## 1 Introduction
[29] adopted Routh-Pade approximation for order reduction of interval systems and also proposed two simple expressions for computing time moments (TMs) and Markov parameters (MPs). A thorough literature survey on model order reduction (MOR) methods based on Routh approximation for discrete and continuous interval systems is given in [30]. Later, the order reduction of higher order continuous interval systems is carried out using frequency domain reduction techniques [31] where the denominator of the model is derived from a differentiation method and the numerator is achieved using factor division, differentiation and Pade approximation methods. Another concept of order reduction of interval systems is stated depending on ISE minimization and impulse response energy using modified particle swarm optimization (PSO) algorithm by [PERSON] et al. [32]. Model order reduction of linear time-invariant continuous and discrete interval systems based on [PERSON]'s theorem using the differential method was presented by [PERSON] and [PERSON] [33]. The resulting interval model preserves all the dominant characteristics of original interval system. The frequency domain fractional-order controllers are designed generally based on the specifications [34] like phase margin, gain crossover frequency, steady-state error cancellation, high-frequency noise rejection and good output disturbance rejection. Authors in [35] extended the differentiation method for reduced order modeling where the stability of lower order models is always guaranteed. In addition, it retains the initial TMs of higher order systems. Further, a new approach of order reduction based on the concept of impulse response gramian is developed in the frequency domain in [36]. Recently, [PERSON] et al. [37] proposed a multipoint Routh-Pade approximation technique for single-input-single-output (SISO) and multi-input-multi-output (MIMO) interval systems. These methods assure a stable reduced model for higher order continuous interval systems. This research presents a design of a robust fractional-order proportional-integral-derivative (FOPID) controller for reduced-order continuous interval plant (ROCIP) of higher-order continuous interval plant (HOCIP) based on considered frequency domain specifications using an advanced sine-cosine algorithm (SCA). Later, this controller is cascaded with HOCIP. Here, PM and \(\omega_{gc}\) are considered as performance criteria of HOCIP for tuning the five parameters of FOPID controller. Since the design of the controller is done in the frequency domain, this approach guarantees the robustness and stability of the system. An improved Routh-Pade approximation method is utilized for reduced order modeling of HOCIP. The modified Routh table is utilized to derive the denominator polynomial of interval model whereas the coefficients of numerator polynomial are obtained by matching few TMs and MPs of system with those of the model. This order reduction technique always yields stable ROCIPs for stable HOCIPs. A third order interval plant is considered as an example to demonstrate the whole procedure of FOPID controller tuning and model order reduction. The efficiency of the proposed method for design of the FOPID controller is proved from the simulation results obtained. The article is structured into seven sections: Section 1 includes the detailed literature survey on the present investigation. The preliminaries of fractional calculus and the structure of FOPID controller are given in Section 2. Problem formulation is briefed in Section 3. The design procedure for the proposed FOPID controller is illustrated in Section 4. In Section 5, the order reduction method for HOCIP is presented. Section 6 explains the SCA in detail. Section 7 includes the test case and its simulation results. Finally, the conclusions are reported in Section 8.