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Abstract

In this paper, a tuning procedure of the parameters of fractional order proportional integral controller is presented for time delay system with arbitrary order. On the basis of obtaining stable solution space by utilizing some stability-domain boundaries which include real-root boundary and σ-stability boundary, the optimal solution can be achieved by applying specific frequency-domain specifications which include gain crossing frequency, phase margin, and robustness constraint. Using the same synthesis by traversing phase angle and phase margin in a given interval, the complete solution space of gain crossing frequency can be obtained and visualized, which is equivalent to demonstrate the complete solution space of optimal parameters of fractional order proportional integral controllers. As a comparison, the tuning procedure of the parameters of integral order proportional integral derivative controller is also discussed. At last, the proposed algorithm is validated by numerical simulation and experimental illustrations. The results show the robustness and advantage of the proposed fractional order proportional integral controller over other controllers.

Keywords

Fractional order proportional integral controller stability surface stability curve robustness

Introduction

Due to the merits of fractional calculus in the area of modeling and controlling complex dynamical processes, it gradually replaces integer order calculus as one of the research focus in the control field.¹ However, compared to integer order controller, the introduction of fractional calculus operators increases the design difficulty of fractional order controller, which affects its promotion in the practical application. To overcome this problem, the tuning method of the parameters of fractional order controller has attracted lots of attention in recent years.

For simplifying the complexity of a control system, a simple methodology for the tuning of integral order proportional integral derivative (IOPID) controllers was first introduced by Barbosa et al.;² the problem of configuring proportional integral derivative (PID) parameters is transformed into determining the order and gain crossing frequency of ideal open-loop transfer function, which can be realized by some frequency-domain or time-domain specifications. Based on this, the application of this method is further extended to fractional order PID (FOPID) controller, in which the fractional orders can be taken as two degrees of freedom (2-DOF) to enlarge the limitation of such method.^3–5 Moreover, some advanced technologies are further considered, such as the idea of 2-DOF PID controller is introduced to ensure disturbance rejection,⁶ or Smith predictor is introduced to overcome the influence of longtime delay in the system.⁷ However, regardless of how advanced the chosen method is, system performance will only be determined by the parameters of the ideal transfer function, which loses flexibility of the method. From another point of view, to achieve the robustness of the system, some studies that focused on the flat phase method are first introduced⁸ to obtain optimal parameters of fractional order controller with several frequency-domain specifications. However, this method only considers the characteristics in the medium frequency band of Bode diagram, which neglects the requirement for the interference suppression capability in high-frequency band and low-frequency band. In order to overcome this problem, two more specifications are added based on sensitivity and complementary sensitivity analysis methods.⁹ Based on this method, a generalized iso-damping approach is proposed to define the invariance of the phase margin with respect to the free parameter variations,¹⁰ which supplies a more comprehensive analysis for the controller to suppress uncertainties. Although such methods provide an exact resolution process for finding the parameters of an FOPID controller, some of the predefined parameters in the algorithm are difficult to select, such as phase margin and gain crossing frequency. With this in mind, the selection of feasible or even optimal predefined parameters has received widespread attention recently. For example, differential evolution algorithm¹¹ or dynamic programming method^{12, 13} is adopted to get a desired solution in a specific area, and the original equality constraint is modified to an inequality constraint to increase the searching probability of the feasible solution. However, the determination of initial search point and search space is a difficult topic. Therefore, the main task at present is, “How to determine an appropriate search space of controller parameters?”

For solving the problem presented above, some researchers focused on applying Pontryagin and Hermite–Biehler theorems to derive theoretical bounds on controller parameters; the collection of the parameters restricted by these bounds can form an appropriate search space.¹⁴ Moreover, an extension of Hermite–Biehler theorem has been found with a combination of the generalized Kharitonov theorem to determine the entire set of stabilizing proportional integral (PI) parameters and obtain robustly stabilizing controller.¹⁵ However, the procedure of finding the optimal solution from a given search space is not discussed. Subsequently, some frequency-domain specifications and stability-domain boundaries in complex plane¹⁶ are both considered to determine the appropriate search space, in which the optimal solution can be found.¹⁷ Moreover, such methods are popularized and applied in the system with other controllers, such as FOPD controller¹⁸ or FO[PI] and FO[PD] controller,^{12, 13} which also validate the effectiveness of the method. However, there still remain two questions. First, these methods only consider the assignment of closed-loop poles on the imaginary axis, and then find the parameters of fractional order controllers to satisfy specific frequency-domain specifications with desired poles in the closed-loop system. Remarkably, on one hand, the solution space obtained in these methods is incomplete, because the desired closed-loop poles on left half plane (LHP) are not considered. On the other hand, the systems with desired frequency-domain characteristics may not contain the closed-loop poles on imaginary axis, which means the frequency-domain specifications and stability-domain boundaries cannot be satisfied simultaneously, namely, these methods have certain limitations. Second, these methods are only applied to a few plants, so it is necessary to extend them to more general areas and enhance their applicability.

