https://lup.lub.lu.se/search/ws/files/5357880/8867248.pdf

System identification-A survey

We present several new algorithms, and more generally a new approach, to recursive estimation algorithms for linear dynamical systems. Earlier results in this area have been obtained by several others, especially Potter, Golub, Dyer and McReynolds, Kaminski, Schmidt, Bryson, and Bierman on what are known as square-root algorithms. Our results are more comprehensive. They also show bow constancy of parameters can be exploited to reduce the number of computations and to obtain new forms of the Chandrasekhar-type equations for computing the filter gain. Our approach is essentially based on certain simple geometric interpretations of the overall estimation problem. One of our goals is to attract attention to non-Riccati-based studies of estimation problems.

/pdf/square-root-algorithms-for-least-squares-estimation-4nbl66zqa4.pdf

Square-root algorithms for least-squares estimation

Certain recently developed fast algorithms for recursive estimation in constant continuous-time linear systems are extended to discrete-time systems. The main feature is the replacement of the Riccati-type difference equation that is generally used for such problems by another set of difference equations that we call of Chandrasekhar-type. The total number of operations in the new algorithm is in general less than with the Riccati-equation based Kalman filter, with significant reductions being obtained in several important special cases. The algorithms are derived via a factorization of increments of the Riccati equation variable, a method that can be extended to nonsymmetric Riccati equations as well.

Some new algorithms for recursive estimation in constant, linear, discrete-time systems

This paper catalogues formulas that are useful for estimating dynamic linear economic models. We describe algorithms for computing equilibria of an economic model and for recursively computing a Gaussian likelihood function and its gradient with respect to parameters. We display an application to Rosen, Murphy, and Scheinkman's (1994) model of cattle cycles.

/pdf/mechanics-of-forming-and-estimating-dynamic-linear-economies-3sya9vu74v.pdf

Mechanics of forming and estimating dynamic linear economies

Gives a comprehensive treatment of several important aspects of the discrete-time periodic Riccati equation (DPRE) arising from the prediction problem for linear discrete-time periodic systems. The authors analyze the symmetric periodic positive semidefinite (SPPS) solution of the DPRE under appropriate assumptions of stabilizability and detectability of the periodic system. Among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the SPPS solution and the stability of the resulting closed-loop system. Some of these results can be seen as extensions of the corresponding results for the time-invariant case; however, a number of them contain contributions to the time-invariant case as well. The paper also gives a numerical algorithm based on an iterative linearization procedure for computing the SPPS solution. The algorithm is a periodic version of Kleinman's algorithm for the time-invariant case. >

The difference periodic Ricati equation for the periodic prediction problem

A new variant of observability is introduced, minimal observability, which has an interesting practical meaning. This concept is then exploited to obtain a simple and completely specified canonical form for multioutput nonautonomous analytic systems.

https://ieeexplore.ieee.org/iel5/9/24103/01098977.pdf

Canonical forms for multivariable systems

The Riccati equation associated with a class of discrete-time correlated noise problems is examined, and the concept of invariant directions for this equation is introduced. For single-output systems the set of such directions is completely characterized. Deletion of these directions by an appropriate transformation of the Riccati equation results in a minimal order equation for computation. This transformation also reveals the underlying structure of the optimal filter for the correlated noise problem.

Correlated noise filtering and invariant directions for the Riccati equation

A priori bounds are derived for a particular nonlinear filtering problem. Upper bounds are generated by construction of a suboptimal filter using sensor orbit coordinates. Further Monte Carlo evaluation of several suboptimal filters is given. Optimal performance is contrasted with various upper and lower bounds.

https://ieeexplore.ieee.org/iel5/9/24164/01101684.pdf

R. Bucy

Papers

Canonical forms for multivariable systems

Correlated noise filtering and invariant directions for the Riccati equation

A priori error bounds for the cubic sensor problem

Periodic equilibria for matrix riccati equations

Remarks on "A note on bounds on solutions of the Riccati equation"