Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 818353, 22 pages

http://dx.doi.org/10.1155/2015/818353

## Singularly Perturbation Method Applied To Multivariable PID Controller Design

^{1}Department of Control and Mechatronics Engineering, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia^{2}Department of Industrial Systems Engineering and Design, Jaume I University of Castello, 12080 Castello de la Plana, Spain^{3}Department of Industrial Electronics, Faculty of Electronics and Computer Engineering, Universiti Teknikal Malaysia Melaka, 76100 Durian Tunggal, Melaka, Malaysia

Received 8 April 2015; Accepted 4 June 2015

Academic Editor: Herve G. E. Kadji

Copyright © 2015 Mashitah Che Razali et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Proportional integral derivative (PID) controllers are commonly used in process industries due to their simple structure and high reliability. Efficient tuning is one of the relevant issues of PID controller type. The tuning process always becomes a challenging matter especially for multivariable system and to obtain the best control tuning for different time scales system. This motivates the use of singularly perturbation method into the multivariable PID (MPID) controller designs. In this work, wastewater treatment plant and Newell and Lee evaporator were considered as system case studies. Four MPID control strategies, Davison, Penttinen-Koivo, Maciejowski, and Combined methods, were applied into the systems. The singularly perturbation method based on Naidu and Jian Niu algorithms was applied into MPID control design. It was found that the singularly perturbed system obtained by Naidu method was able to maintain the system characteristic and hence was applied into the design of MPID controllers. The closed loop performance and process interactions were analyzed. It is observed that less computation time is required for singularly perturbed MPID controller compared to the conventional MPID controller. The closed loop performance shows good transient responses, low steady state error, and less process interaction when using singularly perturbed MPID controller.

#### 1. Introduction

*Multivariable PID Control*. Among the controller variety, PID becomes the controller that is most applied in a physical system [1]. The reason is that it has a characteristic that offers simplicity, clear functionality, and ease of use [2]. However, Ho et al. [3] reported that only one-fifth of PID control loops are in good condition. The others are not, where 30% of PID controllers are not able to perform well due to lack of tuning parameters, 30% due to the installation of a controller system operating manual, and 20% due to the use of default controller parameters.

In recent years, many researchers have paid attention to the MPID controller design for various systems such as in Industrial Scale-Polymerization Reactor [4], Coupled Pilot Plant Distillation Column [5], Narmada Main Canal [6], Quadruple-Tank Process [7], Boiler-Turbine Unit [8], and Wood-Berry Distillation Column [9]. A research by Kumar et al. [4] had proposed a synthesis method of PI controllers based on approximation of relative gain array (RGA) concept to multivariable process. The method was further improved by relative normalize gain array concept (RNGA). Controller based on RNGA concept provides a better performance than RGA concept. Both concepts use the nonstandard PID controller which requires Maclaurin series expansion [10]. In the work by Sarma and Chidambaram [5], PI/PID controllers based on Davison and Tanttu-Lieslehto method extended to nonsquare systems with right-half plane zero were applied. Results show that the Davison method gives better performance with less settling time than Tanttu-Lieslehto method. However, both methods are not applicable for square system.

Essentially, there are many integral controllers that are designed for nonlinear system [11, 12]. However, most of the existing techniques do not guarantee the desired transient performances in the presence of nonlinear parameter variations and unknown external disturbances [13]. In a previous study, Martin and Katebi [14] had proposed Davison, Penttinen-Koivo, Maciejowski, and Combined method as a control tuning design for ship positioning. These controls strategies are based on PID controller which is used to control multivariable system. Due to the effectiveness and simplicity of those proposed controllers, Wahab et al. [15] had used those methods as tuning strategies for wastewater treatment plant (WWTP). The controllers were designed based on steady state of a linear system and static model inverse. The reliability of the proposed method was tested to a nonlinear WWTP. The response shows that good result was obtained from Davison until the Combined method. In the work by Balaguer et al. [16], a comparison between MPID controller with figure of merit controller was done for WWTP based on open-loop, closed-loop, and open-closed loop controller structure analysis. The MPID control tuning based on Davison and Penttinen-Koivo method was carried out by minimizing the residuals of both controllers obtained from the data. However, the dynamic nature of WWTP which involves ill condition characteristic causes difficulties in finding the optimum MPID tuning parameter. The system’s behavior that involves slow and fast variables causes the control tuning strategies to not easily meet specification for multiple control variables at the same time.