In order to address and overcome the challenges presented above, this paper provides its main contributions as follows:

A method is proposed to determine the complete solution space of the parameters of FOPI controller for arbitrary integer order plus time delay system.

The proposed method, as an extension of the work¹⁷ by extending the stability-domain boundaries from imaginary axis to the whole LHP, ensures that the optimal solution of the parameters of FOPI controller can be found for arbitrary integer or fractional order plus time delay system to satisfy desired frequency-domain specifications and stability-domain boundaries simultaneously, while the previous one can only be suitable for parts of such system. For the same reason, the proposed method has a greater probability of obtaining the global optimal solution than the previous one.

Gain crossing frequency is an important parameter in the application of frequency-domain specifications, which can be determined by automatic calculation in the proposed method other than by area searching in the traditional method. It turns out that the proposed method is more convenient than the former.

Numerical simulation and experimental results are presented to validate the advantage of the proposed method.

The content of this paper is organized as follows. The structure of the control system is presented in the next section, then the stability-domain boundaries are applied to determine the stable solution space for the parameters of FOPI controller, based on which the frequency-domain specifications are applied to determine the optimal solution. After that, the complete solution space of gain crossing frequency is collected, which is equivalent to demonstrate the complete solution space of parameters of the FOPI controller. In the “Generalization of the method with an example” section, an example is demonstrated to generalize the proposed algorithm both for FOPI controller and IOPID controller. In the “Simulations” and “Experimental validation” sections, numerical simulations and experiments have been validated in a variety of systems; the results all show the advantages of the proposed method.

Procedure analysis

The structure of the system

Considering a closed-loop system shown in Figure 1, $P (s)$ and $C (s)$ represent the plant and described controller. The transfer function of $P (s)$ can be described as equation (1)

P (s) = \frac{b_{0} s^{n} + b_{1} s^{n - 1} + \dots + b_{n - 1} s + b_{n}}{s^{m} + a_{1} s^{m - 1} + \dots + a_{m - 1} s + a_{m}} e^{- Ls}

(1)

where $n, m$ is the highest order of numerator and denominator polynomials; $m > n$ , $b_{0}, b_{1}, \dots, b_{n}$ , and $a_{0}, a_{1}, \dots, a_{m}$ are corresponding coefficients of each; and $L$ is time delay.

Figure 1.

Structure of the system.

In this paper, the proposed controller $C (s)$ adopts fractional order proportional integral (FOPI) control strategy, and its form can be described as equation (2)

C (s) = K_{p} + K_{i} s^{- λ}

(2)

where $K_{p}$ and $K_{i}$ represent the proportional gain and the integral gain and $λ \in (0, 2)$ is a fractional order.

Combining equations (1) and (2), the transfer function of closed-loop system can be obtained

\begin{matrix} Φ (s) = \frac{P (s) C (s)}{1 + P (s) C (s)} = \\ \frac{(K_{p} + K_{i} s^{- λ}) (b_{0} s^{n} + b_{1} s^{n - 1} + \dots + b_{n - 1} s + b_{n})}{(K_{p} + K_{i} s^{- λ}) (b_{0} s^{n} + b_{1} s^{n - 1} + \dots + b_{n - 1} s + b_{n}) + e^{Ls} (s^{m} + a_{1} s^{m - 1} + \dots + a_{m - 1} s + a_{m})} \end{matrix}

(3)

According to equation (3), the characteristic equation $D (s)$ can be described as equation (4)

\begin{matrix} D (s) = (K_{p} + K_{i} s^{- λ}) (b_{0} s^{n} + b_{1} s^{n - 1} + \dots + b_{n}) + \\ e^{Ls} (s^{m} + a_{1} s^{m - 1} + \dots + a_{m}) = 0 \end{matrix}

