Science > Physics > Causation/Causality, Memory, and Convolution 6: New Interpretation of Riccati Equation
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
19 Feb 2006 05:56:49 PM |
| Object: |
Causation/Causality, Memory, and Convolution 6: New Interpretation of Riccati Equation |
From Osher Doctorow
The Riccati Differential equation is:
1) dy/dt = A(t) + B(t)y + C(t)y^2
with all variables/constants real scalars, although they can be
generalized to matrices with slight changes. In the latter case,
feedback/control such as Kalman filter-predictors and optimal control
using dynamical systems involve propagation of error-covariance or
variance-covariance matrices y = "P" using (1).
The most important Riccati Differential equation subtypes are the
Logistic Differential equation and the Exponential Growth/Decay
Differential equations, respectively:
2) dy/dt = ky(1 - y) (y "normalized" or "standardized" to be between 0
and 1 here)
3) dy/dt = ky
If we multiply out the right hand side of (2) and collect terms, and
likewise for (3), we get respectively:
4) dy/dt - ky + ky^2 = 0
5) dy/dt - ky = 0
SInce k > 0 usually in (4), the second and third terms have alternating
signs, and so up to a constant the three terms can be converted to
Probable Influence/Causation form:
6) 1 + y - x, x and y > = 0
In the case of (5) if k > 0 (exponential growth) and dy/dt > 0, the
terms are alternating in sign and can be easily converted to (6) for
the left-hand-side of (5) by adding 1 to both sides. The case k < 0
and dy/dt < 0 (exponential decay) can just be considered as a
"reflection" geometrically from (5) with k > 0.
Even if we don't standardize y in (2), we still get an equation similar
to (4) and (6) as follows:
7) dy/dt = ay(b - y) = aby - ay^2, a > 0, b > 0
This has alternating signs for the two terms on the right hand side
with the usual condition that y > = 0, so it can be converted to (6) up
to a constant.
So the Riccati Differential equation in its most common forms reflects
(Probable) Causation, and even the matrix feedback form can be regarded
as derived from those most common forms which gives a (Probable)
Causal/Influence interpretation to Feedback Control..
Osher Doctorow
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