Abstract
In this research, the auto-mutual information function and correlation dimension are used for determination of lag and embedding dimension needed for reconstruction of the attractor of in-cylinder pressure of an internal combustion engine. Subsequently, a locally constant model is used for a 20-step prediction. The time series are acquired at three different test conditions and consisting of succeeding pressures at compression Top Dead Center (TDC) position of one of four cylinders. We have concluded that at least at relatively low engine speeds (below 2000 rpm) this method produces acceptable results.
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Abbreviations
- y t :
-
time series general representation
- d :
-
embedding dimension
- m :
-
attractor manifold dimension
- ψ :
-
smooth map
- z t :
-
embedded state
- P s (s i ):
-
probability of message s i belonging to S
- P sq (s i , q j ):
-
joint probability of s i belonging to S and q j belonging to Q
- H(S):
-
entropy of system S
- H(S|Q):
-
conditional entropy of S if Q
- H(S,Q):
-
joint entropy of S and Q
- I(S,Q):
-
mutual information of S and Q
- D q :
-
correlation dimension of order q
- d sat :
-
saturation embedding dimension
- D q (d) :
-
correlation dimension of order q in the d dimensional embedding space
- C N (d) :
-
correlation sum of reconstructed time series of length N in d dimensional space
- h :
-
Heaviside function
- r :
-
box size
- nnr:
-
number of nearest neighbors
- ω :
-
frequency
- rpm:
-
angular speed in revolutions per minute
- ϕ :
-
equivalence ratio
- ρ :
-
density
- C p :
-
constant-pressure-specific heat
- E :
-
ignition energy
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Haji, A.H., Mahzoon, M. & Emdad, H. Reconstruction and prediction of the in-cylinder pressure attractor of an internal combustion engine using locally constant models. Nonlinear Dyn 48, 437–447 (2007). https://doi.org/10.1007/s11071-006-9097-x
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DOI: https://doi.org/10.1007/s11071-006-9097-x