# A state-space method of parameter identification

- Paper ID
IAF-70-45

- author
- company
McDonnell Douglas Astronautics Company - Western Division

- country
U.S.A.

- year
1970

- abstract
Discussion of the use of a Kalman filter to estimate ah augmented state vector is necessary as background and as a means to establish some notation. The augmented state vector consists of the actual linear-plant states plus the parameters arranged as a vector. This idea was originally proposed in 1963 by Kopp and Orford [1] who reported results for a second-order continuous system with two parameters. The general equations for discrete and continuous systems were stated by Kumar in 1964 [2], but these equations are not valid to the degree of generality claimed because it may not be possible (or desirable) to estimate all the plant parameters. This technique of combined (state and parameter) estimation via a Kalman filter (CE/KF) was modified and then applied to a launch vehicle in 1966 [3]. The modification consisted of taking the parameter vector to be a selected number of plant parameters, rather than all of them. It was realized that the approach was still impractical in this modified form even before the technique was used in the latter application. This is because a matrix Riccati differential equation has to be solved on-line for filter implementation, which amounts to solving j(N(N +1)) differential equations where N is the sum of the numbers of plant states and unknown parameters. For example, to estimate two parameters of a third-order plant requires the real-time solution of 15 differential equations, of which 10 are non-linear. Adding one more state raises the number of equations to 21, which illustrates the rapid growth of complexity for such a method.