Analytical Design of Fractional-Order PI Controller for Parallel Cascade Control Systems

Abstract

The fractional-order proportional-integral (FOPI) controller tuning rules based on the fractional calculus for the parallel cascade control systems are systematically proposed in this paper. The modified parallel cascade control structure (PCCS) with the Smith predictor is addressed for stable, unstable, and integrating process models with time delays. Normally, the PCCS consists of three controllers, including a stabilized controller, for a class of unstable and integrating models, a disturbance rejection controller in the secondary loop, and a primary servomechanism controller. Accordingly, the ideal controller is obtained by using the internal model control (IMC) approach for the inner loop. The proportional-derivative (PD) controller is suggested for the stabilized controller and is designed based on a stability criterion. Based on the fractional calculus, the analytical tuning rules of the FOPI controller for the outer loop can be established in the frequency domain. The simulation study is considered for three mentioned cases of process models and the results demonstrate the flexibility and effectiveness of the proposed method for the PCCS in comparison with the other methods. The robustness of the proposed method is also justified by perturbed process models with ±20% of process parameters including gain, time constant, and delay time.

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 u₂) simultaneously affect the primary and secondary output (y₁ and y₂). Luyben 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 large overshoot 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]. Rao 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. Padhan and Majhi [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^λD^µ [13], is involved in two extra parameters as the fractional-order integrator (λ) and fractional-order differentiator (μ). The fractional-order PID (FOPID) controller affords more flexibility in PID controller design and more robustness than the integer-order one [14,15,16,17,18,19,20,21,22]. The tuning rules of the FOPI/PID controller have been reported in some of the literature for single-input single-output (SISO) systems and multi-input multi-output (MIMO) processes. However, there are only a few studies that use fractional-order controllers for PCCS in the literature. In Ref. [9], Pashaei and Bagheri proposed fractional order controllers for the parallel cascade scheme including the Smith predictor for the large time delay. The design concept was based on IMC structure and an extra controller was also suggested to stabilize unstable, integrating primary processes. However, the authors used the Adam-Bashforth-Moulton (ABM) algorithm, which is quite different from most of the other works in the fractional-order control field that are used to simulate fractional-order systems.

In accordance with the literature, the tuning methods of PI^λD^µ can be generally classified as analytic and heuristic ones [14,15,16,17,18,19,20,21,22,23,24,25,26]. In fact, most of the analytic methods are often tuned by considering the objective functions, which are normally considered as phase and gain margin specifications as well as constraints using the maximum sensitivity function (M_s) for robustness [14,15,16,17,18,19,20,21,22]. On the other hand, heuristic methods including genetic algorithms (GA) and particle swarm optimization (PSO) are often adopted to find out suitable parameters for the fractional-order PID controller [23,24,25,26]. In this work, our aim is to design an analytic method of a generalized FOPI controller for the primary loop of a PCCS with a time delay to enhance the performances of designed systems. It is mainly based on the concepts of fractional calculus and the IMC approach by using the frequency domain [18]. The tuning rule of the FOPI controller for primary processes can be derived directly without introducing any nonlinear objective function, and the proposed method can be applied for many typical process models including stable, unstable, and integrating primary processes.

2. Materials and Methods

2.1. Fractional Calculus

Fractional calculus is considered to be a generalization of integral and differential calculus. In the literature, there are some definitions of fractional calculus, but the most common one is the Riemann-Liouville definition [10,12], which is well-known in the control field:

$${}_{a}D_t^{v}f\left(t\right)=\frac{1}{\Gamma \left(n-v\right)}\frac{d^n}{d{t}^n}{\displaystyle {\int }_a^t\frac{f\left(\tau \right)}{{\left(t-\tau \right)}^{v-n+1}}}dt, n-1<v<n$$

(1)

where D is a functional operator; a and t are the limits; $v \left(v\in \Re \right)$ is the fractional-order that denotes derivative for $v>0$ and integration for $v<0$; $\Gamma (•)$ is Euler’s gamma function.

It is trivial to see that the order $v$ can be chosen $0<v<1$ and Laplace transformation is applied to (1) and denoted by Equation (2) as follow:

$$L\left\{{}_{a}D_t^{\pm v}f\left(t\right)\right\}=s^{\pm v}F\left(s\right)$$

(2)

Note that the initial conditions, in this case, still have to equal to zero as in the integer-order one.

2.2. Crone Approximation

For purposes of simulation and hardware implementation, the well-known approximation known as the Crone approximation, proposed by Oustaloup, is adopted to obtain the equivalent transfer function of fractional order (s^v) in the specific range of frequency $\left[{\omega }_l,{\omega }_h\right]$. This approximation uses finite numbers of poles and zeros in the form of the recursive distribution and describes by the following equation [10]:

$$s^v\cong k{\displaystyle \prod _{n=1}^N\frac{1+\left(\frac{s}{{\omega }_{z,n}}\right)}{1+\left(\frac{s}{{\omega }_{p,n}}\right)}}$$

(3)

where the gain, k, should be adjusted for both sides to have unit gain at crossover frequency ${\omega }_c=1$ rad/s; ${\omega }_l$ and ${\omega }_h$ are chosen as a low range of frequency $0.001{\omega }_c$ and $1000{\omega }_c$, respectively; N = 8 is an appropriate choice because a lower value reduces the computational cost, but may cause ripples in both gain and phase behaviors.

