Optimization-based robust fractional-order controllers design for multiple-input single-output systems with time delays

Abstract

This paper deals with the design of fractional order PID controllers based on nonlinear optimization for Multi-Input Single-Output (MISO) systems with time delays. Two approaches are used to obtain optimal controllers’ parameters. The first uses time-domain specifications and consists of minimizing time-domain criteria without constraints. The second uses frequency-domain specifications and imposes the open-loop phase to be flat in a specified frequency band. Simulations results and a comparative study are presented to verify the effectiveness of each approach.

Introduction

Since its development, Proportional Integral Derivative (PID) controller has been frequently used in industrial control because of its ease of application and design. ¹ Nowadays, it still widely used in most control systems even though control theory has advanced considerably. Design, implementation and tuning of PID controllers have all been extensively studied in the literature. ² Nevertheless, PID efficiency dramatically decreases when it deals with uncertain systems, nonlinear systems or dead-time systems. Therefore, despite all its benefits, the PID controller became unable to provide robustness and disturbance rejection in same cases.

To solve this problem, many researchers have developed several approaches and most of them are based on the numerical optimization. ³ However, obtaining the requested robustness is not possible even when the optimization is applied to the PID design. One of the well-known solution to this issue is proposed by Podlubny ⁴ who has developed an extension of the PID. This extension, known as the fractional PID, has really been successful in research due to its robustness characteristics and flexibility in design.^5,6

To obtain robustness to load disturbances, high-frequency noise, and plant model uncertainty, a tuning technique can be used to determine the optimum controller parameters satisfying all constraints. Finding the fractional controllers’ optimum parameters within the specified limitations is the aim of the tuning approach. Two main approaches exist in the literature for the design of a fractional order controller:

• Time-domain design approach: it is based on time-domain specifications, including the dominant pole placement or time-domain integral performance index optimization techniques.

• Frequency-domain design approach: it is based on frequency-domain specifications and optimization-based design techniques.

To improve fractional control efficiency, researchers have presented a design approach based on a constrained numerical optimization to obtain some desired frequency specifications such as phase margin, unity-gain crossover frequency, and sensitivity to high-frequency noises and disturbances.^7–9 Moreover, in Saidi et al. ⁵ an extended Monje’s method have been developed to enlarge the desired frequency band where the phase is considered constant.

In literature, many works treated fractional PID controllers design for Single Input Single Output (SISO) and Multi-Input Multi-Output (MIMO) systems.^10–13 Nowadays, various design methods have been proposed for FOPID controller tuning. The well-known Ziegler-Nichols tuning rule has been expanded to include an S-shaped step response in the FOPID controller, nevertheless, it is only effective in certain lag-dominant processes. ¹⁴ Some tuning techniques are proposed to minimize integrated absolute error (IAE) for a typical first-order-plus dead-time model, subject to constraint on the maximum sensitivity. ¹⁵ The internal-model-control (IMC) tuning technique has also been included into the FOPID controller design in recent years. ¹⁶

In real life industrial applications, Multi-Input Single-Output (MISO) systems are frequently encountered in the fields of petrochemical engineering, fluid transportation and automotive systems.^17–19 The literature presents some works that have treated the MISO rational control problem: in Lupu et al., ²⁰ multi-model method was implemented. In addition, Rico Azagra et al. ²¹ developed the quantitative feedback theory technique to design rational controllers for MISO system. In Zhang et al., ²² an alternating beam forming optimization algorithm based on the Lagrangian method and MM technique was proposed. Eskandari et al. ²³ treated the MISO system problem control with model predictive control (MPC) method.

Multi-Input Single-Output system robust control is a challenging problem. In fact, imposing some frequency-domain or time-domain specifications to sub-system with different dynamics is a difficult problem. That’s why, to the best of our knowledge, MISO robust fractional control problem is still open for research and there exist only few works on it.^24,25

This paper presents robust design methods developed for MISO systems. The first is based on time-domain specifications while the second is based on frequency-domain specifications. Three cases of MISO systems with delay are treated. The first consists of SISO subsystems without time delays. The second is for SISO subsystems with equal time-delays. The last treats SISO subsystems with different time delays.

The paper is organized as follows: the next section presents the problem statement of MISO systems control. The two design approaches are then developed. Next, to show the efficiency of these approaches, numerical simulations are given. The last section presents some conclusions and perspectives.

Problem statement of MISO systems control

A MISO system can be presented, as shown in Figure 1, with $G_m\left(s\right)(m=1,...,M)$ is the transfer function of the $m^{\mathrm{th}}$ SISO subsystem and $M$ is the number of subsystems.

Figure 1.

Closed-loop MISO system.

The control and the output signal are denoted, respectively, by $u_m\left(t\right)$ and $y_m\left(t\right)$ . The reference output signal is $y_{\mathrm{sp}}\left(t\right)$ , the global output signal of the MISO system is $y\left(t\right)$ and $\epsilon \left(t\right)$ is the output error.

