Optimal Stationary Control of a Linear System with State-Dependent Noise

Abstract¶

Consider the linear control system described by the formal, vector stochastic differential equation x_dot = A x - B u + C w_dot_1 + G(x) w_dot_2. In this equation, u is the control and w_dot_1, w_dot_2 are independent Gaussian white noise disturbances. The elements of the matrix G are assumed to be linear in x; and so the term G(x) w_dot_2 represents a disturbance of which the intensity is roughly proportional to the deviation of x from the origin x = 0. Equivalently, the disturbance can be regarded as a wideband random perturbation of the system matrix A. Now consider the problem of choosing a feedback control u = phi(x) such that, in the steady state, the expected quadratic cost E{x'Mx + u'Nu} is a minimum. If G(x) = 0, the solution of this problem is well known. Under mild restrictions the optimal control always exists and is a linear function of x which is independent of the intensity of the additive disturbance C w_dot_1. In the present article it is shown that an optimal control exists for the more general system above, provided the state-dependent noise G(x) w_dot_2 is sufficiently small. The optimal control is again linear, but is now rather critically dependent on the coefficients of G. Examples are provided to show that instability may result if this dependence is ignored. The problem is stated precisely in section 2; the proof of existence is given in sections 3 and 4; and some examples are studied in sections 5 and 6. We conclude with some remarks on the interpretation of the system equation and discuss alternative optimization problems which are closely related.

Summary¶

Extends LQR to linear systems subject to multiplicative (state-dependent) Gaussian noise, deriving a modified algebraic Riccati equation whose stabilizing solution yields the optimal stationary feedback gain. Establishes existence and uniqueness conditions and lays the groundwork for stochastic LQR theory.

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