2.3. Fractional Linear Model

Consider a linear time invariant (LTI) system with a single-input u(t), single-output y(t) (SISO), the fractional-order differential equation (FODE) can be expressed as:

$${\displaystyle \sum _{i=0}^n{a}_i{D}_0^{{\alpha }_i}}y\left(t\right)={\displaystyle \sum _{j=0}^m{b}_j{D}_0^{{\beta }_j}}u\left(t\right)$$

(4)

Using Laplace transform in Equation (2), Equation (4) can be described by the following transfer function:

$$G\left(s\right)=\frac{\mathrm{Y}\left(s\right)}{\mathrm{U}\left(s\right)}=\frac{b_m{s}^{{\beta }_m}+b_{m-1}{s}^{{\beta }_{m-1}}+\dots +b_0{s}^{{\beta }_0}}{a_n{s}^{{\alpha }_n}+a_{n-1}{s}^{{\alpha }_{n-1}}+\dots +a_0{s}^{{\alpha }_0}}$$

(5)

where ${\alpha }_i$ and ${\beta }_i$ are arbitrary, real, and positive.

2.4. The FOPI Controller in the Frequency Domain

The general form of FOPI controller using fractional operator can be expressed as:

$$u\left(t\right)=K_{c}e\left(t\right)+K_I{D}_t^{-\lambda }e\left(t\right),\left(\lambda >0\right)$$

(6)

where $K_c$ and $K_I$ represent the proportional and integral terms respectively; $\lambda$ is the fractional order of the integration.

Using Laplace transform Equation (2), Equation (6) can be rewritten in the form of transfer function:

$$G_c\left(s\right)=K_c+\frac{K_I}{s^{\lambda }}$$

(7)

In special case $\lambda =1$, Equation (7) becomes the normal PI controller (integer order). Therefore, the FOPI controller has an additional tuning parameter ($\lambda$) compared to conventional one. The advantages of fractional-order are indisputable, but it also leads to difficulties in the tuning procedure. In this work, a design approach in frequency domain firstly proposed in [18] is used to derive the analytical tuning rules of the FOPI.

The FOPI controller is converted to the frequency domain by substituting $s=j\omega$ into (7):

$$G_c\left(j\omega \right)=K_c+\frac{K_I}{{\left(j\omega \right)}^{\lambda }}$$

(8)

The fractional power in Equation (8) can be expressed as follows:

$${\left(j\omega \right)}^{\lambda }={\omega }^{\lambda }{j}^{\lambda }={\omega }^{\lambda }{\left[e^{j\left[\frac{\pi }{2}+2n\pi \right]}\right]}^{{}^{\lambda }}={\omega }^{\lambda }\left[e^{j\left[\frac{\pi }{2}\lambda +2n\lambda \pi \right]}\right]$$

(9)

where $n=0,\pm \frac{1}{\lambda },\pm \frac{2}{\lambda },\dots ,\pm \frac{m}{\lambda }$. Finally, it can be rewritten in the following equation:

$${\left(j\omega \right)}^{\lambda }={\omega }^{\lambda }\left(\cos {\gamma }_I+j\sin {\gamma }_I\right),{ \gamma }_I=\frac{\pi \lambda }{2}$$

(10)

Substituting (10) into (8) and transforming into a complex equation, the FOPI controller in the frequency domain is obtained as Equation (11):

$$G_c\left(j\omega \right)=\left(K_c+\frac{K_I\cos {\gamma }_I}{{\omega }^{\lambda }}\right)-j\frac{\left(K_I\sin {\gamma }_I\right)}{{\omega }^{\lambda }}$$

(11)

2.5. Maximum Sensitivity Value

Consider a single-input single-output system, where the maximum peak of sensitivity function is defined as:

$$M_s=\underset{\omega }{\max }\left|S(j\omega )\right|$$

(12)

where $S={\left(1+L\right)}^{-1}$, and L is an open-loop transfer function of the system.

$M_s$ is the inverse of the shortest distance from the Nyquist curve of the open-loop function to the critical point (−1, j0). It is obvious that the higher value of $M_s$ is, the less robust the system is to modeling uncertainties. In general, both for robust stability and performance, the value $M_s$ should be close to 1. Therefore, in most of the literature, the typical range $M_s$ is chosen from 1.4 to 2 for integer-order control systems [27,28]. In this work, for the case of an unstable primary process, this value is considered as a tuning criterion for fractional-order control systems.

2.6. Analytical Design of FOPI Controller for PCCS Combining with Smith Predictor

The general parallel cascade control system is shown in Figure 2, where $G_{p_1}(s)$ and $G_{p_2}(s)$ are the transfer functions of the primary and secondary processes, respectively. $G_{c_1}(s)$ and $G_{c_2}(s)$ are the primary and secondary controllers; $G_d(s)$ is used to stabilize the primary control loop in case the transfer function $G_{p_1}(s)$ is kind of an unstable or integrating process. $G_{d_1}(s)$ and $G_{d_2}(s)$ denote the transfer functions of disturbances affecting the outputs. $G_p(s)$ denotes the equivalent closed-loop transfer function for servo response of the secondary loop (from the input ${r^{\prime }}_2$ to the output $y_1$), and it can be calculated as follows:

$$G_p(s)=\frac{G_{p_1}(s)}{1+G_{c_2}(s)\left(G_{p_2}(s)-{\tilde{G}}_{p_2}(s)\right)+G_d{G}_{p_1}(s)}$$

(13)

Assuming that $G_{p_2}(s)={\tilde{G}}_{p_2}(s)$ (perfect model), Equation (13) becomes:

$$G_p(s)=\frac{G_{p_1}(s)}{1+G_d{G}_{p_1}(s)}=G_{pm}(s)e^{-{\theta }_{p}s}$$

(14)

where ${\theta }_p$ is the delay time at the primary output, and normally, it equals the delay time of the primary transfer function ($G_{p_1}$); $G_{pm}$ is the delay-free part of $G_p$.