A lot of approaches have been proposed to control multivariable system. Some of the approaches are able to deal with a high order multivariable system. However, a simple controller design has always become a desired controller where it is certainly can be accepted by the industry. By that, the required cost to run the system will be minimized as well. Realizing the simple controller design by other researchers [14–16], those methods were applied in this project and improved by adopting singularly perturbation method (SPM) to the controller designs by considering the dynamic matrix inverse.

*Singularly Perturbed Multivariable Controller*. Analysis and synthesis of singularly perturbed control have received much thoughtfulness over the past decades by many people from numerous arenas of studies [17–21]. Singularly perturbation method is able to decompose and simplify the higher order of the full order system into slow and fast subsystems [17, 22, 23], which are known as singularly perturbation system. Definitely, most of the control systems are dynamic, where the decomposition into stages is dictated by multitime scale. In this situation, the slow subsystem corresponds to the slowest phenomena and the fast subsystem corresponds to the fastest phenomena. It basically has two different parts of eigenvalue represented for slow and fast dynamic subsystems [24], where slow subsystem corresponds to small eigenvalue and fast subsystem corresponds to large eigenvalue [25].

This work is focused on the analysis of singularly perturbation system on two different case studies given. Singularly perturbed control of multivariable system is comprised of two steps. First, the multivariable system is decomposed into slow and fast subsystems. Then, the optimal composite singularly perturbed controller is designed [25–28]. There are many approaches that have been developed concerning the control of singularly perturbation system. The approaches use different conditions on the properties of the used functions, different assumptions, different theorems, and different lemma [13, 21, 29] which are specifically based on the systems behaviour.

In a study by Rabah and Aldhaheri [24], singularly perturbation system has been modelled by using a real Schur form method. It shows that any two-time scale system can be altered into the singularly perturbed form via a transformation into an order real Schur form (ORSF). It is based on two steps, transformation of matrix A into an ORSF using an orthogonal matrix and then application of balancing algorithm to an ORSF. Li and Lin [17] had addressed the composite fuzzy multivariable controller to nonlinear singularly perturbation system. The composite controller was obtained from the combination of slow and fast subsystems. It was tested to a DC motor driven inverted pendulum system and it provides realistic and satisfactory simulation results. Multivariable control by Kim et al. [30] used successive Galerkin approximation (SGA) method. This method causes the complexity in computations to increase with respect to the order of the system. Therefore, singularly perturbation method was adopted to decompose the original system into slow and fast subsystems. Result shows that the use of the method greatly reduces the computation complexity and it is more effective than the original SGA method.

To the best of author knowledge, there are two methods to obtain the singularly perturbation system, which are by analytical [21, 29–31] and linear analysis [32–35]. Singularly perturbation system obtained based on linear analysis is discussed and has been applied in this research. By exploiting the properties of singularly perturbation system to the dynamic matrix inverse of MPID control tuning methodology, an easy multivariable tuning method should be established. In Section 2, the time scale analysis is presented to determine the behavior of the system. Section 3 described the methods to obtain singularly perturbation system based on Naidu and Jian Niu method. The sequences of MPID controller based on Davison, Penttinen-Koivo, Maciejowski, and Combined methods are discussed in Section 4. Section 5 presented the optimization method which is based on particle swarm optimization (PSO). The case studies and the performance of the proposed methods for two case studies are investigated and discussed thoroughly in Sections 6 and 7. Finally, conclusions are given in Section 8.

#### 2. Time Scale Analysis

To apply singularly perturbation method to the controller designs, the considered system must consist of a two-time scale characteristic. The two-time scale characteristic can be determine by rearranging the eigenvalue of the system in increasing order which will givewhere , , and are a total, slow, and fast eigenspectrum of the system, respectively. is a smallest eigenvalue of the slow eigenspectrum, is a largest eigenvalue of the slow eigenspectrum, is a smallest eigenvalue of the fast eigenspectrum, and is a largest eigenvalue of the fast eigenspectrum:The system is said to possess a two-time scale characteristic, if the largest absolute eigenvalue of the slow eigenspectrum is much smaller than the smallest absolute eigenvalue of the fast eigenspectrum. This is proven bywhere is a measure of separation of time scales that represents an intrinsic property.

