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Abstract

The fractional order PID (FOPID) controller has been found to have the potential to provide more flexibility in the design options than the integer order PID (IOPID) controller. However, in the FOPID controller design practice, investigations are still required in selecting the achievable design specifications and controller parameters. In order to obtain a practical tuning method for the FOPID controller, taking the first order plus integral systems as the plants, the complete achievable regions of the design specifications are collected and analyzed. In addition, the selectable integral and derivative orders of the FOPID controller are also collected and modeled. With the prior knowledge obtained, a synthesis method for the FOPID controller is proposed to make the control system achieve the desired flat-phase characteristic. To verify the effectiveness of the proposed method, the synthesis method is applied to design the FOPID controller for the permanent magnet synchronous motor (PMSM) speed servo system. Simulation and experimental results show that system with the FOPID controller can achieve satisfying tracking performance and disturbance rejection performance.

Keywords

Fractional order PID control achievable region flat-phase specification neural network

Introduction

Fractional calculus obtains the extended description and control scope by extending the integral and derivative orders to be real number.^1–3 As a result, fractional order controller provides the possibility to offer better control performance than the conventional integer order controller.^4,5 It has been found that the fractional order PID (FOPID) controller can provide more control options and better control performance than the PID controller because of the introduction of two tunable orders.^6–8 However, due to two extra degrees of freedom, more sophisticated strategies need to be considered in designing the FOPID controller.

In general, the tuning methods for the FOPID controller can be summarized into the optimization methods and the analytic methods. The optimization methods tune the controller parameters by optimizing the objective functions, which are often customized according to the requirements on system performance, such as the dynamic performance, stability, and robustness.⁹ In addition, the objective functions are often optimized under the constraints characterizing the specific requirements in actual systems. For example, a fuzzy FOPID controller is tuned by a dynamic particle swarm optimization method.¹⁰ An optimal FOPID controller is designed by minimizing the time-domain performance index under the frequency-domain constraints.¹¹ In the engineering applications of these optimization methods, sufficient time and high-performance micro-controllers may be needed for the optimization process.

The “flat-phase” method is an analytic design method for the FOPI/D controller. Under the flat-phase specification, the frequency characteristic of the control system is configured to a desired shape with nearly constant phase at the middle frequency range.¹² The FOPI/D controller parameters can be obtained by solving three equations derived from the design specifications. However, to tune the FOPID controller with five parameters, the flat-phase design method needs modifying. At present, some simplification strategies have been developed to establish the relations between the parameters of the FOPID methods and then reduce the degrees of freedom of the FOPID controller from five to three.^13,14 Taking advantage of these simplification strategies, the flat-phase design method can be applied to tune the FOPID controller.¹⁵ However, with the reduced degrees of freedom, the tuning flexibility and control performance of the FOPID controller may be limited. Another practical strategy is to select the integral and derivative orders of the controller before applying the flat-phase specification. However, there may be little prior knowledge about the selection of the fractional orders. Moreover, although the FOPID controller has been found to obtain more flexibility in the design options, rules of thumb are still required for the selection of the design specifications.

In this paper, taking the typical first order plus integral system as the plant, the selection of the design specifications and fractional orders in tuning the FOPID controller to achieve the flat-phase characteristic is studied. The complete achievable regions of the design specifications for the FOPID controller design are collected. In addition, the complete selectable regions of the controller’s orders are also collected and analyzed with a neural network based scheme. In this way, with the prior knowledge obtained, a synthesis method for the FOPID controller is proposed to make the system achieve the specified design specifications and flat-phase characteristic. Compared with the existing optimization methods,^9–11 the proposed method is simpler in implementation because the optimization process is avoided. Moreover, the FOPID controller tuned by the proposed method may achieve better dynamic performance over those tuned by the existing simplified analytic methods^13–15 because all the tunable components of the FOPID controller are kept. To verify the effectiveness of the proposed method, the synthesis method is applied to design the FOPID controller for the permanent magnet synchronous motor (PMSM) speed servo system. The robustness and control performance of the obtained FOPID controller are tested by simulations and experiments. In addition, comparative studies are performed between the FOPID controller tuned by the proposed method, the optimal FOPID tuned by the existing methods, and the commonly used CRONE controller.

