Array algorithms for H^2 and H^∞ estimation

Currently, the preferred method for implementing H^2 estimation algorithms is what is called the array form, and includes two main families: square-root array algorithms, that are typically more stable than conventional ones, and fast array algorithms, which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Using our recent observation that H^∞ filtering coincides with Kalman filtering in Krein space, in this chapter we develop array algorithms for H^∞ filtering. These can be regarded as natural generalizations of their H^2 counterparts, and involve propagating the indefinite square roots of the quantities of interest. The H^∞ square-root and fast array algorithms both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H^∞ filters. These conditions are built into the algorithms themselves so that an H^∞ estimator of the desired level exists if, and only if, the algorithms can be executed. However, since H^∞ square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H^2 case, further investigation is needed to determine the numerical behavior of such algorithms

Stochastic H-infinity tracking with preview for state-multiplicative systems

The problem of finite-horizon H∞ tracking for linear time-varying systems with stochastic parameter uncertainties is investigated. We consider three tracking patterns depending on the nature of the reference signal, i.e., whether it is perfectly known in advance, measured on line or previewed in a fixed time-interval ahead. The stochastic uncertainties appear in both the dynamic and measurement matrices of the system. For each of the above three cases a game theory approach is applied for the state-feedback case where, given a specific reference signal, the controller plays against nature which chooses the initial condition and the energy-bounded disturbance. The problems are solved using an expected value of the standard performance index over the stochastic parameters, where necessary and sufficient conditions are found for the existence of a saddle-point equilibrium. The infinite-horizon time-invariant tracking problem is also solved. The theory developed is demonstrated by a simple tracking example. © 2004 IEEE

Robust H-infinity filtering of stationary continuous-time linear systems with stochastic uncertainties

The problem of applying H∞-filters on stationary, continuous-time, linear systems with stochastic uncertainties in the state-space signal model is addressed. These uncertainties are modeled via white noise processes. The relevant cost function is the expected value of the standard H∞ performance index with respect to the uncertain parameters. The solution is obtained via a stochastic bounded real lemma that results in a modified Riccati inequality. This inequality is expressed in the form of a linear matrix inequality whose solution provides the filter parameters. The method proposed is also applied to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H2/H∞ filtering for the above system is also treated. The theory developed is demonstrated by a practical example

Robust H-infinity filtering of stationary continuous-time linear systems with stochastic uncertainties

The problem of applying H∞-filters on stationary, continuous-time, linear systems with stochastic uncertainties in the state-space signal model is addressed. These uncertainties are modeled via white noise processes. The relevant cost function is the expected value of the standard H∞ performance index with respect to the uncertain parameters. The solution is obtained via a stochastic bounded real lemma that results in a modified Riccati inequality. This inequality is expressed in the form of a linear matrix inequality whose solution provides the filter parameters. The method proposed is also applied to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H2/H∞ filtering for the above system is also treated. The theory developed is demonstrated by a practical example