#### 3. Singularly Perturbation Method (SPM) for MIMO System

Industrial processes possess “” number of inputs and outputs variables, where interaction phenomena exist. Interaction phenomena that occur among the inputs and outputs variables of multivariable process cause great difficulties in MPID controller design. Usually, it is solved by tuning the most important loop whereas other loops are detuned by keeping the interactions of that loop adequate. To compensate the interaction phenomena, one of the loops is forced to perform poorly. This detuning method is far from the optimal [4]. In this work, to account for the interaction phenomena, instead of using the original process transfer function to the MPID controller design, that transfer function is rearranged by separating the slow and fast eigenvalues using SPM.

##### 3.1. Naidu Method

In this section we propose a procedure for a separation of slow and fast subsystem. The considered linear equations for two-time scale continuous system with the output vector possessing two widely separated groups of eigenvalues arewhere and are slow and fast variables in and dimension and is a measured output. Matrices , , and are constant matrices of appropriate dimensions. Consider the problem as in (4a) to (4c). The system possesses a two-time scale property. Preliminary to separation of slow and fast subsystem, the system consists of number of small eigenvalue (close to the origin) for slow subsystem and number of fast eigenvalue (far from the origin) for the fast subsystem. The number of slow and fast eigenvalues needs to be identified based on eigenvalue location. Fast eigenvalue of the system is only essential during a short period of time. Then, it is insignificant and the characteristic of the system can be described by degenerating system known as a slow subsystem.

By letting , slow subsystem is obtained asBy assuming as a nonsingular matrix, (5b) becomesUsing equation (6) in (5a), is represented aswhereUsing (6) in (5c), is represented aswhereTo obtain fast subsystem, it can be assumed that the slow variables are constant at fast transients, whereFrom (4b) and (6), Let The fast subsystem is obtained asThe composite system which consists of slow and fast subsystem is achieved using two-stage linear transformation which can be referred in an article written by Chang [36]:whereThe state space form of composite system is represented in (15a) to (15f).

##### 3.2. Jian Niu Method

The two-time scale system can also be solved using other method. This section presents singularly perturbation method based on Jian Niu. In order to apply Jian Niu method, transfer function matrix should be transform into a state space model:To illustrate the two-time scale decomposition, (16) is considered. Equation (16) can have this formwhere is a state space form of . Equations (17a) and (17b) can be represented aswhere is a very small positive constant. Equations (18) can be considered asEquations (19a) to (19c) are the linear equations for two-time scale continuous system, similar just like (1a) to (1c) where This method is discussed in several literatures [33, 35]. The slow subsystem is denoted asAnd the fast subsystem iswhereThe transfer functions for slow and fast subsystem are denoted by (30) and (31), respectively. The composite model is signified as a sum of slow and fast subsystem and a very little item [37]where

#### 4. Multivariable PID Controller Design

Owing to the industrial process control involved with multivariable system, MPID controller design technique is necessary. It is a powerful control technique for solving coupling nonlinear system [38]. The conventional MPID controller designs technique is based on static inverse model [15]. This technique is difficult to obtain the desired control performance. Therefore, an enhancement is presented based on the dynamic inverse matrix and singularly perturbation method to the designs of MPID controller. Essentially, this enhancement has been considered in the previous work reported in [39] and it shows that the enhancement is able to control dynamic system where the output is able to track the set point change and produced less proses interaction. Nevertheless, it only considered that three controller designs instead of four and the selection of parameter tuning are done without optimization technique. In this paper, there are four enhanced MPID controller designs which are Davison, Penttinen-Koivo, Maciejowski, and Combined method where it is applied to the both original and singularly perturbed system with all of the parameter tuning being obtained based on particle swarm optimization. All of these four designs technique are applied to wastewater treatment plant and Newell and Lee evaporator.