Many works have treated the control problem of MISO systems. In Yazdizadeh et al., ²⁶ a new decentralized resilient optimum MISO PID controller based on Characteristic Matrix Eigenvalues and the Lyapunov approach is proposed. Moreover, Kluska ²⁷ has developed an approach to design rational PID for MISO system with time delay. Some works, presented in Moussa et al.,^24,25 have developed fractional order controllers design methods for MISO systems. In this paper, we consider only time delay subsystems with the same structure. The objective is to design fractional-order controllers $C_m(m=1,...M)$ , to obtain closed-loop time-domain or frequency-domain specifications.

The transfer function of the $m^{\mathrm{th}}$ fractional order sub-controller $C_m\left(s\right)$ is given by:

$$C_m\left(s\right)=k_{p,m}+\frac{k_{i,m}}{s^{{\alpha }_m}}+k_{d,m}{s}^{{\beta }_m},m=1,...,M$$

(1)

where $k_{p,m}$ , $k_{i,m}$ , and $k_{d,m}$ are, respectively, the proportional, integrator and derivative gains. ${\alpha }_m$ and ${\beta }_m$ define, respectively, the integration and differentiation order. ⁴

The aim of this work is to find the optimal parameters of all controllers while using the output of the MISO system and the desired performances. Thus, the $m$ fractional PID controllers have $m\times 5$ parameters to be optimized. The large number of these parameters let the problem of optimization be more difficult.

Design of fractional PID controllers

To design $C_m\left(s\right)$ a cost function is minimized. This function is nonlinear with respect to the fractional operators ( $D^{\beta }$ and $I^{\alpha }$ ). Consequently, the parameters of $C_m$ controllers should be determined through nonlinear optimization techniques.

The parameters of $m$ controllers to be optimized are:

$$V=[V_1,...,V_m,...,V_M]$$

(2)

with

$$V_m=[k_{p,m},k_{i,m},k_{d,m},{\alpha }_m,{\beta }_m],m=1,...,M.$$

(3)

Time-domain design approach

For this approach, a time-domain cost function is considered, and no constraints are used. The choice of the cost function is very crucial in solving the design problem. The time-domain cost function $J\left(t\right)$ can be one of the following functions:

• Integral Squared Error:

  $$J_{\mathrm{ISE}}\left(t\right)={\int }_0^{\infty }{\epsilon }^2\left(t\right)\mathrm{dt}$$

  (4)

• Integral Absolute Error:

  $$J_{\mathrm{IAE}}\left(t\right)={\int }_0^{\infty }\left|\epsilon \left(t\right)\right|\mathrm{dt}$$

  (5)

• Integral Time-weighted Squared Error:

  $$J_{\mathrm{ITSE}}\left(t\right)={\int }_0^{\infty }t{\epsilon }^2\left(t\right)\mathrm{dt}$$

  (6)

• Integral Time-weighted Absolute Error:

  $$J_{\mathrm{ITAE}}\left(t\right)={\int }_0^{\infty }t\left|\epsilon \left(t\right)\right|\mathrm{dt}$$

  (7)

The algorithm of time-domain design approach can be summarized as follows:

• ★ Step 1:

• ★ Step 2:

• ★ Step 3: If a maximum number of iterations ( $j=j_{\max }$ ) is achieved or the cost function $J_t^j$ is considered minimal ( $J_t^j<ϵ$ , $ϵ<<1$ ), then the algorithm is stopped. Else, go back to step 2.

The time-domain approach can ensure some performances (stability, disturbance rejection, rapidity, precision). But, in several cases, it cannot ensure robust stability. Consequently, a frequency-domain approach is proposed in the following section.

Frequency-domain based design method

In the frequency-domain, the $m^{\mathrm{th}}$ fractional sub-controller transfer function can be written as:

$$C_m\left(j\omega \right)=k_{p,m}+\frac{k_{i,m}}{{\left(j\omega \right)}^{{\alpha }_m}}+k_{d,m}{\left(j\omega \right)}^{{\beta }_m},m=1,...,M.$$

(8)

Consider a desired phase margin ${\mathcal{P}}_{\mathcal{M}}$ for several frequencies ${\omega }_l$ in the band $[{\omega }_{\min },{\omega }_{\max }]$ including unity-gain crossover frequency ${\omega }_u$ . The parameters of the $P{I}^{\alpha }{D}^{\beta }$ controllers are determined by minimizing the subsequent frequency-domain criterion $J_{{\omega }_u}$ using a nonlinear optimization technique:

$$J_{{\omega }_u}={\left|{\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_u\right)\right|}_{\mathrm{dB}}$$

(9)

where ${\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)=\sum _{m=1}^M\left(C_m\left(j\omega \right)G_m\left(j\omega \right)\right)$ is the MISO system open-loop transfer function.