(4)

Determine stable solution space by stability-domain boundaries

As we know, the design target of the controller is to achieve the best performance while ensuring the stability of the system. The composition of all feasible controller parameters which meets the requirements mentioned above can be called a stable solution space. For the previous researches, real-root boundary (RRB) and complex-root boundary (CRB) are applied as the stability-domain boundaries to acquire the stable solution space of the controller. However, the closed-loop poles can only be assigned on the origin or imaginary axis in this case. Therefore, the stable solution space should be obtained by assigning the poles on the whole LHP, which can be shown in Figure 2.

Figure 2.

Stable region determined by RRB and σ-stability boundary.

In Figure 2, the gray filled part represents the stable region; the dashed line represents the boundary of stable region; $σ$ and $ω$ represent the real part and the imaginary part of the point on the boundary, respectively; and phase angle $θ$ can be described as

θ = \arctan (ω / σ)

(5)

where $θ \in [- (π / 2), 0]$ .

According to Figure 2, the stability-domain boundaries are described as follows:

RRB: The closed-loop poles configured on positive real axis will undoubtedly result in the instability of the system; thus, we can denote the origin for RRB, which can be described as ${D (s) |}_{s = 0} = 0$ . Solving the equation, one can obtain $K_{i} = 0$ .

σ-stability boundary: The closed-loop poles assigned on LHP can be expressed by $s = σ + j ω$ . With a fixed $θ$ , all $σ$ and $ω$ that satisfy equation (5) will constitute σ-stability boundary. Considering equation (4), then the formula $D (σ + j ω) = 0$ can be deduced as equation (6)

K_{p} + K_{i} {(σ + j ω)}^{- λ} = - {\frac{s^{m} + a_{1} s^{m - 1} + \dots + a_{m - 1} s + a_{m}}{b_{0} s^{n} + b_{1} s^{n - 1} + \dots + b_{n - 1} s + b_{n}} e^{Ls} |}_{s = σ + j ω}

(6)

where

{(σ + j ω)}^{- λ} = {(\sqrt{σ^{2} + ω^{2}})}^{- λ} e^{- j λ \arctan \frac{ω}{σ}} = {(\sqrt{σ^{2} + ω^{2}})}^{- λ} [\cos (λ θ) - j \sin (λ θ)]

(7)

Taking the real part and the imaginary part of equation (6), equation (8) can be obtained

\begin{matrix} K_{p} + K_{i} {(σ^{2} + ω^{2})}^{- \frac{λ}{2}} \cos (λ θ) = - U \\ - K_{i} {(σ^{2} + ω^{2})}^{- \frac{λ}{2}} \sin (λ θ) = - V \end{matrix}

(8)

where

\begin{matrix} U = Re {[\frac{s^{m} + a_{1} s^{m - 1} + \dots + a_{m - 1} s + a_{m}}{b_{0} s^{n} + b_{1} s^{n - 1} + \dots + b_{n - 1} s + b_{n}} e^{Ls}] |}_{s = σ + j ω} \\ V = Im {[\frac{s^{m} + a_{1} s^{m - 1} + \dots + a_{m - 1} s + a_{m}}{b_{0} s^{n} + b_{1} s^{n - 1} + \dots + b_{n - 1} s + b_{n}} e^{Ls}] |}_{s = σ + j ω} \end{matrix}

(9)

From equations (8) and (9), equations (10) and (11) are obtained

K_{i} = \frac{V {(σ^{2} + ω^{2})}^{\frac{λ}{2}}}{\sin (λ θ)}

(10)

K_{p} = - U - V \cos (λ θ)

(11)

By setting $λ$ and $θ$ to a fixed value, the variations of $K_{p}$ and $K_{i}$ can be obtained with $σ$ and $ω$ changing along σ-stability boundary from 0. Moreover, by enabling $λ$ to traverse all values in $(0, 2)$ , a three-dimensional region formed by a cluster of curves can be determined, which can be called the feasible stability surface. Therefore, compared to the traditional methods, the desired closed-loop poles can be assigned freely on whole LHP with $θ$ changes in $[- (π / 2), 0]$ in this method, so that one can choose the specified closed-loop poles to achieve expected dynamic characteristics.

