This expository paper sets out the principal results in ${\bf H}_\infty $ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods.

Linear control theory with an H ∞ 0E optimality criterion

A sufficient condition is derived for the existence of a stable suboptimal /spl Hscr//sup /spl infin// controller. It is shown that if a certain algebraic Riccati equation, determined from a parameterization of all suboptimal /spl Hscr//sup /spl infin// controllers, has a positive definite stabilizing solution, then a stable /spl Hscr//sup /spl infin// controller of order 2n can be constructed, where n is the order of the generalized plant.

On the synthesis of stable /spl Hscr//sup /spl infin// controllers

The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optima) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L∞ norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp ( —τs)/(Ts + 1). Numerical experience indicates that L°° optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L∞ approximation errors than Hankel optim...

Approximation of delay systems—a case study

We present two constructions of controllers that globally stabilize linear systems subject to control saturation. The only conditions imposed are the obvious necessary ones, namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition holds. We use essentially arbitrary saturations /spl sigma/, subject only to the requirement that: (i) /spl sigma/ is locally Lipschitz, (ii) s/spl sigma/(s)>0 whenever s/spl ne/0, (iii) /spl sigma/ is differentiable at 0 and /spl sigma/'(0)>0, and (iv) lim inf/sub |s|/spl rarr//spl infin//|/spl sigma/(s)|>0. >

A general result on the stabilization of linear systems using bounded controls

A state space based design method is given to find strongly stabilizing controllers for multiple-input-multiple-output plants (MIMO). A sufficient condition is derived for the existence of suboptimal stable H/sup /spl infin// controller in terms of linear matrix inequalities (LMIs) and the controller order is twice that of the plant. A new parameterization of strongly stabilizing controllers is determined using linear fractional transformations (LFTs).

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Remarks on strong stabilization and stable H/sup /spl infin// controller design

The problem of stabilizing a single-input single-output system using stable feedback is considered. The performance index that is minimized is the weighted sensitivity of the closed-loop system to external disturbances. However, the resulting feedback associated with the minimum value of the optimality criterion turns out to be irrational. A design algorithm has been developed to find an approximation whose deviation from a sub-optimal system lies within a pre-specified error bound.

C. Ganesh

Papers

Design of Optimal Control Systems with Stable Feedback