In this paper the standard problem of $H^\infty $ control theory for finite-dimensional linear time-varying continuous-time plants is considered. The problem is: given a real number $\gamma > 0$, find (if one exists) an internally stabilizing controller such that the closed-loop operator norm is less than $\gamma $. Under rather weak assumptions on the plant model, it is shown that a solution to this problem exists if and only if a pair of matrix Riccati differential equations admits positive semidefinite stabilizing solutions. State-space formulae for one solution to the problem are also given.

H_∞ control of linear timevarying system : A state space approach

In this chapter we consider the finite horizon H ∞ performance robustness problem with parameter variations. Assuming the adequacy of linear expressions for performance variation, an iterative procedure is given to synthesize a suboptimal H ∞ controller, which yields the required performance even under parameter variations. As a by-product, an expression for the variation of performance due to parameter variations is given for this specific controller by making use of variational theory. An example which illustrates the methodology is worked out under parameter uncertainties.

Finite horizon H∞ with parameter variations

Considers the finite horizon H/sub /spl infin// performance robustness problem with parameter variations. Assuming a nominal model, an iterative procedure is given to synthesize a suboptimal H/sub /spl infin// controller, which yields the required performance even under parameter variations. As a by-product, an expression for the variation of performance owing to parameter variations is given for this specific controller. An example which illustrates the methodology is worked out under parameter uncertainties. >

Finite horizon H/sub /spl infin// with parameter variations

The author presents a criterion for the evaluation of the infimal H/sub infinity / norm in both continuous- and discrete-time systems. The criterion consists of choosing the first positive value of a discrete set of values for which a certain two-point boundary value problem has a nontrivial solution. This characterization can be used to develop a fast efficient algorithm for the computation of the infimal H/sub infinity / norm. The problem formulation in the continuous case is very general in that time-varying systems are considered and nonzero initial and final conditions are incorporated into the performance index. >

http://ieeexplore.ieee.org/iel5/1040/8509/00371175.pdf

Evaluation of the infimal H/sub infinity / norm in continuous and discrete systems

In this paper we develop necessary conditions for a minimax problem involving control and exogenous inputs. The problem can be regarded as a finite horizon version of the H∞ optimal control problem. We consider problems involving generalized cost functional and non-zero initial conditions. A criterion for the evaluation of the performance index is given in these cases. Our computational experience shows that the finite horizon performance is useful in computing the infimal H∞ norm in the infinite horizon case, as the final time becomes large. Also, expressions are derived for the variation in performance in terms of system parameter variations. These linear expressions are useful in the evaluation of the robustness of the proposed control strategy.

M. B. Subrahmanyam

Papers

Computation of infimal H∞ norm over a finite horizon