Optimal observers for continuous time linear stochastic systems

In the past, a major objection to application of the well known solution to the linear stochastic optimal control problem has been that the complexity, or dimension, of the solution exceeds that which is necessary for satisfactory feedback control. For a system described by n first order linear differential equations excited by white noise, the solution requires a filter of dimension n. If non white noise sources excite the system, an even higher dimensional filter is required. This paper presents a technique which alleviates the above objection. A recursive algorithm similar to the Kalman filter algorithm is presented which permits design of a reduced order linear estimator to replace the well known Kalman filter. The new estimator, called an observer, is stochastically optimal subject to its reduced order dimensionality constraint, but its performance is not as good as a full Kalman filter. The observer algorithm is general in that it applies to time variable, multivariable systems. A minimum order = n − m 1 exists for the observer. Here n = state dimension and m 1 = dimension of the non white measurement. Non white noise sources of any order q may exist and need not contribute to the dimension of the optimal observer. Optimal observers of all orders between n − m 1 and n − m 1 + q may be designed, the latter case being a Kalman filter. Detailed examples are given to illustrate the theory.