The main contributions of this paper can be summarized as follows.

The complete achievable regions of the design specifications in designing the FOPID controller for the first order plus integral systems are collected and analyzed, providing the prior knowledge in selecting the design specifications.

The complete selectable regions of the integral and derivative orders of the FOPID controller are also collected and analyzed, presenting a practical guideline on tuning the FOPID controller.

A synthesis method is developed according to the priori knowledge to simplify the tuning of the FOPID controller, creating a wide potential application scope of the proposed method.

The rest of this paper is organized as follows: the problems of the flat-phase design method in tuning the FOPID controller are discussed in Section 2; the achievable regions of the design specifications and fractional orders for the FOPID controller design are collected and analyzed in Section 3, and then a FOPID controller synthesis method is proposed; application studies of the proposed method to the PMSM speed servo system are presented in Section 4; the conclusion is presented in Section 5.

Problem description

Consider a unit negative feedback system with a plant $G (s)$ and controller $C (s)$ , as shown in Figure 1, where $r$ and $y$ are the reference and output signals, respectively.

Figure 1.

The feedback control system.

The FOPID controller is introduced as the feedback controller, which can be represented as

C (s) = K_{p} (1 + \frac{K_{i}}{s^{λ}} + K_{d} s^{μ}),

(1)

where $K_{p}$ is the proportional gain, $K_{i}$ is the integral gain, $K_{d}$ is the derivative gain, $λ$ and $μ$ represent the integral and derivative orders, respectively, satisfying $0 < λ < 2$ and $0 < μ < 2$ .¹⁶

The flat-phase specification is represented as (2),

\frac{d [Arg [G (j ω) C (j ω)]]}{d ω} |_{ω = ω_{c}} = 0,

(2)

where $ω_{c}$ is the gain crossover frequency, $Arg [G (j ω)]$ and $Arg [C (j ω)]$ represent the phases of the plant model and the controller, respectively. The flat-phase specification ensures the phase of the control system to have a zero slope at $ω_{c}$ . In addition, the phase margin $φ_{m}$ is given as the design specification. Thus, two more equations are derived,

| C (j ω_{c}) G (j ω_{c}) | = 1,

(3)

Arg [C (j ω_{c})] + Arg [G (j ω_{c})] = φ_{m} - π .

(4)

As can be seen, three equations can be derived according to the flat-phase design method, but the FOPID controller has five degrees of freedom. To obtain the controller parameters, a simple way is to determine the fractional orders $λ$ and $μ$ in advance. In this way, the remaining parameters $K_{p}$ , $K_{i}$ , and $K_{d}$ can be calculated by solving (2), (3), and (4), according to the design specifications $ω_{c}$ and $φ_{m}$ .

However, two problems should be considered in tuning the FOPID controller. First, for some plants and design specifications, the feasible solution of the controller parameters cannot be obtained. Therefore, the design specifications $ω_{c}$ and $φ_{m}$ should be carefully selected to ensure the existence of the feasible controller parameters. Second, when the design specifications $ω_{c}$ and $φ_{m}$ are given, the fractional orders $λ$ and $μ$ should also be selected to obtain the feasible controller parameters.

This paper targets the problems in selecting the design specifications ( $ω_{c}$ and $φ_{m}$ ) and fractional orders ( $λ$ and $μ$ ) in designing the FOPID controller for the first order plus integral system, which can be represented as the following transfer function,

G (s) = \frac{K}{s (Ts + 1)},

(5)

where $T$ and $K$ are the time constant and gain, respectively.

Synthesis method for the FOPID controller

Design specifications selection

The selection of the design specifications is studied by collecting the achievable regions of $ω_{c}$ and $φ_{m}$ for different plants. For a plant, the achievable region of $ω_{c}$ and $φ_{m}$ is composed of all the ( $ω_{c}$ , $φ_{m}$ ) pairs, based on which the feasible FOPID controllers can be designed. The achievable region is constructed within the space specified by the ranges of $ω_{c}$ and $φ_{m}$ . In this paper, the range of $φ_{m}$ is selected as 30°–60°, in accordance with the general requirement in engineering applications.^17,18 In addition, the range of $ω_{c}$ is selected as 1–100 rad/s.