To obtain desired performances, the following constraints are used:

• Desired phase margin ${\mathcal{P}}_{\mathcal{M}}$ :

  $$\begin{array}{l}arg\left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_l\right)\right)=-\pi +{\mathcal{P}}_{\mathcal{M}},\\ {\omega }_l\in \left\{{\omega }_{min},\dots ,{\omega }_u,\dots ,{\omega }_{max}\right\}\end{array}$$

  (10)

• Stability robustness:

  $$\frac{d\arg \left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_l\right)\right)}{d{\omega }_l}|=0,{\omega }_l\in \{{\omega }_{\min },...,{\omega }_u,...,{\omega }_{\max }\}$$

  (11)

To satisfy constraints (10) and (11), there will be really so many constraints to be solved by the optimization algorithm. This will lead to a huge computing time and less effective method. To solve this problem, the constraints are reformulated as follows:

$$\begin{array}{c}\sum _{l=0}^L{(\arg \left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_l\right)\right)+\pi -{\mathcal{P}}_{\mathcal{M}})}^2=0,\hfill \\ \multicolumn{1}{c}{{\omega }_l\in \{{\omega }_{\min },...,{\omega }_{\max }\}}\end{array}$$

(12)

$$\sum _{l=0}^L{\left(\frac{d\arg \left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_l\right)\right)}{d{\omega }_l}\right)}^2=0,$$

(13)

where $L$ is the number of the considered frequencies in $[{\omega }_{\min },{\omega }_{\max }]$ .

• High frequency noise attenuation is achieved by the following constraint:

  $${|T\left(j\omega \right)=\frac{{\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)}{1+{\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)}|}_{\mathrm{dB}}\le A,\forall \omega \le {\omega }_t$$

  (14)

  with $A$ is the desired noise attenuation for $\omega ={\omega }_t$ .

• Output disturbance rejection using an additional constraint on the sensibility function $S\left(j\omega \right)$ :

  $${|S\left(j\omega \right)=\frac{1}{1+{\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)}|}_{\mathrm{dB}}\le B,\forall \omega \le {\omega }_s$$

  (15)

  with $B$ is the desired value of the sensitivity for $\omega ={\omega }_s$ .

  The numerical constrained optimization problem is given as:

  $$\min J_{{\omega }_u}={\left|{\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_u\right)\right|}_{\mathrm{dB}}$$

  subject to

  $$\begin{array}{c}\sum _{l=0}^L{(\arg \left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_l\right)\right)+\pi -{\mathcal{P}}_{\mathcal{M}})}^2=0\hfill \\ {\sum _{l=0}^{L-1}{\left(\frac{d\arg \left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_l\right)\right)}{d{\omega }_l}\right)}^2|}_{{\omega }_l\in \{{\omega }_{\min },...,{\omega }_{\max }\}}=0\hfill \\ {|T\left(j\omega \right)=\frac{{\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)}{1+{\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)}|}_{\mathrm{dB}}\le A\hfill \\ {|S\left(j\omega \right)=\frac{1}{1+{\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)}|}_{\mathrm{dB}}\le B\hfill \end{array}$$

  The cost function to be minimized is defined as:

  $${\left|{\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)\right|}_{\mathrm{dB}}=20\log \sqrt{\mathrm{Re}\left({\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)\right)+\mathrm{Im}\left({\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)\right)}$$

  (16)

  with $\mathrm{Re}\left({\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)\right)$ is real part of the open-loop function:

  $$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Re}\left({\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)\right)=\sum _{m=1}^M\mathrm{Re}\left(C_m\left(j\omega \right)\right)\mathrm{Re}\left(G_m\left(j\omega \right)\right)}\\ -\mathrm{Im}\left(C_m\left(j\omega \right)\right)\mathrm{Im}\left(G_m\left(j\omega \right)\right)\hfill \end{array}$$

  (17)

  and $\mathrm{Im}\left({\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)\right)$ is the imaginary part of the open-loop function:

  $$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Im}\left({\mathcal{O}}_{\mathcal{P}}\left(j\omega \right)\right)=\sum _{m=1}^M\mathrm{Re}\left(C_m\left(j\omega \right)\right)\mathrm{Im}\left(G_m\left(j\omega \right)\right)}\\ -\mathrm{Im}\left(C_m\left(j\omega \right)\right)\mathrm{Re}\left(G_m\left(j\omega \right)\right)\hfill \end{array}$$

  (18)

  For example, we consider that all subsystems composing the MISO system are represented by first order with time delay transfer functions as:

  $$G_m\left(s\right)=\frac{k_m}{1+{\tau }_{m}s}{e}^{-T_{m}s}$$

  (19)

  where $k_m$ is the static gain, ${\tau }_m$ and $T_m$ define respectively the time constant and delay, then the real and imaginary parts of the open-loop are defined as following:

  $$\begin{array}{c}\mathrm{Re}\left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_u\right)\right)=\hfill \\ \frac{(k_{p,m}+k_{i,m}{\omega }_u^{-{\alpha }_m}\cos \left({\alpha }_m\frac{\pi }{2}\right))(k_m\cos \left(T_m{\omega }_u\right)-k_m{\tau }_m{\omega }_u\sin \left(T_m{\omega }_u\right))}{1+{\tau }_m^2{\omega }_u^2}\hfill \\ -\frac{(k_{i,m}{\omega }_u^{-{\alpha }_m}\sin \left({\alpha }_m\frac{\pi }{2}\right)-k_m{\tau }_m{\omega }_u\cos \left(T_m{\omega }_u\right)-k_m\sin \left(T_m{\omega }_u\right))}{1+{\tau }_m^2{\omega }_u^2}\hfill \end{array}$$

