A constructive optimization algorithm using Chebyshev spectral collocation and quadratic programming is proposed for unknown parameter estimation in nonlinear time-varying dynamic system models to be constructed from available data. The parameters to be estimated are assumed to be identifiable from the data, which also implies that the assumed system models with known parameter values have a unique solution corresponding to every initial condition and parameter set. The nonlinear terms in the dynamic system models are assumed to have a known form, and the models are assumed to be parameter affine. Using an equivalent algebraic description of dynamical systems by Chebyshev spectral collocation and data, a residual quadratic cost is set up, which is a function of unknown parameters only. The minimization of this cost yields the unique solution for the unknown parameters since the models are assumed to have a unique solution for a particular parameter set. An efficient algorithm is presented stepwise and is illustrated using suitable examples. The case of parameter estimation with incomplete or partial data availability is also illustrated with an example.

1.
Voit
,
E. O.
,
Marino
,
S.
, and
Lall
,
R.
, 2005, “
Challenges for the Identification of Biological Systems From In Vivo Time Series Data
,”
In Silico Biology
,
5
, pp.
83
92
.
2.
Cruz-Pacheco
,
G.
,
Esteva
,
L.
,
Montano-Hirose
,
J. A.
, and
Vargas
,
C.
, 2005, “
Modelling the Dynamics of West Nile Virus
,”
Bull. Math. Biol.
0092-8240,
67
, pp.
1157
1172
.
3.
Bowman
,
C.
,
Gumel
,
A. B.
,
van den Driessche
,
P.
,
Wu
,
J.
, and
Zhu
,
H.
, 2005, “
Mathematical Model of Assessing Control Strategies Against West Nile Virus
,”
Bull. Math. Biol.
0092-8240,
67
, pp.
1107
1133
.
4.
Capasso
,
V.
, and
Morale
,
D.
, 2005, “
Mathematical Models for HIV Transmission Among Injecting Drug Users
,”
Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections With Intervention
,
W.-Y.
Tan
and
H.
Wu
, eds., pp.
1
30
.
5.
Bajaria
,
S. H.
, and
Kirschner
,
D. E.
, 2005, “
CTL Action During HIV-1 Is Determined Via Interactions With Multiple Cell Types
,”
Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections With Intervention
,
W.-Y.
Tan
and
H.
Wu
, eds., Ref. 4, pp.
219
254
.
6.
Kansal
,
A. R.
, 2004, “
Modelling Approaches to Type 2 Diabetes
,”
Diabetes Technology and Therapeutics
,
6
(
1
), pp.
39
47
.
7.
Cobelli
,
C.
,
Bier
,
D. M.
, and
Ferraninni
,
E.
, 1990, “
Modelling Glucose Metabolism in Man: Theory and Practice
,”
Horm. Metab. Res., Suppl. Ser.
0170-5903,
24
, pp.
1
10
.
8.
Topp
,
B.
,
Promislow
,
K.
,
DE Vries
,
G.
,
Miura
,
R. M.
, and
Finegood
,
D. T.
, 2006, “
A Model of Beta-Cell Mass, Insulin and Glucose Kinetics: Pathways to Diabetes
,”
J. Theor. Biol.
0022-5193,
206
, pp.
605
619
.
9.
Goldbeter
,
A.
, 1995, “
A Model for Circadian Oscillations in the Drosophila Period Protein
,”
Proc. R. Soc. London, Ser. B
0962-8452,
261
, pp.
319
324
.
10.
Chen
,
L.
, and
Wang
,
R.
, 2004, “
Dynamics of Gene Regulatory Networks With Cell Division Cycle
,”
Phys. Rev. E
1063-651X,
70
, p.
011909
.
11.
Smolen
,
P.
,
Baxter
,
D. A.
, and
Byrne
,
J. H.
, 2000, “
Mathematical Modeling of Gene Networks
,”
Neuron
0896-6273,
26
, pp.
567
580
.
12.
Covert
,
M. W.
,
Schilling
,
C. H.
, and
Palsson
,
B.
, 2001, “
Regulation of Gene Expression in Flux Balance Models of Metabolism
,”
J. Theor. Biol.
0022-5193,
213
, pp.
73
88
.
13.