An enumeration based algorithm is designed to judge whether a design specification ( $ω_{c}$ , $φ_{m}$ ) is achievable, as shown in Algorithm 1. Firstly, the sequences of $λ$ and $μ$ are generated by uniformly selecting $N$ values within their ranges, obtaining $λ_{1}$ , $λ_{2}$ , …, $λ_{N}$ and $μ_{1}$ , $μ_{2}$ , …, $μ_{N}$ , respectively. Then the ( $λ$ , $μ$ ) combinations are generated by combining the elements in the sequences of $λ$ and $μ$ , respectively. In this way, $N \times N$ combinations of $λ$ and $μ$ are initialized. Secondly, the parameters of the FOPID controller corresponding to each ( $λ$ , $μ$ ) combination are calculated by solving (2), (3), and (4), according to the design specifications $ω_{c}$ and $φ_{m}$ . Thirdly, the FOPID controllers obtained are sorted randomly and then used for feasibility check. In order to ensure the control systems obtained are reasonable and close to the desired open-loop transfer function in real applications,¹⁹ the following two restrictions are adopted in the feasibility check: first, the controller parameters $K_{p}$ , $K_{i}$ , and $K_{d}$ should be positive real numbers; second, the open-loop magnitude characteristic curve of the control system should have negative slope. The feasibility check is implemented in the following way: if there is a controller satisfying the restrictions, the design specification ( $ω_{c}$ , $φ_{m}$ ) is marked as achievable; on the contrary, if all the controllers do not satisfy the restrictions, the design specification is marked as unachievable.

Algorithm 1 Judging method for each design specification

1: function JUDGE

(ω_{c}, φ_{m})

2: Initialize (

λ_{1}

μ_{1}

), (

λ_{2}

μ_{2}

), …, (

λ_{N}

μ_{N}

).
3:

result \leftarrow unachievable

4: for

i = 1 \to N

do
5: for

j = 1 \to N

do
6: Calculate

K_{p}

, $K_{i}$ , and

K_{d}

based on (

λ_{i}

μ_{j}

) and (

ω_{c}, φ_{m}

)
7: Calculate the magnitude characteristic of the control system.
8: if Feasibility condition is satisfied then
9:

result \leftarrow achievable

10: break
11: end if
12: end for
13: end for
14: return

result

15: end function

Setting $N$ to be 498, applying the judging method above on the design specifications in the space specified by the ranges of $ω_{c}$ and $φ_{m}$ , the achievable region of the design specifications for a plant is obtained. In this paper, the achievable design specifications are collected for the plants represented by (5), with the time constant $T$ ranging from 0.001 to 0.01. For example, Figure 2 shows the achievable regions of the design specifications for the plants with $T$ = 0.003 and $T$ = 0.004, where each blue spot represents an achievable design specification, while the blank part means the corresponding design specifications are unachievable. It can be observed that, the achievable region varies with the time constant of the plant. In addition, the achievable regions of the design specifications in designing the IOPID controller are also collected for the plants with $T$ = 0.003 and $T$ = 0.004, as shown in Figure 3. According to Figures 2 and 3, it can be clearly observed that the achievable regions for the FOPID controller design are much larger than those for the IOPID controller design. Therefore, the flexibility in designing the FOPID controller is demonstrated.

Figure 2.

The achievable regions of $ω_{c}$ and $φ_{m}$ for the FOPID design for the plants with different time constants: (a) $T$ = 0.003 and (b) $T$ = 0.004.

Figure 3.

The achievable regions of $ω_{c}$ and $φ_{m}$ for the IOPID design for the plants with different time constants: (a) T = 0.003 and (b) T = 0.004.