Determine optimal solution by frequency-domain specifications

On the basis of the former section, the scope of feasible solution space should be narrowed to find the optimal solution of FOPI controller, then the best performance of the system will be acquired. In order to accomplish this target, specific frequency-domain specifications will be described in this section. Considering the open-loop transfer function $G (s)$ of Figure 1, and substituting $j ω$ for $s$ , one can obtain

\begin{matrix} {G (s) |}_{s = j ω} \\ = C (j ω) P (j ω) \\ = \frac{e^{- j ω L} [K_{p} + K_{i} {(j ω)}^{- λ}] [b_{0} {(j ω)}^{n} + b_{1} {(j ω)}^{n - 1} + \dots + b_{n}]}{{(j ω)}^{m} + a_{1} {(j ω)}^{m - 1} + \dots + a_{m}} \\ = \frac{e^{- j ω L} (K_{p} + K_{i} ω^{- λ} \cos \frac{λ π}{2} - j K_{i} ω^{- λ} \sin \frac{λ π}{2}) (N_{r} + j N_{i})}{M_{r} + j M_{i}} \end{matrix}

(12)

where $N_{r}, M_{r}$ and $N_{i}, M_{i}$ represent the real part and the imaginary part of the numerator and the denominator polynomials of $P (s)$ , respectively, which can be expressed as follows

\begin{matrix} N_{r} = b_{0} ω^{n} \cos \frac{n π}{2} + b_{1} ω^{n - 1} \cos \frac{(n - 1) π}{2} + \dots + b_{n} \\ N_{i} = b_{0} ω^{n} \sin \frac{n π}{2} + b_{1} ω^{n - 1} \sin \frac{(n - 1) π}{2} + \dots + b_{n - 1} ω \\ M_{r} = ω^{m} \cos \frac{m π}{2} + a_{1} ω^{m - 1} \cos \frac{(m - 1) π}{2} + \dots + a_{m} \\ M_{i} = ω^{m} \sin \frac{m π}{2} + a_{1} ω^{m - 1} \sin \frac{(m - 1) π}{2} + \dots + a_{m - 1} ω \end{matrix}

Then, by following two specifications, gain crossing frequency and phase margin, equations (13) and (14) can be described

{L (ω) |}_{ω = ω_{c}} = \sqrt{\frac{N_{r}^{2} + N_{i}^{2}}{M_{r}^{2} + M_{i}^{2}} [{(K_{p} + K_{i} ω_{c}^{- λ} \cos \frac{λ π}{2})}^{2} + {(K_{i} ω_{c}^{- λ} \sin \frac{λ π}{2})}^{2}]} = 1

(13)

\begin{matrix} {φ (ω) |}_{ω = ω_{c}} = - \arctan \frac{K_{i} ω_{c}^{- λ} \sin \frac{λ π}{2}}{K_{p} + K_{i} ω_{c}^{- λ} \cos \frac{λ π}{2}} \\ - ω_{c} L + \arctan \frac{N_{i}}{N_{r}} - \arctan \frac{M_{i}}{M_{r}} \\ = - \arctan \frac{K_{i} ω_{c}^{- λ} \sin \frac{λ π}{2}}{K_{p} + K_{i} ω_{c}^{- λ} \cos \frac{λ π}{2}} - ω_{c} L \\ + α - β = - π + ϕ_{m} \end{matrix}

(14)

where $L (ω)$ and $φ (ω)$ are gain and phase of open-loop transfer function, respectively; $ω_{c}$ is gain crossing frequency; and $ϕ_{m}$ is phase margin.

According to equation (14), with fixed $ω_{c}$ , $ϕ_{m}$ , and $λ$ , $K_{p}$ can be uniquely determined by $K_{i}$ through equation (15)

K_{p} = Q K_{i}

(15)

where

Q = \frac{ω_{c}^{- λ} \sin \frac{λ π}{2} - \tan (α - β - ω_{c} L + π - ϕ_{m}) ω_{c}^{- λ} \cos \frac{λ π}{2}}{\tan (α - β - ω_{c} L + π - ϕ_{m})}

Substituting equation (15) into equation (13) to eliminate $K_{p}$ , $K_{i}$ can be solved as equation (16)

