## 1 Introduction  
In recent years, the fractional-order proportional-integral-derivative (FOPID) controller, which is first proposed by [PERSON] [1], has attracted more attention of many researchers in the field of control systems. The FOPID has five tuning parameters including proportional, integral, derivative gain and fractional orders of the integral and derivative terms. In the special case, when the fractional orders are equal to unity, the controller becomes a conventional proportional-integral-derivative (PID). Therefore, it is considered as a generalization of the PID controller to non-integer orders, and it provides more flexibility in controller design as well as better dynamic performances and robustness compared with the conventional one [1; 2; 3; 4]. Due to more tuning parameters, it is harder to derive analytical tuning rules for the controller. Different tuning methods have been suggested to solve this kind of problem [5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. However, most of them are used to deal with single-input single-output (SISO) systems. In this paper, a novel structure of FOPID is proposed to apply for two-input two-output (TITO) processes. In this work, the simplified decoupling Smith predictor structure (SDSP) for multivariable processes proposed by [PERSON] et al. [15] is adopted to deal with interactions between process variables as well as multiple time delays of the processes. The realizability problem plays a crucial role in implementing a decoupler because of the requirements of being stable and proper of all its internal transfer functions. To overcome this issue, therefore, many researchers proposed the approximation approaches such as prediction error method (PEM), linear least square in frequency domain [16], and coefficient matching (CM) [17; 18]. However, these mentioned methods are only suitable for reducing to integer-order transfer functions. In this paper, to enhance dynamic behaviors of decoupled systems, fractional order processes are addressed to be the equivalent transfer functions of the decoupled elements. Therefore, the whole controller structure used in this work is called fractional simplified decoupling Smith predictor (F-SDSP). The particle swarm optimization (PSO) algorithm for approximation procedure proposed in [15] is employed to find out the parameters of approximated fractional functions. [PERSON] et al. [14] performed a similar work by using the Genetic Algorithm (GA) to reduce a high integer-order transfer function of a SISO system to a lower fractional-order one. Besides, to improve the performances of a system, a generalized PI/PID controller, known as fractional-order PI/PID (FOPI/FOPID) controller, is suggested for decoupled systems. In the last two decades, various methods for FOPI/FOPID design were proposed in the field of process control. Among those, there are two prominent approaches: The internal model control (IMC)-based procedure and the gain and phase margin-based frequency domain design. The first one uses IMC scheme to reduce the number of tuning parameters; and normally, there is only one parameter left needs to be tuned based on some criteria of robust performances such as maximum peak (\(M_{p}\)) or maximum sensitivity (\(M_{s}\)) [11; 12; 13; 14]. The second one uses constraints on phase margin, gain crossover frequency and the flat phase around the gain crossover frequency to ensure robustness performance [5; 6; 7; 8; 9; 10]. The most disadvantage of the latter is that those constraints are only enough to solve three tuning parameters which means merely to be appropriate to a FOPI controller (a FOPID has five tuning parameters). Furthermore, most of the previous works only deal with SISO processes. Therefore, expanding some existing design methods to multivariable processes is necessary and deserves to attract more attention from researchers. In this paper, the IMC-based FOPI/FOPID design is adopted to find out analytical tuning rules of both FOPI and FOPID controllers for TITO processes.