In actual applications, the achievable design specifications should be selected before the FOPID controller is designed. However, it may be impossible to collect all the achievable design specifications. Based on the collected samples, various kinds of tools can be adopted to estimate the feasibility of the candidate design specifications, for example, the look-up table, and the advanced classifiers based on learning machines.²⁰

Selection of the fractional orders

According to the flat-phase design method, when the design specification ( $ω_{c}$ , $φ_{m}$ ) is given, there may be various combinations of $λ$ and $μ$ available for the FOPID controller design. In this section, the distribution of the selectable fractional orders is studied in order to obtain some rules of thumb for selecting the fractional orders.

Similarly, an enumeration based algorithm is designed as follows to collect all the selectable combinations of ( $λ$ , $μ$ ) for each design specification.

Initialize the ( $λ$ , $μ$ ) combinations by selecting $N$ values uniformly from their ranges, respectively, and combining the values of $λ$ and $μ$ in sequence.

Solve the FOPID controller for each ( $λ$ , $μ$ ) combination according to $ω_{c}$ and $φ_{m}$ .

Check whether the FOPID controller obtained satisfies the given restrictions (the same as mentioned in Section 3.1). If so, the corresponding ( $λ$ , $μ$ ) combination will be included in the selectable region; otherwise, it will be discarded.

Applying the collecting method above on each achievable design specification, the selectable regions of the fractional orders for different design specifications and plants can be obtained. Figure 4(a) and (b) show two selectable regions of $λ$ and $μ$ for the FOPID controller design with different design specifications and plants, where each blue spot represents a selectable combination of ( $λ$ , $μ$ ), while the blank part means the corresponding combinations are unselectable.

Figure 4.

The selectable region of $λ$ and $μ$ for different design specifications and plants: (a) $ω_{c}$ = 40 rad/s, $φ_{m}$ = 45°, $T$ = 0.003 and(b) $ω_{c}$ = 35 rad/s, $φ_{m}$ = 55°, $T$ = 0.005.

From Figure 4(a) and (b), it can be observed that the shape of the selectable region varies with $ω_{c}$ , $φ_{m}$ , and $T$ . When designing the FOPID controller to fulfill the flat-phase specification, the fractional orders should be selected from the selectable region. However, it maybe impossible to collect all the selectable fractional orders for all cases. A practical way is to establish a model to estimate the selectable regions based on the design specifications and the time constants of the plants. But it may also be difficult to model all the ( $λ$ , $μ$ ) combinations for each design specification and plant. In this paper, a simplified scheme is proposed for the estimation of the fractional orders, that is, only the maximum and minimum values of $λ$ and $μ$ in the selectable region are considered and fitted by the estimation model. In this way, the obtained model will be used to estimate the lower and upper limits of the fractional orders.

Firstly, the maximum and minimum values of $λ$ in the selectable regions corresponding to different combinations of $ω_{c}$ , $φ_{m}$ , and $T$ are collected. Figure 5(a) and (b) show the distributions of the minimum and maximum values of $λ$ in the selectable regions for different design specifications ( $ω_{c}$ , $φ_{m}$ ) and fixed time constant ( $T$ = 0.0017). From Figure 5(a) and (b), it can be observed that the distribution of the maximum and minimum values of $λ$ may obey to a certain law. Therefore, a neural network based model is used to fit the distribution of $λ_{\min}$ and $λ_{\max}$ .

Figure 5.

The distributions of $λ_{\min}$ and $λ_{\max}$ for different design specifications with $T$ = 0.0017: (a) $λ_{\min}$ and (b) $λ_{\max}$ .

The input vector of the model is selected as [ $T$ , $ω_{c}$ , $φ_{m}$ $]^{T}$ , while the output vector is selected as [ $λ_{\min}$ , $λ_{\max}$ $]^{T}$ . The input vectors of the training samples are generated by combining the elements in the sequences of $T$ , $ω_{c}$ , and $φ_{m}$ , respectively. The sequence of $T$ is composed of 10 values selected uniformly in the logarithmic space specified by the range of $T$ , that is, $T$ = 0.001, 0.0013, 0.0017, …, 0.01. Similarly, the sequence of $ω_{c}$ is composed of 20 values selected uniformly in the logarithmic space specified by the range of $ω_{c}$ , that is, $ω_{c}$ = 1, 6.21, 11.42, …, 100 rad/s. In addition, the sequence of $φ_{m}$ is composed of 11 values selected uniformly in the linear space specified by the range of $φ_{m}$ , that is, $φ_{m}$ = 30, 33, …, 60°. In this way, except for the unfeasible samples, there are more than 1300 samples used for the neural network training. According to a common rule of thumb,²¹ the training samples are divided into the training set, the validation set and the testing set. The sample size ratio of three sample sets is 70%:15%:15%.

