# An Analytical Design of Simplified Decoupling Smith Predictors for Multivariable Processes
## 1 Introduction  
Multivariable processes with multiple time delays are common in most industrial systems. For such systems, interactions between process variables cause difficulties in feedback controller design. Most advanced control strategies deal with that kind of problem by two different approaches: decentralized and centralized control. To design a decentralized control, most control engineers normally adopt traditional single-loop PID controllers for multivariable systems, known as multi-loop PI/PID controllers, because of their simplicity, effectiveness, and easy implementation [1; 2; 3]. However, they only work well for modest interactive processes. In systems with strong interactions between channels, centralized control is advisable. Its scheme can be classified into two kinds: a decoupling network in combination with a diagonal matrix controller [4; 5; 6] or a pure centralized strategy [7; 8; 9]. In this work, therefore, the centralized control with the decoupling network is addressed to convert a multi-input multi-output (MIMO) system into a multi-loop single-input single-output (SISO) system and conventional design approaches are used to control the whole system. In practice, simplified decoupling is the favorite choice because of its simple decoupling network, the unitary diagonal elements, and robustness [4; 5]. When solving decoupling problems, one has to approximate very complicated transfer functions to some standard forms and also deal withrealizability issues, which are a major cause of improper (non-causal) transfer functions [5; 6; 7; 10; 11; 12]. Numerous approximation techniques or reduced-order models are available, such as the prediction error method (PEM), the least-squares algorithm, the linear least-squares approximation [7] or Gaussian approach in the frequency domain, and the coefficient matching (CM) method [2; 5]. However, the major disadvantage of these mentioned method is the computational burden when the order of MIMO systems increases. In this work, the particle swarm optimization (PSO) algorithm was adopted to approximate the complicated functions of decoupling elements to some well-known forms of industrial processes, such as first-order plus time delay (FOPTD), second-order plus time delay (SOPTD), and second-order plus time delay with negative zero (SOPTDNZ) systems. A minor modification was proposed to handle the improper transfer functions which may be obtained from the proposed algorithm. However, time delays still exist in the diagonal elements of the apparent decoupled matrix. It is obvious that processes with significant dead time are hard to analyze and design based on standard feedback controllers and also result in sluggish responses in outputs. The Smith predictor (SP) scheme, known as the dead time compensator (DTC), is an appropriate method to deal with this situation by removing the delay terms from the closed-loop transfer functions [13]. However, the original scheme is only applied for SISO systems with their controllers tuned for a tradeoff between robustness and performance. To overcome this drawback, many variations of the SP have been proposed, and the most frequently used structure is a first-order lag filter (FOLF) on the feedback loop, also called the filter Smith predictor (FSP) [14; 15]. From this method, the DTC structure can handle integrating and unstable processes [16; 17]. In addition, other scholars have also extended this scheme to MIMO cases as well [18; 19; 20; 21; 22; 23]. In order to expand the SP to MIMO systems, two approaches normally can be used. The first one is an FSP structure which is applied directly to the whole multivariable system [18; 19; 21; 22]. The second one is done in combination with decoupling techniques which simplify the transfer function matrix of processes and then address an SP scheme for each single loop like that of SISO systems [20]. In the present work, the latter was adopted by using the simplified decoupling technique and the original SP for the sake of simplicity.