Bradshaw
,
P. C.
, and
Samuels
,
D. C.
, 2005, “
A Computational Model of Mitochondrial Deoxynucleotide Metabolism and DNA Replication
,”
Am. J. Physiol.: Cell Physiol.
0363-6143,
288
, pp.
989
1002
.
14.
Mendes
,
P.
, and
Kell
,
D. B.
, 1998, “
Nonlinear Optimization of Biochemical Pathways: Applications to Metabolic Engineering and Parameter Estimation
,”
Bioinformatics
1367-4803,
14
(
10
), pp.
869
883
.
15.
de Mottoni
,
P.
, and
Schiaffino
,
A.
, 1981, “
Competetion Systems With Periodic Coefficients: A Geometric Approach
,”
J. Math. Biol.
0303-6812,
11
, pp.
319
335
.
16.
Butcher
,
E. C.
,
Berg
,
E. L.
, and
Kunkel
,
E. J.
, 2004, “
Systems Biology in Drug Discovery
,”
Nat. Biotechnol.
1087-0156,
22
(
10
), pp.
1253
1259
.
17.
Grass
,
G. M.
, 1997, “
Simulation Models to Predict Oral Drug Absorption From In Vitro Data
,”
Adv. Drug Delivery Rev.
0169-409X,
23
, pp.
199
219
.
18.
Tan
,
W.-Y.
, and
Huang
,
Y.
, 2005, “
Bayesian Estimation of Individual Parameters in a HIV Dynamic Model Using Long Term Viral Load Data
,”
Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections With Intervention
,
W.-Y.
Tan
and
H.
Wu
, eds., Ref. 4, pp.
361
384
.
19.
Mendes
,
P.
, 2001, “
Modeling Large Biological Syetms From Functional Genomic Data: Parameter Estimation
,”
Foundations of Systems Biology
,
H.
Kitano
, ed., pp.
163
186
.
21.
Vera
,
J.
, and
Torres
,
N. V.
, 2004, “
Metmap: An Integrated Matlab Package for Analysis and Optimization of Metabolic Systems
,”
In Silico Biology
,
4
, pp.
97
108
.
22.
Chalidze
,
P.
, and
Cusumano
,
J. P.
, 2004, “
A Dynamical Systems Approach to Failure Prognosis
,”
ASME J. Vibr. Acoust.
0739-3717,
126
(
1
), pp.
2
8
.
23.
Yin
,
S. H.
, and
Epureanu
,
B. I.
, 2005, “
Enhanced Nonlinear Dynamics and Monitoring Bifurcation Morphing for the Identification of Parameter Variations
,”
J. Fluids Struct.
0889-9746,
21
, pp.
543
559
.
24.
Masri
,
S. F.
, and
Caughey
,
T. K.
, 1979, “
A Nonparametric Identification Technique for Nonlinear Dynamic Problems
,”
ASME J. Appl. Mech.
0021-8936,
46
, pp.
433
447
.
25.
Tunali
,
E. T.
, and
Tarn
,
T. J.
, 1987, “
New Results for Identifiability of Nonlinear Systems
,”
IEEE Trans. Autom. Control
0018-9286,
32
, pp.
146
154
.
26.
Trefethen
,
L. N.
, 2000,
Spectral Methods in MATLAB
,
SIAM
,
Philadelphia
.
27.
Hulme
,
B. L.
, 1972, “
One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems
,”
Math. Comput.
0025-5718,
26
(
118
), pp.
415
426
.
28.
de Boor
,
C.
, and
Swartz
,
B.
, 1973, “
Collocation at Gaussian Points
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429,
10
(
4
), pp.
582
606
.
29.
Weiss
,
R.
, 1974, “
The Application of Implicit Runge–Kutta and Collocation Methods to Boundary Value Problems
,”
Math. Comput.
0025-5718,
28
(
126
), pp.
449
464
.
30.
Sinha
,
S. C.
,
Gourdon
,
E.
, and
Zhang
,
Y.
, 2005, “
Control of Time-Periodic Systems Via Symbolic Computation With Applications to Chaos Control
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
10
(
8
), pp.
835
854
.
31.
Sinha
,
S. C.
,
Redkar
,
S.
, and
Butcher
,
E. A.
, 2006, “
On Micromodeling of Nonlinear Systems With Time Periodic Coefficients
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
11
(
4
), pp.
510
530
.
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