The feedforward multi-layer neural network is adopted to estimate the lower and upper limits of $λ$ . The sigmoid function and linear function are adopted as the activation function of the neurons in the hidden and output layers, respectively. In addition, the mean squared error (MSE) between the estimated and actual outputs is selected as the objective function. The weights and biases of the network are trained by the back propagation (BP) algorithm.^22,23 According to a common strategy,²⁴ the number of the layers and neurons of the neural network is determined by comparing the MSE between the networks with different numbers of layers and neurons. The network with one hidden layer is applied at first, with the number of neurons in the hidden layer varies from 3 to 22. The MSE of the testing set with respect to the number of neurons is plotted in Figure 6. As can be seen, the MSE decreases gradually as the number of neurons increases from 3 to 12, and then varies in a steady range around 0.000043 when the number of neurons is around 16–22. Therefore, the number of neurons in the hidden layer is selected to be 17. In addition, the network with two hidden layers is also applied to fit the training samples. The MSE of the network with two hidden layers is around 0.000042–0.00005, which does not show advantage over the network with one hidden layer.

Figure 6.

The MSE of the testing set with respect to the number of neurons.

The training of the neural network stops after 91 cycles of training. The MSE of the training, validation and testing sets during the training process is plotted in Figure 7, respectively. From Figure 7, it can be seen that the estimation error of the network has converged to minimum. In addition, the predicted and actual outputs of $λ$ in the training, validation and testing sets are plotted in Figure 8, respectively. From Figure 8, it can be seen that, the regression ( $R$ ) of the training, validation, and testing sets is close to 1.

Figure 7.

The variations of the MSE of the training, validation and testing sets.

Figure 8.

The predicted and actual outputs of $λ$ : (a) training set, (b) validation set, (c) testing set, and (d) All sets.

Secondly, the maximum and minimum values of $μ$ in the selectable region corresponding to different combinations of $ω_{c}$ , $φ_{m}$ , and $T$ are collected. The distributions of the $μ_{\min}$ and $μ_{\max}$ for different design specifications ( $ω_{c}$ , $φ_{m}$ ) and fixed time constant ( $T$ = 0.0017) are plotted in Figure 9(a) and (b), respectively. As can be seen, the values of $μ_{\min}$ and $μ_{\max}$ may not show obvious distribution law. Simplified schemes should be considered to handle the distributions of $μ_{\min}$ and $μ_{\max}$ .

Figure 9.

The distributions of $μ_{\min}$ and $μ_{\max}$ for different design specifications with $T$ = 0.0017: (a) $μ_{\min}$ and (b) $μ_{\max}$ .

All the samples of $μ_{\min}$ and $μ_{\max}$ for different design specifications and plants are divided into several groups according to their values, respectively. The number of samples in each value range is counted and shown in Table 1, where the number of $μ_{\min}$ samples is denoted as $n_{μ_{\min}}$ and that of the $μ_{\max}$ samples is denoted as $n_{μ_{\max}}$ . It can be seen that, most of the values of $μ_{\min}$ are distributed in the range around 0.2, while most of the values of $μ_{\max}$ are distributed in the range around 1.2. Based on this result, a simplification scheme is proposed: taking the mean values of $μ_{\min}$ and $μ_{\max}$ as the lower and upper limits of $μ$ for all design specifications and plants.

Table 1.

The value distributions of $μ_{\min}$ and $μ_{\max}$ for different design specifications and plants.