In this work, the secondary process model is assumed as a well-known first order plus time delay process (FOPTD), as in Equation (15). The primary process model is investigated by one of the following transfer functions, as in Equations (16)–(18)

$$G_{p_2}(s)=\frac{K_{{}_2}{e}^{-{\theta }_{2}s}}{{\tau }_{2}s+1}$$

(15)

$$G_{p_1}(s)=\frac{K_{{}_1}{e}^{-{\theta }_{1}s}}{{\tau }_{1}s+1}$$

(16)

$$G_{p_1}(s)=\frac{K_{{}_1}{e}^{-{\theta }_{1}s}}{{\tau }_{1}s-1}$$

(17)

$$G_{p_1}(s)=\frac{K_{{}_1}{e}^{-{\theta }_{1}s}}{s({\tau }_{1}s+1)}$$

(18)

where Equation (17) represents the unstable first order plus time delay (UFOPTD) and Equation (18) is the second order integral plus time delay (SOIPTD).

2.6.1. Design of Secondary Controller Based on IMC Approach for Disturbance Rejection

Figure 3 shows the block diagram of the secondary control design based on the IMC approach where $G_{p_2}(s)$ and ${\tilde{G}}_{p_2}(s)$ are the process and the process model respectively; $G_{c_2}(s)$ is the IMC controller. Note that, in this work, the IMC controller is placed on the feedback loop, which is different from the well-known IMC approach. As a result, the role of the controller is to solve the regulator problem in the secondary loop.

For the nominal case ($G_{p_2}(s)={\tilde{G}}_{p_2}(s)$), the set-point and disturbance responses in Figure 3 can be simplified as:

$$y_2=G_{p_2}{r}_2+(1-G_{c_2}{\tilde{G}}_{p_2})G_{d_2}d$$

(19)

As mentioned above, the secondary transfer function $G_{p_2}$ is stable in this study. Therefore, from Equation (19), it is obvious that the controller $G_{c_2}$ is only used for disturbance rejection. The IMC approach [27] is adopted to design $G_{c_2}$. The process model ${\tilde{G}}_{p_2}(s)$ is factored into two parts:

$${\tilde{G}}_{p_2}(s)=p_{2m}{p}_{2a}$$

(20)

where $p_{2a}$ is the portion of the model which includes the dead time and/or right half plane zeros and $p_{2a}(0)=1$; $p_{2m}$ is the remaining portion of the secondary process. Therefore, we have:

$$p_{2a}=e^{-{\theta }_{2}s}; p_{2m}=\frac{K_2}{{\tau }_{2}s+1}$$

(21)

To obtain a good disturbance rejection for the process $G_{d_2}$ which has poles near zero at $z_{d1},z_{d2},\dots ,z_{dm}$, then $(1-G_{c_2}{\tilde{G}}_{p_2})$ should have zeros at $z_{d1},z_{d2},\dots ,z_{dm}$. According to the IMC approach, the IMC controller is given as follows:

$$G_{c_2}=p_{2m}{}^{-1}f=\frac{{\tau }_{2}s+1}{K_2}f$$

(22)

where f is considered as the IMC filter whose transfer function is given in this case as follows:

$$f=\frac{1}{{\lambda }_{2}s+1}$$

(23)

Then the controller is obtained:

$$G_{c_2}=\frac{{\tau }_{2}s+1}{K_2({\lambda }_{2}s+1)}$$

(24)

2.6.2. FOPI Controller Design for the Primary Control Loop

The block diagram of the system is reduced in Figure 4 where $G_p$ and ${\tilde{G}}_p$ represent the equivalent process and the obtained transfer function from ${r^{\prime }}_2$ to $y_1$ in Figure 2, respectively. ${\tilde{G}}_{pm}$ is ${\tilde{G}}_p$ without the delay term. This structure follows the Smith predictor scheme to eliminate the sluggish response at the primary output when the primary process has a long delay time.

The closed loop transfer functions for the outer loop (from r₁ to y₁)

$$\frac{Y_1}{R_1}=\frac{G_{c_1}(s)G_p(s)}{1+G_{c_1}(s)G_{pm}(s)}$$

(25)

From Equation (25), it is obvious that the characteristic equation of the closed-loop system is the delay-free function. In order to design the primary controller, the equivalent process ${\tilde{G}}_p$ first has to be obtained. In this work, depending on the kinds of secondary process transfer functions, several methods are proposed to derive ${\tilde{G}}_p$ in the form of FOPTD.

