As is often the case in practice, classical control theory does not fully provide sufficient tools for solving applied problems of a certain specific class. Such problems include automatic control of waste heat boilers where physical restrictions are imposed on controls, namely, they must belong to the class of non-negative functions with restrictions on amplitude and growth rate. In the theoretical part of this paper, within the framework of the block approach, methods for the synthesis of general linear systems with one-sided restrictions on control actions and state variables were developed. Then, the developed algorithms were applied to the waste heat boiler con-trol system under conditions of incomplete measurements of state variables and the action of parametric and external uncontrolled disturbances. The effectiveness of the proposed algorithms was confirmed by numerical simulation.