Range	$(0, 0.3]$	$(0.3, 0.6]$	$(0.6, 0.9]$	$(0.9, 1.2]$	$(1.2, 1.5]$	$(1.5, 1.8]$	$(1.8, 2)$
$n_{μ_{\min}}$	1001	231	66	24	0	0	0
$n_{μ_{\max}}$	12	29	61	415	783	22	0

Controller design

In actual applications, the FOPID controller can be tuned according to the following steps. First, the design specification ( $ω_{c}$ , $φ_{m}$ ) is selected according to the specified requirement and the obtained prior knowledge of the achievable regions. Second, based on the design specification, the lower and upper limits of $λ$ are estimated by the neural network. Third, $λ$ and $μ$ are selected from the simplified selectable region specified by the lower and upper limits of $λ$ and $μ$ . To avoid an unfeasible ( $λ$ , $μ$ ) combination, a reasonable way is to select the central point of the simplified selectable region, that is, $λ$ = ( $λ_{L}$ + $λ_{H}$ )/2, $μ$ = ( $μ_{L}$ + $μ_{H}$ )/2, where $λ_{L}$ , $λ_{H}$ are the lower and upper limits of $λ$ , $μ_{L}$ , and $μ_{H}$ are those of $μ$ . Finally, the remaining parameters of the FOPID controller are calculated according to (2), (3), and (4).

Application study

PMSM speed servo plant

The proposed synthesis method is applied to the FOPID controller design for a real PMSM servo system. Adopting the commonly used $d$ – $q$ coordinates and the field-oriented vector control scheme,²⁵ the characteristics of a PMSM can be represented as follows,

\frac{d i_{q}}{dt} = \frac{1}{L_{q}} (u_{q} - R i_{q} - n_{p} ω ψ_{f}),

(6)

\frac{d ω}{dt} = \frac{1}{J} (C_{m} i_{q} - T_{d}),

(7)

where $i_{q}$ and $u_{q}$ are the $q$ -axis current and voltage, respectively, $L_{q}$ is the $q$ -axis stator inductance, $R$ is the stator resistance, $ω$ is the motor angular velocity, $n_{p}$ is the pole pairs number, $ψ_{f}$ is the magnetic flux linkage, $C_{m}$ is the torque coefficient, $J$ is the rotor inertia, $T_{d}$ is the disturbance torque. Taking the deviation between the reference and actual $q$ -axis current as the input, a PI controller is applied as the $q$ -axis current controller,

C_{i} (s) = K_{s} (1 + \frac{R}{L_{q} s}) .

(8)

Then the $q$ -axis current loop can be represented as a first order model,

G_{i} (s) = \frac{1}{L_{q} s / K_{s} + 1} .

(9)

Combining (7) and (9), the PMSM speed servo system can be represented as Figure 10, where $ω_{r}$ is the reference velocity, $C (s)$ is the speed controller to be designed, $d$ is the external disturbance, $d = T_{d} / C_{m}$ . Thus, the PMSM speed servo plant model is identical to (5), with $T = L_{q} / K_{s}$ and $K = C_{m} / J$ .

Figure 10.

The dynamic diagram of PMSM speed servo systems.

It is well known that the PMSM servo system is a typical nonlinear system, where various kinds of uncertainties and disturbances exist, such as unmodeled dynamics, parameter uncertainties and disturbance torques.²⁶ The composite control scheme incorporating a feedback controller and feedforward compensation has been known as an effective way to deal with such systems. Currently, feedforward compensation strategies have been developed to eliminate the effect of the uncertainties and disturbances.^27,28 In this paper, we focus on the feedback controller which is often used to guarantee the stability and dynamic performance of the control system.

Simulation results

Robustness to gain variations

According to the specification of the PMSM, the PMSM speed servo plant is represented as follows,

G (s) = \frac{4.8}{s (0.0039 s + 1)} .

