Design of Linear Regulators for Nonlinear Stochastic Systems

The problem of regulating a nonlinear stochastic system with uncertain parameters and nonlinear noisy measurements is considered. The second-order terms in an expansion about the reference path are viewed as multiplicative disturbances in state and control. The linear estimation/control problem is solved by approximating these as wideband disturbances. The Certainty-Equivalence Principle and the Separation Theorem do not hold. The control and filter gains, found by solving a TPBVP, explicitly hedge against the parameter uncertainties and nonlinearity effects. Numerical results are presented for a combined guidance/navigation system for shuttle entry. this paper a new technique for design of linear regutors for nonlinear, time-varying stochastic systems with certain parameters is developed. It is assumed that the desire is to follow a known reference path with minimum error. The available measurements are a set of known nonlinear functions of both the state and uncertain parameters. The performance index is a weighted sum of mean squared state deviations and estimation errors. A widely used solution technique is to transform to a linear variational problem by linearizing about a desired reference path in state space.1'2 The familiar linear-quadratic-Gaussian (LQG) theory is then applied to derive a set of linear time-varying control feedback and filter gains. These gains are derived independently since, by the LQG assumption, both the Certainty-Equivalence Principle2 and the Separation Theorem3'4 hold. The LQG design works well if the effects of nonlinearities and parameter uncertainties are suitably small. However, since the LQG approach explicitly assumes both linearity and perfectly known parameters, it does not attempt to compensate for nonlinearities or parameter uncertainties. The method of approach used here involves consideration of both the first- and second-order terms in the Taylor series expansion about a reference path. By inspection of the second-order terms, it is seen that nonlinearities and parameter uncertainties act as disturbances which multiply the state (path deviations), control, and estimation errors. These disturbances are modeled as wideband processes. With this assumption, the multiplicative uncertainties appear as white state-dependent noise (SDN) and control-dependent noise (CDN). The control of linear systems with SDN and CDN and quadratic cost function with perfect state information has been considered by several authors and a rather complete theory is available.5'8 By assuming a linear filter structure, the theory yields a set of time-varying control feedback and filter gains, which can be computed a priori* by solving a two-point boundary-value problem (TPBVP). An especially interesting aspect of this approach is that control penalties are not necessarily required in the formulation of the optimization problem in order to obtain a solution. The problem is made nonsingular in the continuous case (which is not considered here; see Ref. 9) by the presence of CDN which adds convexity to the problem and acts essentially as additional control penalty. This provides a systematic and natural approach to determination of control weighting functions.