Case 1: The primary process is FOPTD

The transfer function of the primary process is as Equation (16). In this case, the primary transfer function is also stable, so there is no need to use the stabilized controller, $G_d=0$, and from Equation (14):

$${\tilde{G}}_p(s)=G_{p_1}(s)=\frac{K_{{}_1}{e}^{-{\theta }_{1}s}}{{\tau }_{1}s+1}=\frac{K_p{e}^{-{\theta }_p}}{{\tau }_{p}s+1}$$

(26)

Case 2: The primary process is UFOPTD

$$G_{p_1}(s)=\frac{K_{{}_1}{e}^{-{\theta }_{1}s}}{{\tau }_{1}s-1}$$

(27)

In this case, the primary process is unstable, hence, the stabilized controller is suggested as follows:

$$G_d(s)=K_d(1+{\tau }_{d}s)$$

(28)

The equivalent transfer function is obtained:

$${\tilde{G}}_p(s)=\frac{G_{p_1}(s)}{1+G_d{G}_{p_1}(s)}=\frac{K_1{e}^{-{\theta }_{1}s}}{{\tau }_{1}s-1+K_d{K}_1(1+{\tau }_{d}s)e^{-{\theta }_{1}s}}$$

(29)

Using Padé 1/1 approximation: $e^{-{\theta }_{1}s}\approx \frac{1-0.5{\theta }_{1}s}{1+0.5{\theta }_{1}s}$ and choosing

$${\tau }_d=0.5{\theta }_1$$

(30)

$${\tilde{G}}_p(s)=\frac{K_1{e}^{-{\theta }_{1}s}}{{\tau }_{1}s-1+K_d{K}_1(1-0.5{\theta }_{1}s)}=\frac{K_1{e}^{-{\theta }_{1}s}}{({\tau }_1-0.5K_d{K}_1{\theta }_1)s+K_d{K}_1-1}=\frac{K_p{e}^{-{\theta }_p}}{{\tau }_{p}s+1}$$

(31)

where ${\theta }_p={\theta }_1$, $K_p=\frac{K_1}{K_d{K}_1-1}$ and ${\tau }_p=\frac{{\tau }_1-0.5K_d{K}_1{\theta }_1}{K_d{K}_1-1}$.

To guarantee the stability criterion, two parameters $K_p$ and ${\tau }_p$ have to be greater than zero. Therefore, the parameter $K_d$ has to meet the following requirement:

$$\frac{1}{K_1}<K_d<\frac{{\tau }_1}{0.5K_1{\theta }_1}$$

(32)

Case 3: The primary process is SOIPTD

$$G_{p_1}(s)=\frac{K_{{}_1}{e}^{-{\theta }_{1}s}}{s({\tau }_{1}s+1)}$$

(33)

The stabilized controller as in the previous case, Equation (28), is also needed in this case; therefore, the equivalent transfer function is derived as follows:

$${\tilde{G}}_p(s)=\frac{G_{p_1}(s)}{1+G_d{G}_{p_1}(s)}=\frac{K_1{e}^{-{\theta }_{1}s}}{s({\tau }_{1}s+1)+K_d{K}_1(1+{\tau }_{d}s)e^{-{\theta }_{1}s}}$$

(34)

Similar to case 2, using Equation (30) to replace into Equation (34):

$${\tilde{G}}_p(s)=\frac{K_1{e}^{-{\theta }_{1}s}}{s({\tau }_{1}s+1)+K_d{K}_1(1-0.5{\theta }_{1}s)}=\frac{K_1{e}^{-{\theta }_{1}s}}{{\tau }_1{s}^2+(1-0.5K_d{K}_1{\theta }_1)s+K_d{K}_1}$$

(35)

The condition of $K_d$ is obtained:

$$0<K_d<\frac{1}{0.5K_1{\theta }_1}$$

(36)

Moreover, in this case, ${\tilde{G}}_p$ is the second-order transfer function. To ensure a servo response of the closed-loop system, as well as to simplify the design problem, the system has to be the overdamped one which the damping factor has not to be less than 1. From Equation (35), another condition of $K_d$ is also derived:

$$\frac{1-0.5K_1{K}_d{\theta }_1}{2\sqrt{K_1{K}_d{\tau }_1}}\ge 1$$

(37)

In that case, the system transfer function will be approximated to the first-order one. Applying the approximation technique using the PSO algorithm [29], Equation (35) will be reduced to the first order plus time delay process as the two previous cases, Equations (26) and (31).

2.6.3. The General Proposed Design Method for the Above Three Cases of the Primary Controller

After deriving the equivalent transfer functions of primary processes in the three above cases to the general form of FOPTD, Equation (38), the proposed FOPI controller will be designed based on the IMC approach. However, the frequency domain mentioned above is addressed here.

$${\tilde{G}}_p=\frac{K_p{e}^{-{\theta }_{p}s}}{{\tau }_{p}s+1}$$

(38)

In order to choose the fractional order of the controller, the relative dead time parameter is considered [10]:

$$\Delta =\frac{{\theta }_p}{{\tau }_p+{\theta }_p}$$

(39)

In this work, the fractional-order $\lambda$ is determined according to the value of $\Delta$ as Equation (40). It is based on the guideline in [10], however, only the fractional order is considered in this study

$$\lambda =\{\begin{array}{ll}1.1,& \mathrm{if} \Delta \ge 0.6\\ 0.9,& \mathrm{if} 0.1\le \Delta <0.6\\ 0.7,& \mathrm{if} \Delta <0.1\end{array}$$

