Abstarct: This thesis considers a class of time-delayed fractional-order and integer-order examines the stability and controller design methods for these systems‎. ‎One of the phenomena that occur in most physical systems is the existence of time-delay‎, ‎so stability analysis and controller design in the presence of delay are necessary‎. ‎Also in controller design‎, ‎since system input is costly‎, ‎so to reduce costs and the amplitude of the control signal fluctuations‎, ‎the parameter state feedback matrix‎, ‎and determine the parameter values in such a way that the norm matrix of the feedback is minimized and the system time control is optimized‎. ‎This thesis is organized into two parts‎. ‎The first part examines some of the proposed methods in the field of stability analysis and controller design integer-order delay systems and fractional-order systems‎. ‎In the second part‎, ‎we propose a method baxsed on similarity transforms for the stabilization of continuous and discrete-time systems‎, ‎as well as the minimization of the norm feedback matrix‎, ‎we do this by solving a nonlinear programming problem and comparing it with the methods presented‎. ‎Finally‎, ‎the advantages of the proposed methods using simulation are examined on different examples and the results show the simulation‎, ‎effects‎, ‎and efficiency of the proposed method‎.