Abstract
A constructive method is developed to design inverse optimal filters to estimate the states of a class of linear distributed parameter systems (DPSs) based on the calculus of variation approach. Inverse optimality guarantees that the cost functional to be minimized is meaningful in the sense that the symmetric and positive definite weighting kernel matrix on the states is chosen after the filter design instead of being specified at the start of the filter design. Inverse optimal design enables that the Riccati nonlinear partial differential equation (PDE) can be simplified to a Bernoulli PDE, which can be solved analytically. The filter design is based on a new Green matrix formula, a new unique and bounded solution of a linear PDE, and analytical solution of a Bernoulli PDE. The inverse optimal filter design is first developed for the case where the measurements are spatially available, then is extended to the practical case where only a finite number of measurements is available.
Highlights
In many physical processes, the dynamical system one wishes to estimate is described by partial differential equation (PDE) such as chemical reactors, heat exchangers, transmission lines, vibrating beams and electrical, optical or acoustic waves
There have no formal proof of existence and uniqueness of the symmetric and positive definite solution of these Riccati nonlinear PDEs for a sufficiently large class of linear distributed parameter systems (DPSs), which cover the important practical processes
Difficulties arisen in solving the Riccati nonlinear PDEs and two-point-boundary value problems, which are resulted from the classical design of optimal filters for distributed parameter systems, motivate the approach of designing inverse optimal filters in this paper
Summary
The dynamical system one wishes to estimate is described by PDEs such as chemical reactors, heat exchangers, transmission lines, vibrating beams and electrical, optical or acoustic waves. From here the classical optimal filtering results are extended into infinite-dimensional systems [22], [3], [11], [2] This approach eventually results in operator Riccati equations which have similarities to the results presented here. Difficulties arisen in solving the Riccati nonlinear PDEs and two-point-boundary value problems, which are resulted from the classical design of optimal filters for distributed parameter systems, motivate the approach of designing inverse optimal filters in this paper. The inverse approach in [12], [9] uses a control Lyapunov function, which is a solution of Hamilton-JacobiBellman with a meaningful cost, for systems governed by nonlinear ODEs obtained by solving a stabilization problem. For a matrix operator Ax, the notation A∗x denotes its adjoint
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