# Analytical Design of Fractional-Order PI Controller for Parallel Cascade Control Systems
## 1 Introduction  
Cascade control is commonly used in industrial processes. It is adopted to reduce disturbance and improve the servo response of the closed-loop system. The traditional cascade control structure basically consists of the primary (outer) and the secondary (inner) loops where both manipulated and disturbance variables affect the primary output through the secondary output. The parallel cascade control (Figure 1) was proposed based on some real application situations where the manipulated and disturbance variables (\(d\) and \(u_{2}\)) simultaneously affect the primary and secondary output (\(y_{1}\) and \(y_{2}\)). [PERSON] first introduced parallel cascade control structure (PCCS) to solve problems of the composition control of a distillation column, which is the reflux flow rate (manipulated variable) and the feed flow or composition (disturbance) affecting the purity of the overhead product (primary output) and the tray temperature (secondary output) [1]. Currently, although there have been many advanced control techniques, PI/PID controllers are still widely used in controller designs of the inner and outer loop of the cascade control. However, the number of works related to parallel cascade control is still limited due to the complexity of the control structure as well as the tuning procedure. In Ref. [2], a simple method was proposed for unstable processes, in which a proportional (P) controller and a proportional-integral (PI) were considered for the secondary and the primary loop, respectively. However, the obtained results were not good, with a largeovershoot compared to other methods. A modified PCCS was suggested in Ref. [3] to deal with stable, unstable, and integrating processes. The authors adopted a proportional-derivative (PD) controller to stabilize the unstable/integrating processes and used the internal model control (IMC) approach to design the controller for disturbance rejection in the secondary loop. A disadvantage of the cascade control scheme is that if the primary process has a large time delay, some available tuning rules in the literature may not give a good servo performance [1; 2; 4]. Therefore, many researchers proposed to use a time-delay compensator (Smith predictor) for the outer loop [3; 5; 6; 7; 8; 9]. [PERSON] et al. [5] originally combined PCCS and the Smith predictor for stable processes with a large time delay and achieved satisfactory closed-loop performances. In Ref. [5], the authors still used the IMC approach for the inner loop and direct synthesis for the outer loop, and also included a setpoint filter for the secondary loop. [PERSON] and [PERSON] [7] improved the PCCS for a class of stable, unstable, and integrating processes with time delay. The PID controllers in Ref. [7] were designed based on a loop shaping technique and the set-point filter was also included using the integral squared error (ISE) index. In order to enhance the servo performance, a modified Smith predictor was also adopted in the primary loop. Recently, fractional calculus [10] has been paying increased attention to fractional-order processes both from the academic and control engineers for the modeling and control issues. On the one hand, fractional-order dynamic systems are useful in representing various stable physical phenomena which increasing flexibility with less computational cost. For example, in Ref. [11], the authors described imperfections in integrated devices due to parasitic nonlinearities and they would enrich the system dynamics as well as manifest hidden system behaviors. On the other hand, the implementation of fractional-order systems is realizable by several methods which are easy to carry out by control engineers [12]. The generalization of the PID controller, which is called the PI\({}^{\lambda}\)D\({}^{\mu}\)[13], is involved in two extra parameters as the fractional-order integrator (\(\lambda\)) and fractional-order differentiator (\(\mu\)).