  (20)
  $$\begin{array}{c}\mathrm{Im}\left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_u\right)\right)=\hfill \\ \frac{(k_m{\tau }_m{\omega }_u\cos \left(T_m{\omega }_u\right)-k_m\sin \left(T_m{\omega }_u\right))(k_{p,m}+k_{i,m}{\omega }_u^{-{\alpha }_m}\cos \left({\alpha }_m\frac{\pi }{2}\right))}{1+{\tau }_m^2{\omega }_u^2}\hfill \\ -\frac{(k_m\cos \left(T_m{\omega }_u\right)-k_m{\tau }_m{\omega }_u\sin \left(T_m{\omega }_u\right))(-k_{i,m}{\omega }_u^{-{\alpha }_m}\sin \left({\alpha }_m\frac{\pi }{2}\right))}{1+{\tau }_m^2{\omega }_u^2}\hfill \end{array}$$

  (21)

  In the case when $T_m=0$ , we obtain the following equations:

  $$\begin{array}{c}\mathrm{Re}\left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_u\right)\right)=\frac{k_m(k_{p,m}+k_{i,m}{\omega }_u^{-{\alpha }_m}\cos \left({\alpha }_m\frac{\pi }{2}\right))}{1+{\tau }_m^2{\omega }_u^2}\hfill \\ -\frac{(-k_{i,m}{\omega }_u^{-{\alpha }_m}\sin \left({\alpha }_m\frac{\pi }{2}\right))(-k_m{\tau }_m{\omega }_u)}{1+{\tau }_m^2{\omega }_u^2}\hfill \end{array}$$

  (22)
  $$\begin{array}{c}\mathrm{Im}\left({\mathcal{O}}_{\mathcal{P}}\left(j{\omega }_u\right)\right)=\frac{(-k_m{\tau }_m{\omega }_u)(k_{p,m}+k_{i,m}{\omega }_u^{-{\alpha }_m}\cos \left({\alpha }_m\frac{\pi }{2}\right))}{1+{\tau }_m^2{\omega }_u^2}\hfill \\ -\frac{k_m(-k_{i,m}{\omega }_u^{-{\alpha }_m}\sin \left({\alpha }_m\frac{\pi }{2}\right))}{1+{\tau }_m^2{\omega }_u^2}\hfill \end{array}$$

  (23)

Numerical simulations

Consider the following Two-Input Single-Output (TISO) system represented by Figure 2.

Figure 2.

The considered TISO system structure.

With $k_1=1$ , ${\tau }_1=10$  sec, $k_2=0.1$ and ${\tau }_2=0.1$  sec, three cases will be presented depends on the value of time delay $T_m$ . To ensure a feasible implementation of fractional controller for MISO systems, Pade approximation is used mainly when the time delays are not equals:

$$e^{-T_{m}s}\simeq D_m\left(s\right)=\frac{1-\frac{T_m}{2}s}{1+\frac{T_m}{2}s}$$

(24)

For this example, only a $\mathrm{P}{\mathrm{I}}^{\alpha }$ controller is used. The search space of the tuning parameters is initialized as follows: $k_{p,m}>0$ ; $k_{i,m}>0$ ; $0<{\alpha }_m<2$ .

Time-domain design approach

In this section, the parameters of $\mathrm{P}{\mathrm{I}}^{\alpha }$ are optimized by minimizing four performances indices (ISE, IAE, ITSE, and ITAE). The desired behavior of the corrected closed-loop system is chosen as a second order system with ${\xi }^d=0.707$ and ${\omega }_n^d=3$  rad/sec. The time-domain desired performances are presented by Table 1. ²⁸

Table 1.

Time-domain design specifications.

Specifications	${\xi }^d$	${\omega }_n^d$ (rad/s)	$t_r^d$ (sec)	$t_s^d$ (sec)	${\mathcal{O}}_{\mathcal{S}}^d(\%)$
Value	0.707	3	0.721	1.06	4.325

Case 1 $\left(T_1=T_2=0\right)$ : Consider the two subsystems without delays. Figure 3 presents the step responses of MISO system corrected by fractional controllers. Table 2 shows the optimum fractional controllers parameters using different performances indices. Table 3 presents the time-domain MISO system performances: the rise time $t_r$ , the settling time $t_s$ , the damping factor ${\xi }_m$ , the first overshoot ${\mathcal{O}}_{\mathcal{S}}$ and the measured similarities between the desired and measured output signals FIT defined by:

$$\mathrm{FIT}=100\times (1-\sqrt{\frac{{\sum }_{k=1}^{N_s}{(y\left(t_k\right)-y_{\mathrm{sp}}\left(t_k\right))}^2}{{\sum }_{k=1}^{N_s}{(y\left(t_k\right)-\overline{y})}^2}})$$

(25)

where $\overline{y}$ is the mean value of the output signal and $N_s$ is the number of measures.