(40)

Step 1: Design the stabilized controller

Case 1: The primary process transfer function is stable. Therefore, the stabilized loop will not be used in this case. Hence, $G_d$ is assigned to zero ($G_d=0$)

Case 2: Note that when the order is greater than 1, system responses will create an overshoot. Therefore, in this work, to guarantee a good servo tracking of the primary control loop, the relative dead time parameter $\Delta$ will be chosen in the range of $[0.1,0.6]$, it means that $\lambda =0.9$. From Equations (31) and (39), the following condition is obtained:

$$0.1\le \frac{{\theta }_1}{\frac{{\tau }_1-0.5K_d{K}_1{\theta }_1}{K_d{K}_1-1}+{\theta }_1}<0.6\Rightarrow 0.1\le \frac{{\theta }_1(K_d{K}_1-1)}{{\tau }_1-{\theta }_1+0.5{\theta }_1{K}_d{K}_1}<0.6$$

(41)

Case 3: For this case, the condition (36) and (37) are used to choose $K_d$ for the stabilized controller, remember that ${\tau }_d=0.5{\theta }_1$ as in Equation (29).

Step 2: Analytical design of the FOPI controller

The ideal feedback controller that is equivalent to the IMC controller can be expressed in terms of the internal model ${\tilde{G}}_p$ and the IMC controller $q$. Due to the Smith predictor structure, the delay term will be removed in the closed-loop transfer function of the primary loop as in Equation (25). Therefore, in this step, the FOPI controller will be designed for the delay-free process $G_{pm}$.

Figure 5a,b shows the block diagrams of the IMC scheme and equivalent classical feedback control structures, respectively, where $G_{pm}$ is the process, ${\tilde{G}}_{pm}$ the process model, $q$ the IMC controller, and $G_{c_1}$ the equivalent feedback controller. For the nominal case ($G_{pm}={\tilde{G}}_{pm}$), in Figure 5a, the system response can be calculated by the set-point and disturbance signals in the following simplified equation:

$$y=G_{pm}qr+(1-{\tilde{G}}_{pm}q)G_{d_1}d$$

(42)

where ${\tilde{G}}_{pm}=\frac{K_p}{{\tau }_{p}s+1}=p_m$, which is the portion of the model inverted by the controller, and in this case, $p_a=1$ (IMC approach is already mentioned in the previous section).

To obtain a good response for processes with poles near zero, the IMC controller $q$ should be designed to satisfy the following conditions.

1. If the process $G_{pm}$ has poles near zero at $z_1,\hspace{0.17em}{z}_2,\cdots ,\hspace{0.17em}{z}_m$ then $q$ should have zeros at $z_1,\hspace{0.17em}{z}_2,\cdots ,\hspace{0.17em}{z}_m$

2. If the process $G_{d_1}$ has poles near zero, $z_{d1},\hspace{0.17em}{z}_{d2},\cdots ,\hspace{0.17em}{z}_{dm}$ then $1-{\tilde{G}}_{pm}q$ should have zeros at $z_{d1},\hspace{0.17em}{z}_{d2},\cdots ,\hspace{0.17em}{z}_{dm}.$

Since the IMC controller $q$ is designed as $q=p_m^{-1}f$, the first condition is satisfied automatically. The second condition can be fulfilled by designing the IMC filter $f$ as:

$$f=\frac{\beta s+1}{{({\lambda }_{1}s+1)}^2}$$

(43)

where ${\lambda }_1$ is an adjustable parameter which controls the tradeoff between the performance and robustness; $\beta$ is an extra degree of freedom to cancel the poles near zero in $G_{d_1}$, and determined via the following equation:

$${\left(1-{\tilde{G}}_{pm}q\right)|}_{s=-1/{\tau }_p}={\left(1-\frac{\beta s+1}{{({\lambda }_{1}s+1)}^2}\right)|}_{s=-1/{\tau }_p}=0$$

(44)

Thus, the value of $\beta$ is obtained as:

$$\beta =2{\lambda }_1-\frac{{\lambda }_1^2}{{\tau }_p}$$

(45)

Then, the IMC controller is expressed as follows:

$$q=p_m^{-1}f=\frac{{\tau }_{p}s+1}{K_p}\frac{\beta s+1}{{\left({\lambda }_{1}s+1\right)}^2}$$

(46)

From Figure 5a,b, the ideal feedback controller can be derived in terms of the internal model ${\tilde{G}}_{pm}$ and the IMC controller $q$:

$$G_{c_1}=\frac{q}{1-{\tilde{G}}_{pm}q}$$

(47)

Substituting Equation (46) into Equation (47), the ideal feedback controller is finally obtained:

$$G_{c_1}=\frac{({\tau }_{p}s+1)(\beta s+1)}{K_p\left[{({\lambda }_{1}s+1)}^2-(\beta s+1)\right]}$$

(48)

