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     

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     

Simplified adaptive estimation

A simplified adaptive scheme is suggested for the estimation of the state vector of linear systems driven by white process noise that is added to an unknown deterministic signal. The design approach is based on embedding the Kalman filter (KF) within a simplified adaptive control loop that is driven by the innovation process. The simplified adaptive loop is idle during steady-state phases that involve white driving noise only. However, when the deterministic signal is added to the driving noise signal, the simplified adaptive control loop enhances the KF gains and helps in reducing the resulting transients. The stability of the overall estimation scheme is established under strictly passive conditions of a related system. The suggested method is applied to the target acceleration estimation problem in a Theater Missile Defence scenario.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also 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 simple tracking example.     

Nondefinite least squares and its relation toH∞-minimum error state estimation

The problem of recursive nondefinite least-squares state estimation of continuous-time stationary processes is solved, by applying Pontryagin's maximum principle. A comparison of the derived solution to the result that is obtained for the H_∞ -minimum error estimation suggests a new interpretation for the H_∞-optimal estimation mechanism. According to this interpretation, the estimator tries to optimally estimate the required combination of the states, in the l₂-norm sense, against the worst disturbance signal that stems from a fictitious measurement of this combination     

Minimum H∞-norm regulation of lineardiscrete-time systems and its relation to linear quadratic discretegames

A solution is derived to the H_∞-optimization problem that arises in multivariable discrete-time regulation when the controller has full access to the state vector. The solution method is based on the close relations that exist between linear quadratic differential game theory and H_∞-optimization. The existing theory of discrete-time quadratic games is readily applied in order to derive the solution to a finite-time horizon version of the H_∞ -optimization problem. The solution of the infinite-time horizon H_∞-optimization problem is obtained by formally taking the limit of the number of stages to infinity     

H∞ tracking of linear continuous‐time systems with stochastic uncertainties and preview

The problem of finite‐horizon H_∞ tracking for linear continuous time‐invariant systems with stochastic parameter uncertainties is investigated for both, the state‐feedback and the output‐feedback control problems. 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. In the state‐feedback case, for each of the above three cases a game theory approach is applied 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 the expected value of the standard performance index over the stochastic parameters, where, in the state‐feedback case, necessary and sufficient conditions are found for the existence of a saddle‐point equilibrium. The corresponding infinite‐horizon time‐invariant tracking problem is also solved for the latter case, where a dissipativity approach is considered. The output‐feedback control problem is solved as a max–min problem for the three tracking patterns, where necessary and sufficient condition are obtained for the solution. The theory developed is demonstrated by a simple example where we compare our solution with an alternative solution which models the tracking signal as a disturbance. Copyright © 2004 John Wiley & Sons, Ltd.     

Static output-feedback of state-multiplicative systems with application to altitude control

A linear parameter dependent approach for designing a constant output-feedback controller for a linear time-invariant system with stochastic multiplicative Wiener-type noise, that achieves a minimum bound on either the stochastic H ₂ or the H _∞ performance level is introduced. A solution is achieved also for the case where in addition to the stochastic parameters, the system matrices reside in a given polytope. In this case, a parameter dependent Lyapunov function is introduced which enables the derivation of the required constant gain via a solution of a set of linear matrix inequalities that corresponds to the vertices of the uncertainty polytope.

The stochastic uncertainties appear in both the dynamic and the measurement matrices of the system. The problems are solved using the expected value of the standard performance index over the stochastic parameters. The theory developed is applied to an altitude control example.     

Stochastic H/sub /spl infin// tracking with preview for state-multiplicative systems

The problem of finite-horizon H/sub /spl infin// 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.     

Min–max Kalman filtering

The problem of H_∞-optimal state estimation of linear continuous-time systems that are measured with an additive white noise is addressed. The relevant cost function is the expected value of the standard H_∞ performance index, with respect to the measurement noise statistics. The solution is obtained by applying the matrix version of the maximum principle to the solution of the min–max problem in which the estimator tries to minimize the mean square estimation error and the exogenous disturbance tries to maximize it while being penalized for its energy. The solution is given in terms of two coupled Riccati difference equations from which the filter gains are derived. In the case where an infinite penalty is imposed on the energy of the exogenous disturbance, the celebrated Kalman filter is recovered. In the stationary case, where all the signals are stationary, an upper-bound on the solutions of the coupled Riccati equations is obtained via a solution of coupled linear matrix inequalities. The resulting filter then guarantees a bound on the estimation error covariance matrix. An illustrative example is given where the velocity of a maneuvering target has to be estimated utilizing noisy measurements of the position.