(10)

According to (10), an achievable design specification is given as $ω_{c}$ = 50rad/s, $φ_{m}$ = 45°. Thus, the lower and upper limits of $λ$ are estimated by the neural network based model as $λ_{L}$ = 0.4875 and $λ_{H}$ = 0.6627. In addition, the mean values of $μ_{\min}$ and $μ_{\max}$ samples are used as the lower and upper limits of $μ$ , that is, $μ_{L}$ = 0.2325, $μ_{H}$ = 1.2093. According to the center selection strategy proposed in Section 3.3, the fractional orders are selected as $λ$ = 0.5751 and $μ$ = 0.7209. Therefore, the FOPID controller is obtained by solving (2), (3), and (4), yields,

C_{1} (s) = 3.3 (1 + \frac{24.12}{s^{0.5751}} + 0.0132 s^{0.7209}) .

(11)

The open-loop Bode diagram of the system using $C_{1} (s)$ is plotted in Figure 11. It can be observed that, the gain crossover frequency and phase margin of the control system are 50 rad/s and 45°, respectively, in accordance with the design specifications. In addition, the phase characteristic has zero slope at $ω_{c}$ . Therefore, the change of the control system’s phase margin will be small under gain variations.

Figure 11.

The open-loop Bode plot of the system with $C_{1} (s)$ .

To test the system’s robustness to gain variations, step response simulations are performed on the control systems with different plant model gains. The speed responses of the nominal system and the systems with $\pm 20 %$ gain variations are plotted in Figure 12. It can be observed that, the overshoots of the systems with gain variations are similar to that of the nominal system.

Figure 12.

The gain robustness test (simulation).

Comparison with some existing methods

To verify the effectiveness of the proposed FOPID controller, performance comparisons are performed between $C_{1} (s)$ and those tuned by some existing methods. Firstly, taking the integral squared error (ISE) as the objective function, an optimal FOPID controller is designed using the differential evolution (DE) algorithm,²⁹ yields,

C_{2} (s) = 2.78 (1 + \frac{49.74}{s^{0.6996}} + 0.137 s^{0.512}) .

(12)

Secondly, an optimal FOPID controller is designed using the state transition algorithm (STA),³⁰ with the integrated time absolute error (ITAE) adopted as the objective function, yields,

C_{3} (s) = 7.61 (1 + \frac{25.84}{s^{0.8859}} + 0.0022 s^{0.5508}) .

(13)

Thirdly, a second generation CRONE controller is designed according to the same design specification as used to design the FOPID controller $C_{1} (s)$ , that is, $ω_{c}$ = 50 rad/s, $φ_{m}$ = 45°, with the integral part and low-pass part introduced to enhance the rejection to the steady state error and high frequency noise, respectively.³¹ Therefore, the feedback controller is designed as

C_{4} (s) = 45.38 (1 + \frac{5}{s}) \frac{1 + 0.0039 s}{s^{0.373}} \frac{1}{1 + s / 500} .

(14)

Step response simulations are carried out, using $C_{1} (s)$ (denoted as FOPID-Proposed), $C_{2} (s)$ (denoted as FOPID-DE), $C_{3} (s)$ (denoted as FOPID-STA) and $C_{4} (s)$ as the feedback controller of the PMSM servo system. The speed responses are plotted in Figure 13. In addition, the performance indices are presented in Table 2. It can be observed that, the step response of the system using the proposed FOPID controller is close to those of the systems using the optimal FOPID controllers.

Figure 13.

The step response tests (simulation).

Table 2.

The step response performance indices (simulation).

Index	FOPID-DE	FOPID-STA	CRONE	FOPID-Proposed
Settling time (s)	0.228	0.220	0.223	0.229
Overshoot (%)	14.3	14.1	16.9	14.3

Load disturbance response simulations are carried out. A load torque is imposed when the PMSM is running in a steady state. The speed responses of the control systems using the FOPID-DE, FOPID-STA, CRONE, and FOPID-Proposed are plotted in Figure 14. In addition, the performance indices of four systems are presented in Table 3, respectively. It can be observed that, the load disturbance responses of four control systems are close to each other.

Figure 14.

The load disturbance response tests (simulation).

Table 3.

The load disturbance rejection performance indices (simulation).

