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This paper considers the design of output regulators with the use of approximate models. The measure of the approximation between process and model outputs is represented by a bound in norm of the output error signals and it requires the computation of two numbers. The design is achieved with a rain-max approach where control and error signals are the two antagonists. The min-max solution is obtained as a linear function of the model state (open-loop solution). It is shown that no dosed-loop controls can improve the open-loop min-max performance. Conditions are given so as to preserve the min-max performance by means of proportional feedback of the system's output. In this case the min-max feedback law is obtained (closed-loop solution).