It is obvious that the resulting controller, Equation (48), does not have the FOPI-type controller form as in Equation (11) and also not in the frequency domain. Therefore, it is necessary to convert it into the frequency domain by replacing $s=j\omega$ and rearranging it into the complex form. After several calculations, the final result is as follows:

$$\begin{array}{l}{G}_{c_1}\left(j\omega \right)=\frac{1}{K_p}\frac{\left(1+j\omega {\tau }_p\right)\left(1+j\omega \beta \right)}{{\left(1+j{\lambda }_1\omega \right)}^2-\left(1+j\omega \beta \right)}\\ =\frac{\left\{-{\lambda }_1{}^2{\omega }^2\left(1-{\omega }^2{\tau }_p\beta \right)+{\omega }^2\left({\tau }_p+\beta \right)\left(2{\lambda }_1-\beta \right)\right\}}{K_p\left[{\lambda }_1{}^4{\omega }^4+{\omega }^2{\left(2{\lambda }_1-\beta \right)}^2\right]}-j\frac{\left\{\omega \left(1-{\omega }^2{\tau }_p\beta \right)\left(2{\lambda }_1-\beta \right)+{\omega }^3\left({\tau }_p+\beta \right){\lambda }_1{}^2\right\}}{K_p\left[{\lambda }_1{}^4{\omega }^4+{\omega }^2{\left(2{\lambda }_1-\beta \right)}^2\right]}\end{array}$$

(49)

By comparing Equation (49) and Equation (11), the analytical tuning rules of the FOPI controller can be obtained as follows:

$$K_{I_1}=\frac{1}{K_p}\frac{{\omega }^{\lambda }\left\{\omega \left(1-{\omega }^2{\tau }_p\beta \right)\left(2{\lambda }_1-\beta \right)+{\omega }^3\left({\tau }_p+\beta \right){\lambda }_1{}^2\right\}}{\sin {\gamma }_I\left[{\lambda }_1{}^4{\omega }^4+{\omega }^2{\left(2{\lambda }_1-\beta \right)}^2\right]}$$

(50)

$$K_{c_1}=\frac{\left\{-{\lambda }_1{}^2{\omega }^2\left(1-{\omega }^2{\tau }_p\beta \right)+{\omega }^2\left({\tau }_p+\beta \right)\left(2{\lambda }_1-\beta \right)\right\}}{K_p\left[{\lambda }_1{}^4{\omega }^4+{\omega }^2{\left(2{\lambda }_1-\beta \right)}^2\right]}-\frac{K_I\cos {\gamma }_I}{{\omega }^{\lambda }}$$

(51)

From these above equations, two tuning algorithms, Algorithms 1 and 2, for the primary controllers are proposed for case 2, and 3. It is obvious that the control parameters of case 1 are easily obtained.

Algorithm 1: The proposed tuning algorithm for case 2.

1:  Initialization

Calculate $\left[K_{d\min },K_{d\max }\right]$ according to Equations (32) and (41)     $K_d=K_{d\min }$; $\Delta K_d=0.01$; assign the value of $M_{s\_desired}$

$2: \mathbf{while}\left|M_s-M_{s\_desired}\right|>0.01$do

$3: \mathit{Compute} K_{p_1}, {\tau }_{p_1}$according to Equation (31)

$4: \mathit{Determine} {\lambda }_1$based on Equation (40)

$5: \mathit{Compute} K_{c_1}, K_{I_1}$according to Equations (50) and (51)

$6: \mathit{Calculate} M_s$for each set of control parameters

$7: \mathit{Increase} \mathit{the} \mathit{value} \mathit{of} K_d$by$\Delta K_d$:$K_d=K_{d\min }+\Delta K_d$

$8: \mathbf{If} \left(K_d>K_{d\max }\right)$: assign$M_s=M_{s\_desired}$ % avoid an infinite loop

9:  end while

10: end

Algorithm 2: The proposed tuning algorithm for case 3.

1:  Initialization

Calculate $\left[K_{d\min },K_{d\max }\right]$ according to Equations (36) and (37)    Choose $K_d$ from this range

$2: \mathit{Compute} \mathit{the} \mathit{equivalent} \mathit{transfer} \mathit{function} {\tilde{G}}_p$according to Equation (35)

3:  Approximate into ${\tilde{G}}_p$ FOPDT using PSO algorithm

$4: \mathit{Determine}{\lambda }_1$based on Equation (40)

$5: \mathit{Compute} K_{c_1}, K_{I_1}$according to Equations (50) and (51)

6:  end

3. Results

In this section, some typical process models are considered to evaluate the performance of the proposed method and the obtained results of this work are compared with those of some available literature. In this paper, the performance indices including the integral absolute error (IAE), the integral time-weighted absolute error (ITAE), and total variant (TV) are addressed and the smaller values of those indicate the better performance in terms of servo tracking as well as the regulator problem.

3.1. Example 1

Consider the following process models which are stable transfer functions in both secondary and primary loops, and also studied in some literature [3,4].

$$\begin{array}{l}{G}_{p_1}(s)=G_{d_1}(s)=\frac{e^{-4s}}{20s+1}\\ G_{p_2}(s)=G_{d_2}(s)=\frac{1}{10s+1}\end{array}$$

(52)

In this example, the obtained results are compared with the results of [3,4]. According to Equation (24), the secondary controller is obtained:

$$G_{cd2}(s)=\frac{10s+1}{s+1}$$

(53)

For the primary controller, the controller is obtained according to two following steps:

-

Step 1: Determine the stabilized controller, in this example $G_d=0$. Therefore, the equivalent transfer function of the primary loop is derived as in Equation (54):

$${\tilde{G}}_p(s)=G_{p_1}(s)=\frac{e^{-4s}}{20s+1}$$

(54)

-

Step 2: Applying Equations (40), (50), and (51) to derive the control parameters. The obtained controller and is shown in

Table 1. The control parameters of the proposed and other methods of example 1.