Index	FOPID-DE	FOPID-STA	CRONE	FOPID-Proposed
Recovery time (s)	0.058	0.054	0.060	0.066
Speed drop (%)	1.46	1.49	1.57	1.57

Experimental results

The PMSM speed control experimental test rig is shown in Figure 15. The experimental platform consists of a PMSM (model: P10B18200BXS), a DC generator, a resistance box, a servo driver and its PC interface. The motor control programs are implemented by the TMS320F28335 digital signal processor (DSP). Specifically, the fractional operator $s^{r}$ is realized by an impulse response invariant discretization approach.³² The PMSM is driven by an insulated gate bipolar transistor inverter in the servo driver. In addition, a DC generator is connected with the PMSM. The output terminal of the DC generator is connected with a resistance box and a switch. In the experiment, the load disturbance can be imposed on the PMSM by closing the switch.

Figure 15.

The PMSM experimental test rig: (a) configuration and (b) setup.

Robustness to gain variations

Gain robustness test experiments are performed on the system with $C_{1} (s)$ . The gain variations are introduced by multiplying the proportional gain of $C_{1} (s)$ by 1.2 and 0.8, respectively. The speed responses of the nominal system and the systems with gain variations are plotted in Figure 16, respectively. It can be observed that, the overshoots of the systems with gain variations are similar to that of the nominal system.

Figure 16.

The gain robustness test (experiment).

Comparison with some existing methods

Step response experiments are performed, using $C_{1} (s)$ , $C_{2} (s)$ , $C_{3} (s)$ , and $C_{4} (s)$ as the speed controller. The speed responses of four control systems are plotted in Figure 17. In addition, the performance indices are presented in Table 4. It can be observed that, the response of the system using the proposed FOPID controller is close to that of the system using the FOPID-STA.

Figure 17.

The step response tests (experiment).

Table 4.

The step response performance indices (experiment).

Index	FOPID-DE	FOPID-STA	CRONE	FOPID-Proposed
Settling time (s)	0.257	0.296	0.372	0.244
Overshoot (%)	13.44	8.37	15.43	9.36

The load disturbance responses of the systems using the FOPID-DE, FOPID-STA, CRONE, and FOPID-Proposed are plotted in Figure 18. In addition, the performance indices are presented in Table 5. It can be observed that, the load disturbance responses of four systems are close to each other.

Figure 18.

Load disturbance response tests (experiment).

Table 5.

The load disturbance rejection performance indices (experiment).

Index	FOPID-DE	FOPID-STA	CRONE	FOPID-Proposed
Recovery time (s)	0.054	0.052	0.066	0.053
Speed drop (%)	2.42	2.38	2.41	2.14

According to the simulation and experimental results, it can be concluded that the FOPID controller designed by the proposed synthesis method can guarantee the control system to achieve the desired robustness and dynamic performance, which are close to those of the systems using the optimized FOPID controllers.

Conclusion

Considering the feedback control problem of the first order plus integral plant, a synthesis method for the FOPID controller is developed by studying the achievable regions of the design specifications and controller parameters. Applying the synthesis method, the robust control system satisfying the specified design specification can be obtained without complicated optimization. The robustness and control performance of the FOPID controller designed by the proposed method are verified by both simulation and experiments. The advantages of the proposed method is demonstrated by the comparison with some existing methods. For other classes of plants, the prior knowledge may be obtained following the similar strategy and the success of the FOPID controller synthesis method may be duplicated. Some open issues will be studied in the future works, for example, the synthesis strategies to meet more general requirements and the engineering application of the synthesis method.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Natural Science Foundation of Guangdong, China (grant no. 2019A1515110180), the National Natural Science Foundation of China (grant no. 62173150 and 61803087), the Projects of Guangdong Provincial Department of Education (grant no. 2017KQNCX215 and 2019KZDZX1034).

ORCID iD

Weijia Zheng

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Robust fractional order PID controller synthesis for the first order plus integral system

Abstract

Keywords

Introduction

Problem description

Synthesis method for the FOPID controller

Design specifications selection

Selection of the fractional orders

Controller design

Application study

PMSM speed servo plant

Simulation results

Robustness to gain variations

Comparison with some existing methods

Experimental results

Robustness to gain variations

Comparison with some existing methods

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References