, τ)dτ=-cot θ hdom(t),
h˜2(ν1, ν2)=-π/2π/2h˜2,θ(ν1, ν2)f(θ)dθ,
h˜2,θ(ν1, ν2)=x˜(ν1cos θ+ν2sin θ).
h2,θ(t, τ)=--x˜(ν1cos θ+ν2sin θ)×exp[j2π(ν1t+ν2τ)]dν1dν2=-exp(j2πν1t)-x˜(ν1cos θ+ν2sin θ)×exp(j2πν2τ)dν2dν1=1sin θ-exp(j2πν1t)xτsin θ×exp[-j2πν1(cot θ)τ]dν1=1sin θ xτsin θ×-exp{j2πν1[t-(cot θ)τ]}dν1=1sin θ xτsin θδ[t-(cot θ)τ],
x(t)=cos θ-h2,θ(t cos θ, τ)dτ.
H3(t)=---h3(τ1, τ2, τ3)x(t-τ1)x(t-τ2)×x(t-τ3)dτ1dτ2dτ3.
H3(t1, t2, t3)=---h3(τ1, τ2, τ3)x(t1-τ1)×x(t2-τ2)×x(t3-τ3)dτ1dτ2dτ3,
H˜3(ν1, ν2, ν3)=h˜3(-ν1, -ν2, -ν3)x˜(ν1)x˜(ν2)x˜(ν3).
y˜(ν1, ν2)=x˜1(-ν1, -ν2)x˜2(ν1)x˜3(ν2).
H˜proj(ν1, ν2)=-H˜3(ν1, ν2, ν3)dν3=x˜(ν1)x˜(ν2)×-h˜3(-ν1, -ν2, -ν3)x˜(ν3)dν3.
H˜proj(ν1, ν2)=hnew(-ν1, -ν2)x˜(ν1)x˜(ν2),
f(t)=a dV(t)dt+bV2(t),
H1[V]=a ddt [V],
H1(s)=as.
K1(s)=1as.
T(jω)=1jω=1ωexp-j π2.
f(t)=d2x(t)dt2+a dx(t)dt+b sin[x(t)],
sin(x)=x-x33!+
f(x)=H1(x)+H3(x),
H1(x)=d2x(t)dt2+a dx(t)dt+bx(t),
H3(x)=-b x3(t)3!.

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