##### 4.1. Davison Method

Multivariable control design based on Davison method simply applies the integral term, which causes decoupling rise at low frequencies The controller expression is represented by (34) [16], where and are integral feedback gain and controller error, respectively,Since this research is focused on dynamic control, is defined as in (35), where is the only controller tuning parameter, which undoubtedly needs to be tuned progressively until the finest solution is discovered. Due to the involvement of the inverse system, the control design is only applicable for square matrix. If the system involves time delay, the time delay needs to be eliminated.

##### 4.2. Penttinen-Koivo Method

This method is somewhat advanced and then the method proposed by Davison. In Penttinen-Koivo method, a proportional term is introduced. It comprises both integral and proportional term. Indirectly, this causes decoupling to take place at low and high frequencies. Davison and Penttinen-Koivo method are only similar in terms of an integral term which is linearly related to the inverse of plant dynamics The controller expression is represented in (36) [16], where , , and are proportional gain, integral feedback gain, and controller error correspondinglyDynamic terms of and are expressed in (37), where and are the tuning parameters for both proportional and integral feedback gain.

##### 4.3. Maciejowski Method

Maciejowski method applies all proportional, integral, and derivative gains in its controller design. For maciejowski method, the tuning was done around the bandwidth frequency, . Consequently, the interaction is reduced and good decoupling characteristic is provided around the frequency [15]. However, due to the needs of plant frequency analysis experiment, this method is quite difficult to be used throughout the industry [15]The controller expression is represented by (38), where , , , and are proportional, integral feedback, derivative gains, and controller errorDynamic terms of , , and are expressed in (39), where , , and are Maciejowski tuning parameters. Due to a complex gain obtained from the calculation of , a real approximation of is necessary which can be done by solving the following optimization problem:where is a constant that is used to minimize .

##### 4.4. Combined Method

In order to overcome the weakness of the Maciejowski method which requires rigorous frequency analysis, a new method was proposed by Wahab et al. [15]. It is the result of the previous controllers where methods by Davison, Penttinen-Koivo, and Maciejowski are combined together:Equation (41) represents the proposed control design, where , , and are the tuning parameters and controller error: is defined in (42). is also a tuning parameter. This method keeps some properties in Maciejowski method but excludes the needs of frequency analysis [15].

#### 5. Optimized Singularly Perturbed MPID Parameter Tuning

To ensure a fair comparison, the optimum parameter tuning for each of controller designs is measured by using particle swarm optimization (PSO). PSO optimizes a problem by having a population (swarm) of candidate solutions (birds) which is known as particles that are updated from iteration to iteration [40]. These particles are moved into the search space seeking for a food according to its own flying experience and its companion flying experience. It can be expended to multidimensional search. Each particle (solution) is characterized by its position and velocity, and every one of them searches for better positions within the search space by changing its velocity [41]. Each particle preserves the track of its current position within the search space. This value is identified as the particle’s local best known position (*pbest*) and leads to the best known position (*gbest*), which is defined as enhanced positions that are found by the other particles. By that, the finest solution is attainedEquation (43) represents the update equations of new velocity and new position, where , and correspond to the velocity at time , new particle position, inertia weight, current velocity at time , cognitive weight, global weight, random variable within the range of , random variable within the range of ,* pbest*, and* gbest*.

The overall performance of PSO can be increased by proper selection of inertia weight, . Lower value of provides a good ability for local search and higher value of provides a good ability for global search [41]:To achieve a respectable performance, is determined according to (44), where is the maximum value of inertia weight, is the minimum value of inertia weight, is the current number of iteration, and is the maximum number of iteration. Most of the previous researchers have used and , where significant enhancement of PSO is achieved [42, 43]:Fitness function which is also known as cost function is represented by (45), where is a system error. The procedure of PID parameter optimization by using PSO is summarized as follows:(1)Initialization: initialize a population of particles with arbitrary positions and velocities on* X* dimensions in the problem space. Then, randomly initialize* pbest* and* gbest*.(2)Fitness: calculate the desired optimization fitness function in* X* dimensions for every particle.(3)*pbest*: compare calculated fitness function value for every particle in the population. If current value is smaller than* pbest*, and then update* pbest* as current particle position.(4)*gbest*: determine the best success particle position among all of the individual best positions and designate as a* gbest*.(5)New velocity and position: update the velocity and position of the particle based on (43).(6)Repeat step (2) all over again until a criterion is encountered.

