Discrete-time H ∞ and H 2 control with structured disturbances and probability-relaxed requirements

This paper presents a method that reduces the conservatism inherent in the disturbance representation in standard H _∞ theory. It addresses several discrete-time control synthesis problems where the disturbances acting on the system are structured by a convex family of linear filters (‘signal polytope’) and where the system disturbance attenuation level attained over the signal polytope is ensured to (just) a prescribed probability. This setup enables realistic multi-feature disturbance delimiting while accounting for uncertainty in the disturbance model itself by allowing probability waivers. The core of the probability aspects of the proposed solutions is the search for a truncated signal polytope which provides both the required probability and the best robust disturbance attenuation level. Many examples are given, including one of an aircraft output-feedback control with a polytope of low-pass filters representing different wind phenomena. The examples demonstrate that addressing realistic disturbances results in better control designs (hence better performance) and that a small certainty waiver can yield a large performance gain.     

Two-degree-of-freedom H∞-optimization ofmultivariable feedback systems

A solution to the two-degree-of-freedom H_∞ -minimization problem that arises in the design of multivariable optimal continuous-time stochastic control systems is derived. A decoupling approach that enables a partially independent design of the prefilter and the feedback controller and yields a simple solution to the optimization problem is applied. This solution is obtained by transforming the optimization problem into two standard form (four-block) problems     

On regional nonlinear H-filtering

The structure of the nonlinear H^∞-filter in the neighborhood of the estimated trajectory is investigated and a bound on the size of the neighborhood that allows this structure is determined, both for finite and infinite horizons. Riccati inequalities that depend on the estimated trajectory are derived for finding the filter gain matrix and an algorithm for calculating the bound on the size of the above neighborhood is presented. Explicit formulas are obtained in the infinite horizon case for the minimum achievable disturbance attenuation level, the size of the neighborhood, and the corresponding filter gain.     

A simple solution to the singular linear minimum-variance estimation problem

A simple expression is obtained for the transfer function matrix of the minimum-variance estimator for the states of a linear stationary continuous-time right invertible system whose output is measured perfectly. Using thes-domain approach, it is shown that the optimal estimator first finds the driving noise input that achieves, when applied on the minimum-phase image model of the system, an output spectrum which is identical to the measurement spectrum. This input is then applied on a state-space representation of the minimum-phase image to produce the optimal estimate.     

Game theory approach to optimal linear state estimation and itsrelation to the minimum H∞-normestimation

A possible game theory approach to optimal state estimation is presented. It is found that in a certain differential game, the minimizer's policy is identical to the one obtained by optimal estimation in the minimum H_∞-norm sense. This interpretation of H_∞-optimal state estimation provides better insight into the mechanism of H_∞ -optimal filtering, especially in the case where the exogenous signals are not energy bounded     

The design of optimal reduced order stochastic observers fordiscrete-time linear systems

The minimum variance state estimation of linear discrete-time systems with random white noise input and partially noisy measurements is investigated. An observer of minimal-order that attains the minimum-variance estimation error is found. The structure of this observer is shown to depend strongly on the geometry of the system. This geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate. The transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise. An explicit expression is found for the transfer function matrix of the observer, from which a simple solution to the linear discrete-time singular optimal filtering problem is obtained     

The zero properties of linear passive systems

The zeros and the slope of the asymptotes to the root-loci of linear, passive, time-invariant systems are found. Using the geometric approach to the definition of zeros, it is proved that the system finite zeros are all in the left half of the complex plane and that the asymptotes to the system root-loci at infinity are all in the direction of the negative real axis.     

Visibility and Dead-Zones in Digital Terrain Maps

The problem of detecting the visible and hidden areas of a topographic surface from a given viewpoint is well known, and has applications in GIS and other fields. The computational geometry literature is rich with solutions based on a polygonal representation of the surface. This work confronts the problem by working directly on the Digital Elevation Map which represents the terrain surface. We present an algorithm that processes discrete lines of sight from the viewpoint to the surface perimeter, and tests the unit-sized terrain elements along the discrete cross-sections defined by these lines. The algorithm is very efficient, performing O(n) testing operations, consisting of a few additions and no more than one multiplication each, where n is the number of the terrain elements in the map.