K_{i} = \sqrt{\frac{M_{r}^{2} + M_{i}^{2}}{(N_{r}^{2} + N_{i}^{2}) (ω_{c}^{- 2 λ} + 2 ω_{c}^{- λ} \cos \frac{λ π}{2} Q + Q^{2})}}

(16)

By setting $ϕ_{m}$ and $λ$ to a fixed value, the variations of $K_{p}$ and $K_{i}$ can be obtained with gain crossing frequency $ω_{c} \to + \infty$ from 0. Then, by enabling $λ$ to traverse all values in $(0, 2)$ , a three-dimensional region formed by a cluster of curves can be determined, which can be called the reference stability surface. Moreover, the expected performance of the system can be attained in frequency domain by setting $ϕ_{m}$ in a given interval.

Among the above discussion, to acquire the optimal solution of FOPI controller, $λ, ω$ in feasible stability surface and $λ, ω_{c}$ in reference stability surface should be uniquely identified by setting $θ, ϕ_{m}$ to the pre-specified value. Moreover, with a fixed $λ$ , $ω$ and $ω_{c}$ can be obtained by calculating the intersection of two curves belonging to their respective surfaces, and the coordinate of the intersection can be denoted for the desired value of $K_{p}$ , $K_{i}$ , and $λ$ . By following the same way and traversing $λ$ in $(0, 2)$ , the calculated intersections can be connected into a three-dimensional curve, which can be called the sub-optimal stability curve. All the intersections on this curve will satisfy the stability-domain boundaries, gain crossing frequency, and phase margin specifications simultaneously.

In addition, fractional order $λ$ is the last parameter which should be determined in the optimal solution of FOPI controller. Considering the robustness constraint as shown in equations (17) and (18), phase margin should keep constant while open-loop gain changes to maintain the iso-damping property invariant, and it should also be appeared in the peak of phase frequency characteristic curve, which could make the overshoot smaller than the case where phase margin appeared in the trough of phase frequency characteristic curve

{\frac{d φ (ω)}{d ω} |}_{ω = ω_{c}} = \frac{λ K_{p} K_{i} ω_{c}^{- λ - 1} \sin \frac{λ π}{2}}{K_{p}^{2} + K_{i}^{2} ω_{c}^{- 2 λ} + 2 K_{p} K_{i} ω_{c}^{- λ} \cos \frac{λ π}{2}} + \frac{{N_{i}}^{'} N_{r} - {N'}_{r} N_{i}}{N_{r}^{2} + N_{i}^{2}} - \frac{{M_{i}}^{'} M_{r} - {M'}_{r} M_{i}}{M_{r}^{2} + M_{i}^{2}} - L = 0

(17)

{\frac{d^{2} φ (ω)}{d ω^{2}} |}_{ω = ω_{c}} \leq 0

(18)

Substituting all the intersections on the sub-optimal stability curve into equations (17) and (18), the point $(K_{p}, K_{i}, λ)$ that meets the condition is the optimal solution of FOPI controller, which can be called a robustness point.

Complete solution space of gain crossing frequency $ω_{c}$

As can be seen from the above discussion, with a fixed phase margin $ϕ_{m}$ and phase angle $θ$ , the optimal parameter of FOPI controller can be obtained by stability-domain boundaries and frequency-domain specifications. In this section, by testing phase margin $ϕ_{m} \in [(50 π / 180), (70 π / 180)]$ and phase angle $θ \in [- (π / 2), 0]$ , the complete solution space of gain crossing frequency $ω_{c}$ will be expressed in quantity. Meanwhile, the complete solution space of corresponding parameters of FOPI controller can be easily found by the algorithm mentioned above.

To confirm the advantage of the proposed algorithm by comparison, an FOPI controller mentioned in Chen et al.¹⁷ is discussed; besides, the IOPID controller is also discussed by following the same synthesis in this paper. As the control plant subject to such controllers, $P_{1} (s)$ , $P_{2} (s)$ , and $P_{3} (s)$ are all taken into consideration

\begin{matrix} P_{1} (s) = \frac{1}{s + 1} e^{- 0.1 s}, & P_{2} (s) = \frac{1}{(s + 1) (s + 2) (s + 3)} e^{- 0.5 s}, & P_{3} (s) = \frac{1}{3 s^{1.05} + 1} e^{- s} \end{matrix}

Here, $P_{1} (s)$ , $P_{2} (s)$ , and $P_{3} (s)$ represent the first order, third order, and fractional order plant with time delay, respectively.

