A game theoretic controller and its relationship to H∞ and linear-exponential-Gaussian synthesis

A simple derivation of a controller whose structure is identical to that of both the linear-exponential-Gaussian and H_∞ controllers is presented. The controller is determined by a game-theoretic approach, in which the control plays against adversaries consisting of the process and measurement disturbances and the initial state. The game formulation considers a quadratic performance index subject to linear time-varying dynamics and measurements. The resulting solution produces an n-dimensional compensator which compactly expresses the controller as a linear combination of the measurement history. The controller requires the solution to two Riccati differential equations (RDE). Conditions for saddle point strategies are given. If the infinite-time, time-invariant game formulation is considered, and the existence of a nonnegative definite solution to the algebraic Riccati equation (ARE) is assumed, the solution to the RDE with certain initial condition is shown to be nonnegative definite and bounded from above, and to converge to the minimal nonnegative definite symmetric solution of the ARE. Given these results, an H_∞ norm is shown to be bounded from above. By modeling parameter uncertainty as a fictitious input to a linear time-invariant system, the game theory approach becomes applicable