Figure 3.

Closed-loop system responses for a positive unity step unit setpoint $y_{\mathrm{sp}}=1$ with $T_1=T_2=0$  sec.

Table 2.

Parameters of $P{I}^{\alpha }$ controllers obtained by time-domain based design with $T_1=T_2=0\sec$ .

Criteria	Controller	$k_{p,m}$	$k_{i,m}$	${\alpha }_m$
ISE	$C_1$	$3.471\times {10}^{-4}$	$8.911\times {10}^{-4}$	0.416
	$C_2$	$3.410\times {10}^{-6}$	2.218	1.155
IAE	$C_1$	1.542	3.229	0.462
	$C_2$	0.016	1.842	1.154
ITSE	$C_1$	$7.633\times {10}^{-4}$	$14\times {10}^{-4}$	0.435
	$C_2$	$6.745\times {10}^{-6}$	2.296	1.127
ITAE	$C_1$	0.042	0.102	0.270
	$C_2$	$3.404\times {10}^{-4}$	2.411	1.110

Table 3.

Time-domain performances obtained by $P{I}^{\alpha }$ controllers with $T_1=T_2=0$  sec.

Criteria	$t_r\left(\sec \right)$	$t_s\left(\sec \right)$	${\xi }_m$	${\mathcal{O}}_{\mathcal{S}}(\%)$	FIT (%)
ISE	0.8	2.5	0.657	6.45	88.990
IAE	0.8	2.5	0.654	6.59	87.656
ITSE	0.8	1.9	0.689	5.027	89.521
ITAE	0.7	1	0.709	4.256	88.140

It is clear from Table 3 and Figure 3 that the criterion ITAE gives the desired time-specifications. To compare these results with those obtained by PI controllers, minimization of the ITAE criterion is used to tune PID parameters. Table 4 presents time-domain performances obtained by $\mathrm{P}{\mathrm{I}}^{\alpha }$ and PI controllers using ITAE criterion with $T_1=T_2=0$  sec.

Table 4.

Time-domain performances obtained by $\mathrm{P}{\mathrm{I}}^{\alpha }$ and PI controllers using ITAE criterion with $T_1=T_2=0$  sec.

Criteria	$t_r\left(\sec \right)$	$t_s\left(\sec \right)$	${\xi }_m$	${\mathcal{O}}_{\mathcal{S}}(\%)$	FIT (%)
PID	0.8	2.6	0.684	5.241	80.277
FO-PID	0.7	1	0.709	4.256	88.140

As shown by Table 4 the performances obtained by the $\mathrm{P}{\mathrm{I}}^{\alpha }$ controllers are more closer to the desired performances than those obtained by PI controllers. In addition, when a disturbance is added to the system output, the $\mathrm{P}{\mathrm{I}}^{\alpha }$ controllers reject it more faster than PI controllers as shown by Figure 4. These results demonstrate the superiority of the fractional controllers for MISO control.

Figure 4.

Closed-loop system responses for a positive load disturbance $d=0.2$ .

Case 2 $T_1=T_2\ne 0$ : In this section, the two SISO subsystems are with time-delays and their values are $T_1=T_2=0.1$  sec. The optimum parameters of each $\mathrm{P}{\mathrm{I}}^{\alpha }$ controller are given in Table 5 and the time-domain performances for each criterion are summarized in Table 6.

Table 5.

Parameters of $P{I}^{\alpha }$ controllers obtained by time-domain based design with $T_1=T_2=0.1$  sec.

Criteria	Controller	$k_{p,m}$	$k_{i,m}$	${\alpha }_m$
ISE	$C_1$	0.003	0.004	0.373
	$C_2$	1.995 $\times {10}^{-5}$	1.878	1.083
IAE	$C_1$	0.690	1.425	0.293
	$C_2$	0.005	1.755	1.074
ITSE	$C_1$	0.005	0.01	0.280
	$C_2$	3.004 $\times {10}^{-6}$	1.939	1.06
ITAE	$C_1$	1.58	5.477	0.189
	$C_2$	0.006	1.464	1.054

Table 6.

Time-domain performances obtained by $P{I}^{\alpha }$ controllers with $T_1=T_2=0.1$  sec.

Criteria	$t_r\left(\sec \right)$	$t_s\left(\sec \right)$	${\xi }_m$	${\mathcal{O}}_{\mathcal{S}}(\%)$	FIT (%)
ISE	0.9	1.2	0.697	4.735	91.185
IAE	0.9	1.2	0.709	4.247	90.153
ITSE	0.9	1.2	0.730	3.470	90.670
ITAE	0.8	1.1	0.749	2.844	84.560

Figure 5 shows the step responses of the corrected MISO system with $\mathrm{P}{\mathrm{I}}^{\alpha }$ . From Figure 5 and Table 6, IAE give the damping factor measured value ${\xi }_m$ near the desired one.

