Input Design for Nonlinear Stochastic Dynamic Systems – A Particle Filter Approach

Abstract

Abstract:We propose an algorithm for optimal input design in nonlinear stochastic dynamic systems. The approach relies on minimizing a function of the covariance of the parameter estimates of the system with respect to the input. The covariance matrix is approximated using a joint likelihood function of hidden states and measurements, and a combination of state filters and smoothers. The input is parametrized using an autoregressive model. The proposed approach is illustrated through a simulation example.

D-optimal input design for nonlinear FIR-type systems: A dispersion-based approach

“…. , i n−1 , j) = m + (j − 1)A n−1 Remark 7 In order to illustrate that the equations in (13) are not sufficient conditions for the existence of a realizable time sequence for a given a frequency vector, consider a disconnected graph which satisfies the constraints. In such a graph there is no single path connecting all the nodes, meaning there is also no corresponding time sequence.…”

“…In a stochastic framework, the frequency matrix can be considered as mutual discrete probability distribution functions of the n stochastic variables. Imposing that the signal is stationary, will lead to the same constraint as given in (13) [22]. The graph described above can then be seen as a Markov chain used to generate a realization of the frequency matrix.…”

“…Therefore the full search space is contained in a convex polyhedron [2]. Second, the constraints presented in (13) correspond to a set of linear equality constraints, meaning they describe a subspace centered at the origin. Combining these two observations shows that the space of realizable relative frequency vectors consists of the intersection between a polyhedron and a subspace, which in turn is again a polyhedron [2] and therefore a convex set.…”

“…Two observations should be made. First, the design does not obey the constraints (13). As a result, the frequency vector cannot be realized as a time sequence.…”

“…All errors are smaller than 10 −2 , meaning only rounding errors are present. The unconstrained design does not satisfy the constraints (13). Therefore, the design will be slightly altered in order to find a time sequence which realizes the unconstrained design as well as possible.…”