Every stabilizing controller is l<SUB>1</SUB>- and H<SUB>∞</SUB>-optimal

A new formulation is given for the problem of optimally rejecting possibly unbounded disturbances. It is shown that, for a given plant, every particular stabilizing controller is l<SUB>1</SUB>- and also H&#8734;-optimal for some particular classes of appropriately chosen disturbances. An example demonstrates the authors' work

l<SUB>1</SUB>-optimality of feedback control systems: the SISO discrete-time case

Controllers that optimally reject a class of disturbances are considered. The problem of determining when a stabilizing control is l1-optimal for a given plant is studied for some stable weighting function. This problem belongs to the class of inverse problems in optimal control introduced by Kalman. It is shown that for a given plant, the set of all the H&#8734;-optimal controllers (obtained by considering all stable weighting functions with no zeros on the unit circle) is actually contained in the corresponding set of l1-optimal controllers. It is also demonstrated that an l1-optimal controller, unlike an H&#8734;-optimal controller, can remain l1-optimal for the same plant for a wide range of nontrivially different weighting functions