Figure 5.

Closed-loop system responses for a positive unity step unit setpoint $y_{\mathrm{sp}}=1$ with $T_1=T_2=0.1$  sec.

Case 3 $(T_1\ne T_2\ne 0)$ : In this part, the time delays are different as $T_1=1\sec$ and $T_2=0.1\sec$ . Table 7 summarizes the optimum parameters of fractional controllers $C_1$ and $C_2$ . The Time-domain performances of fractional controllers optimized with different criteria are given in Table 8.

Table 7.

Parameters of $P{I}^{\alpha }$ controllers obtained by time-domain based design with $T_1=1$  sec and $T_2=0.1$  sec.

Criteria	Controller	$k_{p,m}$	$k_{i,m}$	${\alpha }_m$
ISE	$C_1$	3.154	2.447	0.260
	$C_2$	0.074	1.963	0.962
IAE	$C_1$	5.807	1.245	0.798
	$C_2$	0.068	1.898	0.933
ITSE	$C_1$	3.170	2.368	0.376
	$C_2$	0.030	1.988	0.942
ITAE	$C_1$	2.494	2.305	0.298
	$C_2$	0.029	2.040	0.960

Table 8.

Time-domain performances obtained by $P{I}^{\alpha }$ controllers with $T_1=1$  sec and $T_2=0.1$  sec.

Criteria	$t_r\left(\sec \right)$	$t_s\left(\sec \right)$	${\xi }_m$	${\mathcal{O}}_{\mathcal{S}}(\%)$	FIT (%)
ISE	0.72	1.06	0.710	4.205	96.577
IAE	0.72	1.06	0.709	4.221	97.116
ITSE	0.72	1.06	0.717	3.93	96.573
ITAE	0.72	1.06	0.714	4.03	96.760

The step responses of MISO system controlled with $\mathrm{P}{\mathrm{I}}^{\alpha }$ optimized by the three criteria are shown in Figure 6. From Figure 6 and Table 7, IAE gives the damping factor measured value ${\xi }_m$ near the desired one.

Figure 6.

Closed-loop system responses for a positive unity step unit setpoint $y_{\mathrm{sp}}=1$ with $T_1=1$  sec and $T_2=0.1$  sec.

Frequency-domain based design

In this section parameters variations are considered to validate the robustness of the proposed frequency-domain approach. To see the efficiency of $\mathrm{P}{\mathrm{I}}^{\alpha }$ in this case, we consider gain variations as: $k_1=[0.5,1.5]$ and $k_2=[0.05,0.15]$ . Three cases are presented: the first where the time delays are zero, the second where the time-delays are different to zero, but they are equal and the last where $T_1\ne T_2\ne 0\sec$ . A comparative study with performances obtained by PI is also presented.

Case 1 $(T_1=T_2=0)$ : The desired performances for the design of the controllers are summarized in Table 9 with an unity-gain crossover frequency ${\mathrm{\omega }}_u\in [{\mathrm{\omega }}_{\min },{\mathrm{\omega }}_{\max }]=[0.5,10]$  rad/sec.

Table 9.

Frequency-domain specifications for $(T_1=T_2=0)$  sec.

${\omega }_u(\mathrm{rad}/\sec )$	$1.5$
${\mathcal{P}}_{\mathcal{M}}(°)$	$45$
${\omega }_s(\mathrm{rad}/\sec )$	0.35
${\omega }_t(\mathrm{rad}/\sec )$	8
$B\left(\mathrm{dB}\right)$	−20
$A\left(\mathrm{dB}\right)$	−20

Table 10 shows the optimum parameters of $\mathrm{P}{\mathrm{I}}^{\alpha }$ and PI controllers. Figure 7 presents the Bode diagrams of the open-loop corrected MISO system. The open-loop phase is flat in the desired frequency band and the frequency-performances are equal to the desired ones as shown by the Figure 7(a). But, Figure 7(b) indicates that the open-loop corrected by PI did not give the desired performances.

Table 10.

Parameters of $\mathrm{P}{\mathrm{I}}^{\alpha }$ and PI obtained by frequency-domain based design with $T_1=T_2=0$  sec.

Type	Controller	$k_{p,m}$	$k_{i,m}$	${\alpha }_m$
$\mathrm{P}{\mathrm{I}}^{\alpha }$	$C_1$	0.1	16.89	0.68745
	$C_2$	0.1	4.7634	1.0144
PI	$C_1$	0.3527	11.709
	$C_2$	0.01	4.7936

Figure 7.

Bode diagrams of the open-loop corrected MISO system with $T_1=T_2=0$  sec: (a) MISO system corrected by $\mathrm{P}{\mathrm{I}}^{\alpha }$ and (b) MISO system corrected by PI.