	Secondary Loop	Primary Loop
Proposed	$G_{c2}(s)=\frac{10s+1}{s+1}$	$G_d(s)=0$	$G_{c1}(s)=3.3467+\frac{0.274}{s^{0.9}}$
Raja	$G_{cd2}(s)=\frac{10s+1}{0.01s+1}$	$G_{cd1}(s)=0$	$G_{c1}(s)=2.4799\left(1+\frac{1}{20.9471s}\right)$
Lee (Case B)	$G_{c2}(s)=10\left(1+\frac{1}{10s}\right)$		$G_{c1}(s)=2.75\left(1+\frac{1}{22s}+1.85s\right)\frac{1}{10s+1}$

.

To justify the effectiveness of the proposed method, the controller settings in Table 1 are simulated by providing a unit step change in the set point at t = 0 and a unit step load disturbance at t = 100 (s). In addition, to prove the robustness of the proposed controller as well as to compare with other methods, the gains and time delays of $G_{p_1},G_{p_2},G_{d_1}$ and $G_{d_2}$ are perturbed by $+20\%$, whereas the time constants of those transfer functions are also perturbed by $-20\%$. Figure 6 and Figure 7 illustrate the nominal and perturbed closed-loop responses respectively at the primary output in this example. From the figures, it can be seen that the proposed method improved the closed-loop performance in the servomechanism problem compared to other methods. The performance indexes in

Table 2. The performance index in nominal and perturbed models for example 1.

	Nominal			Perturbed (±20%)
	IAE	ISE	TV	IAE	ISE	TV
Proposed	6.1723	2.7231	3.6211	6.6548	2.8601	4.3013
Raja	8.8062	6.8088	4.0936	10.8054	7.6263	4.7519
Lee (case B)	10.727	7.4696	7.9901	12.3538	7.8052	12.430

also clarify the effectiveness of the proposed method, although the figures also show that the proposed tuning rules yield oscillatory at the disturbance change. The control signals in the nominal and perturbed cases are shown in Figure 8 and Figure 9, respectively. It is obvious that the proposed method has a smoother control signal compared to other methods. As a result, the TV indexes of the proposed method in Table 2 have the smallest value in all cases.

3.2. Example 2

The following primary and secondary process models, Equation (55), were studied by several researchers [2,3,8] for the case of an unstable process:

$$\begin{array}{l}{G}_{p_1}(s)=G_{d_1}(s)=\frac{e^{-4s}}{20s-1}\\ G_{p_2}(s)=G_{d_2}(s)=\frac{2e^{-2s}}{20s+1}\end{array}$$

(55)

The secondary controller is obtained from Equation (24):

$$G_{cd2}(s)=\frac{20s+1}{2(s+1)}$$

(56)

(where ${\lambda }_2=1$).

The primary controller is also derived using the proposed Algorithm 1 for case 2. From Algorithm 1, the value $M_{s\_desired}$ first needs to be assigned. And in this example, $M_{s\_desired}=1.1$ to ensure the robustness of the proposed fractional-order controller. The obtained controller parameters are tabulated in

Table 3. The control parameters of the proposed and other methods of example 2.

	Secondary Loop	Primary Loop
Proposed	$G_{c2}(s)=\frac{20s+1}{2(s+1)}$	$G_d(s)=2.893\left(1+2s\right)$	$G_{c1}(s)=0.9901+\frac{0.4063}{s^{0.9}}$
Raja	$G_{cd2}(s)=\frac{0.5(20s+1)}{0.6s+1}$	$G_{cd1}(s)=1.002\left(1+\frac{2s}{0.2s+1}\right)$	$G_{c1}(s)=2.589\left(1+\frac{1}{7367s}\right)$
Santosh	$G_{c2}(s)=k_{c2}=5.25$		$G_{c1}(s)=0.2386\left(1+\frac{1}{0.7917s}\right)$

. In this example, the proposed method is compared with other methods reported by Santosh [2] and Raja [8], and their results are also listed in Table 3.

Figure 10 and Figure 11 show the nominal closed-loop responses and the corresponding control signals in example 2. In this example, a negative unit step disturbance is also inserted at t = 125 (s) to evaluate the capability of the proposed control in the regulator problem. In general, the proposed tunning settings have a better performance in set-point change with a faster response and a shorter settling time. However, in the regulator problem, the proposed method yields a slight oscillatory compared to the method proposed by Raja [8]. The performance indexes are summarized in

Table 4. The performance index in nominal and perturbed models for example 2.

		Nominal		Perturbed (±20%)
	IAE	ISE	TV	IAE	ISE	TV
Proposed	8.9633	4.0985	2.3717	11.1245	4.2856	4.5196
Raja	7.9704	3.5565	3.2496	8.9385	3.5034	4.9707
Santosh	28.0025	16.9457	8.2492	37.2077	24.424	11.1628

.