#### 6. Case Studies

In the next subsections, an introduction to the case studies is presented. First, an overview of the wastewater treatment plant (WWTP) is provided and the Newell and Lee evaporator model is explained. These two case studies are considered to demonstrate the performance of the proposed methods.

##### 6.1. Case Study I: Wastewater Treatment Plant (WWTP)

Wastewater treatment plant is designed either for carbon removal or for carbon and nitrogen removal. In this project, the carbon removal scenario is considered. The control plant outputs are substrate and dissolved oxygen. Scanty provision of substrate affects the growth of microorganisms that are responsible for treating the wastewater and too many provisions lead to a drop in the microorganisms growth rate. The standard amount of substrate is around 51 mg/L [44]. Meanwhile, insufficient dissolved oxygen will cause the degradation of the pollutants and the plant to become less efficient. Too much dissolved oxygen can cause excessive consumption of energy where it will increase the cost for the treatment. Other than that, it also can cause too much sludge production. The amount of dissolved oxygen concentration needs to be controlled so that it is in the range of 1.5 mg/L–4.0 mg/L [45]. The aim of this case study is to control the concentration of substrate and dissolved oxygen at the desired value by manipulating the manipulate variables of dilution rate and air flow rate, respectively. The state space form of the nonlinear wastewater treatment plant is linearized from [39] as follows:where the state is composed by , the biomass, , the substrate, , the dissolved oxygen, and , the recycled biomass. The input variables are , the dilution rate, and , an air flow rate. Matrices , and are given byThese , , , and matrices are used in the design of singularly perturbed MPID control and tested into the nonlinear wastewater treatment plant.

##### 6.2. Case Study II: Newell and Lee Evaporator

This subsection presents the Newell and Lee evaporator system which is considered as a second case study. The objective is to evaluate the effectiveness and the performance of the proposed singularly perturbed MPID controllers for different system. Here, unstable system is considered. Similar to the first case study, the four different methods of MPID are implemented, which is Davison, Penttinen-Koivo, Maciejowski, and Combined methods. The plant to be controlled is given by the following state space model [46]: where the state is composed by , the separator level, , the product composition, and , an operating pressure. The input variables are , the product flow rate, , the steam pressure, and , the cooling water flow rate. The outputs to be controlled are , separator level, and , operating pressure. Matrices , and are given by

#### 7. Results and Discussion

This section presents the results and discussion for both case studies. It will include the results of singularly perturbed MPID control based on four proposed methods, which are Davison, Penttinen-Koivo, Maciejowski, and Combined method for both full order and singularly perturbed system. The first section shows the MPID control based on particle swarm optimization. The obtained optimum tuning parameters are presented. The second section shows the simulation results for control of the closed loop system during substrate and dissolved oxygen set point change, while the last section provides the result which shows the stability of the closed loop system.

##### 7.1. Results for the Case Study 1: Wastewater Treatment Plant (WWTP)

The eigenvalue of the open loop wastewater treatment plant is as follows:

As a result Since is less than 1, the system is said to possess a two-time scale characteristic. The eigenvalue at is considered as a fast responseBy using algorithm discussed in Section 3.1, the original system which refers to (47a) to (47d) is represented in state space form of singularly perturbed system as indicated in (52a) to (52d). This state space form is used to design the controller tuning, while the simulation and performance of the system are based on the original system. From the state space form in (52a) to (52d), it is clearly shown that the eigenvalues of the system are grouped into two distinct and separate sets, which causes the time consumed to obtain the MPID tuning parameters reduce. All eigenvalues of the singularly perturbed system are remained at the left-half plane, which is , , , and . It is indicated that the system is boundary input boundary output (BIBO) stable Based on the algorithms explained in Section 3.2, the Jian Niu method is successfully able to represent the original system into composite of singularly perturbed system as in (53a) to (53d), where the only existing eigenvalue is located at . Since it is in the left-half plane, the system is BIBO stable. To validate the models from both methods, singularly perturbation method obtained by Naidu and Jian Niu, the magnitude and phase plot between the original system and singularly perturbed system are plotted as shown in Figure 1.