In Figure 8(a), the Bode diagrams show that the controllers satisfies a frequency noise attenuation $A$ equal to $-20$  dB at the frequency ${\omega }_t=8$  rad/sec. Concerning the disturbances rejection, the Bode diagrams depicted in Figure 8(b) show that the desired performances are well respected with an attenuation $B$ of $-20$  dB at the frequency ${\omega }_s=0.35$ rad/sec. But, in Figure 8(c) attenuation $B$ of $-20$  dB at the frequency ${\omega }_s=0.25$  rad/sec, is not satisfied. Moreover, noise attenuation $A$ equal to $-20$  dB at the frequency ${\omega }_t=8$  rad/sec.

Figure 8.

Bode diagrams of the sensitivity and the complementary sensitivity for the corrected MISO system with $T_1=T_2=0$  sec: (a) sensitivity with $\mathrm{P}{\mathrm{I}}^{\alpha }$ , (b) complementary sensitivity with $\mathrm{P}{\mathrm{I}}^{\alpha }$ , (c) sensitivity with PI, and (d) complementary sensitivity with PI.

Figure 9(a) and (b) show the step responses of the corrected MISO system by $\mathrm{P}{\mathrm{I}}^{\alpha }$ and PI with varying the gain. These figures indicate that the fractional controllers are more robust to gain variation.

Figure 9.

Step responses of closed-loop MISO system with gain variation and $T_1=T_2=0$  sec: (a) with $\mathrm{P}{\mathrm{I}}^{\alpha }$ controllers and (b) with PI controllers.

Figure 10 depicts the control signals produced by the fractional and rational controllers tuned with optimization algorithms. It is obvious that control signals of the fractional controllers provide high energy intensity to make the system faster. Figure 10 indicates that the fractional control signals are more steady.

Figure 10.

Control signals of the closed-loop MISO system with $T_1=T_2=0$  sec: (a) with ${\mathrm{PI}}^{\alpha }$ controllers and (b) with PI controllers.

In Figure 11, the responses of the corrected system with $\mathrm{P}{\mathrm{I}}^{\alpha }$ and PI controllers, in the presence of an additive disturbance, are plotted. Despite the presence of additive disturbance, the fractional controllers ensure the precision and stability of MISO system. From the figure, the fractional controllers are more robust to the disturbances.

Figure 11.

Step responses of the closed-loop corrected MISO systems in the presence of additive disturbance with $T_1=T_2=0$  sec.

Case 2 $(T_1=T_2\ne 0)$ : Consider that the time-delays are different to 0 but equals: $T_1=T_2=0.5$  sec. The desired frequency-performances used to design $\mathrm{P}{\mathrm{I}}^{\mathrm{\alpha }}$ are summarized in Table 11.

Table 11.

Frequency-domain specifications for $T_1=T_2\ne 0\sec$ .

${\omega }_u(\mathrm{rad}/\sec )$	$0.8$
${\mathcal{P}}_{\mathcal{M}}(°)$	$65$
${\omega }_s(\mathrm{rad}/\sec )$	0.26
${\omega }_t(\mathrm{rad}/\sec )$	10
$B\left(\mathrm{dB}\right)$	−10
$A\left(\mathrm{dB}\right)$	−15

The optimum parameters of fractional controllers $\mathrm{P}{\mathrm{I}}^{\alpha }$ are defined by the following transfer functions:

$$\begin{array}{c}{C}_1\left(s\right)=1.6538+\frac{0.2504}{{\mathrm{s}}^{1.2923}}\hfill \\ C_2\left(s\right)=2.0791+\frac{6.6107}{{\mathrm{s}}^{1.1314}}\hfill \end{array}$$

(26)

The optimum parameters of PI controllers are defined by the following transfer functions

$$\begin{array}{c}{C}_1\left(s\right)=2.8677+\frac{0.7073}{s}\hfill \\ C_2\left(s\right)=1.8228+\frac{11.412}{s}\hfill \end{array}$$

(27)

The Bode diagrams of the open-loop MISO system corrected by both fractional and rational controllers are presented by Figure 12. In the frequency band $[0.11]$  rad/sec the phase is flat as shown in Figure 12(a). However, the phase is not flat at the specified frequency band as shown in Figure 12(b).

Figure 12.

Bode diagrams of the open-loop corrected MISO system with $T_1=T_2\ne 0\sec$ : (a) with $\mathrm{P}{\mathrm{I}}^{\alpha }$ controllers and (b) with PI controllers.

The Bode diagrams shown in Figure 13(a) demonstrate that the fractional controllers attenuate efficiently noises where $\left(A\right)$ equal to $-15$  dB at the frequency ${\omega }_t=10$  rad/sec. Concerning the disturbances rejection, the Bode diagrams depicted in Figure 13(b) demonstrate that the desired performances are well respected with an attenuation $\left(B\right)$ of $-10$  dB at the frequency ${\omega }_s=0.26$  rad/sec. The Bode diagrams in Figure 13(c) indicates that PID controllers did not give the desired noise attenuation because $\left(A\right)$ is not equal to $-15$  dB at the desired frequency. Also, the desired performances are not well respected with an attenuation $\left(B\right)$ of $-10$  dB.

Figure 13.

