Dynamical State and Parameter Estimation

Fast Sampling of Evolving Systems with Periodic Trajectories

“…which bears some similarity with [1,16]. Here, however, no discretization of the original continuous-time dynamical model is required and ∂ŷ(λ, t i )/∂λ are computable as definite integrals.…”

“…This is a standard inverse problem, and many methods for finding solutions to this problem have been developed to date (sensitivity functions [20], splines [6], interval analysis [15], adaptive observers [19], [5], [9], [12], [24], [25], [8] and particle filters and Bayesian inference methods [1]). Despite these methods are based on different mathematical frameworks, they share a common feature: one is generally required to repeatedly find numerical solutions of nonlinear ordinary differential equations (ODEs) over given intervals of time (solve the direct problem).…”

Parameter Identification in Dynamic Systems Using the Homotopy Optimization Approach

“…We then decrease λ further to 1 − 2 δλ; since the parameter guesses have been refined, the observer gain can be 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 5 reduced without increasing the error beyond ε. At each stage in this process, we use the converged result from the previous stage as the initial guess for p. This process is repeated until λ = 0, and equation (9) has morphed back into equation (1). In summary, the homotopy optimization approach follows the path of minimal error as the observer gain is decreased.…”

“…The optimization problems are usually solved using deterministic methods, which require the solution of differential equations at each optimization step. The solution of these ODEs can be obtained using initial-value methods [10,26], shooting methods [1], or collocation methods [2]. When deterministic approaches like the steepest descent [22], Gauss-Newton [22], and Levenberg-Marquardt [20] algorithms are used in the optimization procedure, it is not uncommon to converge to a local minimum rather than the global minimum [7].…”

“…Homotopy was successfully applied to ARMAX models [13,14] for the identification of linear parameters. In the work of Abarbanel et al [1], the authors have coupled the mathematical model to the experimental data for identifying parameters from a chaotic time series for first-order systems, which is related to the homotopy method. The application of homotopy to the general nonlinear parameter identification problem has not been studied in the literature.…”