Moreover, the robustness of the proposed controller is testified by perturbing the gains and time delays of $G_{p_1},G_{p_2},G_{d_1}$ and $G_{d_2}$ $+20\%$, whereas the time constants of those transfer functions $-20\%$. Figure 12 and Figure 13 illustrate the process responses and control signals respectively in the perturbed case. The performance indexes, in this case, are also listed in Table 4. It is obvious that the proposed method is comparable to the method proposed by Raja and has a much better performance compared to the settings proposed by Santosh in both servomechanism and regulator problems.

3.3. Example 3

In this example, the process models include a SOIPTD as a primary process and a stable FOPTD secondary process, Equation (57). These models were also investigated by several researchers [3,8,9].

$$\begin{array}{l}{G}_{p_1}(s)=G_{d_1}(s)=\frac{e^{-6.5672s}}{s(3.4945s+1)}\\ G_{p_2}(s)=G_{d_2}(s)=\frac{2e^{-2s}}{s+1}\end{array}$$

(57)

The secondary controller is obtained similarly to two previous examples:

$$G_{c2}(s)=\frac{s+1}{2(0.6s+1)}$$

(58)

The primary controller, in this example, is derived according to Algorithm 2. $K_d$ is chosen from the range of Equations (36) and (37), and in this example $K_d=0.02$. From Equation (35) the equivalent transfer function ${\tilde{G}}_p$ is obtained:

$${\tilde{G}}_p=\frac{e^{-6.5672s}}{3.494s^2+0.9343s+0.02}$$

(59)

Using the PSO algorithm proposed in [29] to reduce ${\tilde{G}}_p$ to the FOPTD. Figure 14 illustrates the unit step response of the original Equation (59) and the approximated transfer function Equation (60). It can be seen that the reduced-order transfer function approximates well to the second-order one, and it could be adopted for the next step of the design procedure.

$${\tilde{G}}_p\simeq \frac{50.237e^{-6.5672s}}{47.5566s+1}$$

(60)

The control parameters are obtained and tabulated in

Table 5. The control parameters of the proposed and other methods of example 3.

	Secondary Loop	Primary Loop
Proposed	$G_{c2}(s)=\frac{s+1}{2(0.6s+1)}$	$G_d(s)=0.02\left(1+3.2836s\right)$	$G_{c1}(s)=0.0977+\frac{0.0038}{s^{0.9}}$
Raja (2016)	$G_{cd2}(s)=\frac{0.5(s+1)}{0.5s+1}$	$G_{cd1}(s)=0.003\left(1+3.2836s\right)$	$G_{c1}(s)=0.042\left(1+\frac{1}{322.964s}\right)$
Raja (2017)	$G_{cd2}(s)=\frac{0.5(s+1)}{0.6s+1}$	$G_{cd1}(s)=0.003\left(1+3.2836s\right)$	$G_{c1}(s)=0.0939\left(1+\frac{1}{321.6012s}\right)$

. In this example, the proposed method is compared with other methods suggested by Raja et al. [3] (2016) and [8] (2017). Figure 15 and Figure 16 show the nominal closed-loop responses as well as the corresponding control signals. The proposed method yields a fast response accompanied by an overshoot compared to other methods. However, the methods proposed by Raja have an issue in the case of the regulator problem in which the steady-state has a deviation after a load disturbance. In contrast, the proposed tuning settings still keep a good steady-state response after the load disturbance. The performance indexes listed in

Table 6. The performance index in nominal and perturbed models for example 3.

		Nominal		Perturbed (±20%)
	IAE	ISE	TV	IAE	ISE	TV
Proposed	17.9520	7.5790	0.1845	18.5801	7.9852	0.2038
Raja (2016)	39.9687	21.848	0.0819	39.8605	22.166	0.0851
Raja (2017)	17.9977	8.1274	0.1591	17.9531	8.3991	0.1703

prove that the proposed method has comparable results to other methods.

In order to evaluate the robustness of the proposed controller to uncertainties in process dynamics, the gains and time delays of $G_{p_1},G_{p_2},G_{d_1}$ and $G_{d_2}$ are perturbed by $+20\%$ whereas the time constants of those transfer functions are changed by $-20\%$. Figure 17 and Figure 18 show the process responses and control signals respectively in the perturbed case and prove that the proposed controller still keeps robust stability and performs a good steady-state response due to the load disturbance. The performance indexes in the perturbed case listed in Table 6 state the effectiveness of the proposed settings.

4. Conclusions

In this paper, the fractional-order proportional-integral (FOPI) controller is proposed and designed analytically in the frequency domain for the parallel cascade control structures (PCCS) combined with the Smith predictor which are adopted for stable, unstable, and integrating process models with time delays. The control structure consists of three controllers including a stabilized controller for a class of unstable and integrating models, a disturbance rejection controller in the secondary loop (inner), and a primary servomechanism controller. Accordingly, the ideal controller is obtained by using the internal model control (IMC) approach for the inner loop. The proportional-derivative (PD) controller is suggested for the stabilized controller and is designed based on a stability criterion. The tuning procedures are summarized in Algorithms 1 and 2 for unstable and integrating processes, respectively. The simulation studies investigated for three mentioned cases of process models demonstrate the flexibility and effectiveness of the proposed method for the PCCS in comparison with the other methods. The robust stability is also justified by perturbed process models with ±20% of process parameters. The simulations prove that the proposed controller still keeps good responses in terms of set-point tracking and the regulatory problem in all cases.