Bode diagrams of the sensitivity and the complementary sensitivity for the corrected MISO system with $T_1=T_2\ne 0\sec$ : (a) sensitivity with $\mathrm{P}{\mathrm{I}}^{\alpha }$ , (b) complementary sensitivity with $\mathrm{P}{\mathrm{I}}^{\alpha }$ , (c) sensitivity with PI, and (d) complementary sensitivity with PI.

Figure 14 shows the step responses of the corrected closed-loop MISO system by $\mathrm{P}{\mathrm{I}}^{\alpha }$ and PI with an approximate delay $T_1=T_2\ne 0$ . Figure 14(a) shows that despite the gain variation the responses are stable and precise. But in Figure 14(b) PI do not ensure the stability of different responses in the presence of gain variations.

Figure 14.

Step responses of corrected plant with gain variation (closed loop) with $T_1=T_2\ne 0\sec$ : (a) with ${\mathrm{PI}}^{\alpha }$ controllers and (b) with PI controllers.

Case 3 $(T_1\ne T_2\ne 0)$ : In this part, the time-delays are $T_1=1\sec$ and $T_2=0.1$  sec. The desired performances used in this case are given in Table 12.

Table 12.

Parameters of $\mathrm{P}{\mathrm{I}}^{\alpha }$ for the MISO system with $T_1\ne T_2\ne 0$ .

	Controller	$k_{p,m}$	$k_{i,m}$	${\alpha }_m$
${\mathrm{PI}}^{\alpha }$	$C_1$	6.0093	0.9808	1.0786
	$C_2$	2.5624	1.7993	1.3122
PI	$C_1$	2.5198	0.5237
	$C_2$	0.3517	6.0194

Figure 15 shows the Bode diagrams of corrected open-loop MISO system by fractional and rational controllers. In Figure 15(a) the open-loop phase is flat in the frequency band $[0.11]$  rad/sec. But in Figure 15(b) the open-loop phase is not flat in the desired frequency band. So, $\mathrm{P}{\mathrm{I}}^{\alpha }$ is more robust.

Figure 15.

Bode diagrams of the open-loop corrected MISO system with $T_1\ne T_2\ne 0\sec$ : (a) with ${\mathrm{PI}}^{\alpha }$ controllers and (b) with PI controllers.

Table 13 summarize the optimized parameters of ${\mathrm{PI}}^{\alpha }$ and PI controllers.

Figure 16(a) depicts the Bode diagrams of the sensitivity of MISO system corrected by ${\mathrm{PI}}^{\alpha })$ with an attenuation $\left(B\right)$ equal to $-10$  dB for the frequency ${\omega }_s=0.28$  rad/sec. In addition, the noise attenuation $\left(A\right)$ is equal to $-15$  dB for the frequency ${\omega }_t=3$  rad/sec as indicated in Figure 16(b). Figure 16(c) presents the Bode diagrams of the sensitivity of MISO system corrected by PI with an attenuation $\left(B\right)$ equal to $-10$  dB for the frequency ${\omega }_s=0.27$  rad/sec. Also, the noise attenuation $\left(A\right)$ is equal to $-15$  dB for the frequency ${\omega }_t=3.84$  rad/sec as indicated in Figure 16(d). These results indicate the efficiency of fractional controllers which achieved all the frequency specifications.

Figure 16.

Bode diagrams of the sensitivity and the complementary sensitivity for the corrected MISO system with $T_1\ne T_2\ne 0\sec$ : (a) sensitivity for the MISO system corrected by $\mathrm{P}{\mathrm{I}}^{\alpha }$ , (b) complementary sensitivity for the MISO system corrected by $\mathrm{P}{\mathrm{I}}^{\alpha }$ , (c) sensitivity for the MISO system corrected by PI, and (d) complementary sensitivity for the MISO system corrected by PI.

Figure 17 presents the step responses of the corrected MISO system by the fractional and rational controllers with gain variation. By Figure 17 the $\mathrm{P}{\mathrm{I}}^{\alpha }$ are more robust to this variation. From $T_1$ to $T_2$ the MISO signal take the dynamic of subsystem SISO with little time-delay. However, PI do not show the difference time-delays as shown in Figure 17(b).

Figure 17.

Step responses of corrected plant with gain variation (closed loop) with $T_1\ne T_2\ne 0\sec$ : (a) ${\mathrm{PI}}^{\alpha }$ and (b) PI.

Conclusion

In this paper, two fractional order controllers design approaches are developed for MISO systems. The first is based on time-domain specifications and uses a nonlinear optimization algorithm without constraints. The second approach is based on frequency-domain specifications and uses a nonlinear optimization algorithm with constraints.

Numerical simulations are presented to discuss results. Three cases based on the value of the time-delays are considered in both time-domain design and frequency-domain design. In frequency-domain, to see more the efficiency of fractional controllers a comparative study with rational controllers is presented. Besides the difficulty of MISO system control design, it has been shown that the proposed approaches lead to the desired performances and give more robustness.

The controllers designed in this work are robust in stability despite the complexity of MISO system. This can be an incentive for applying to real system in future work.