Both the parameters of FOPI and IOPID controller have the characteristics of three degrees of freedom (3-DOF), so the derivative gain $K_{d}$ in IOPID controller can play an equivalent role as the fractional order $λ$ in FOPI controller. As well as FOPI controller, with pre-specified $ϕ_{m}$ , theta and K_d, the feasible stability surface and reference stability surface can be successively constructed for IOPID controller as K_p and K_i changes. Then obtaining the sub-optimal stability curve from the intersection of two surfaces, the robustness point can be found, and the gain crossing frequency $ω_{c}$ of each point will compose a complete solution space which satisfies stability-domain boundaries and frequency-domain specifications simultaneously.

Analogous to $λ$ in FOPI controller, the change interval of $K_{d}$ in IOPID controller should also be defined. Considering the transfer function of IOPID controller as equation (19)

C_{iopid} (s) = K_{p} + \frac{K_{i}}{s} + K_{d} s

(19)

where $K_{p}, K_{i}, K_{d} > 0$ represent the proportional gain, integral gain, and derivative gain of the IOPID controller, respectively.

According to $P (s)$ and the IOPID controller, the variations of $K_{p}, K_{i}, K_{d}$ in IOPID controller can be easily determined by Routh–Hurwitz stability criterion. Considering the first-order Pade approximation of $e^{- Ls}$ in $P (s)$ , the closed-loop characteristic polynomial can be obtained. Without loss of generality, $P_{1} (s)$ is discussed in detail

P_{1} (s) = \frac{1}{s + 1} e^{- 0.1 s} \approx \frac{1}{s + 1} \frac{1 - τ s}{1 + τ s}

(20)

where $τ$ is the time constant. By setting $τ$ to 0.05, the characteristic polynomial can be written as equation (21)

D (s) = (1 - K_{d}) s^{3} + (21 - K_{p} + 20 K_{d}) s^{2} + (20 + 20 K_{p} - K_{i}) s + 20 K_{i} = 0

(21)

According to equation (21), Table 1 can be listed by the Routh–Hurwitz stability criterion.

Table 1.

Routh–Hurwitz criterion.

$s^{3}$	$1 - K_{d}$	$20 + 20 K_{p} - K_{i}$
$s^{2}$	$21 - K_{p} + 20 K_{d}$	$20 K_{i}$
$s^{1}$	$\frac{420 + 400 K_{p} - 41 K_{i} + 400 K_{d} - 20 K_{p}^{2} + K_{p} K_{i} + 400 K_{p} K_{d}}{21 - K_{p} + 20 K_{d}}$	0
$s^{0}$	$20 K_{i}$	0

Considering the necessary condition of the Routh–Hurwitz stability criterion—namely, the coefficient of equation (21) needs to be all positive—equation (22) can be obtained as follows

\begin{matrix} 1 - K_{d} > 0 \Rightarrow K_{d} < 1 \\ 21 - K_{p} + 20 K_{d} > 0 \Rightarrow K_{p} < 21 + 20 K_{d} \\ 20 + 20 K_{p} - K_{i} > 0 \Rightarrow K_{i} < 20 + 20 K_{p} \\ 20 K_{i} > 0 \Rightarrow K_{i} > 0 \end{matrix}

(22)

Then considering the sufficient condition of the Routh–Hurwitz stability criterion—namely, all elements in first line of Table 1 should be positive—one can conclude that $Z = 420 + 400 K_{p} - 41 K_{i} + 400 K_{d} - 20 K_{p}^{2} + K_{p} K_{i} + 400 K_{p} K_{d} > 0$ . Defining $Z = Y_{1} + Y_{2}$ , one can judge the sign of $Y_{1}$ and $Y_{2}$ in terms of equation (22)

\begin{matrix} Y_{1} = 420 + 400 K_{p} + 400 K_{d} - 20 K_{p}^{2} + 400 K_{p} K_{d} \\ = 20 (21 + 20 K_{d}) + 20 K_{p} (20 - K_{p} + 20 K_{d}) \\ > 20 K_{p} + 20 K_{p} (20 - K_{p} + 20 K_{d}) \\ > 20 K_{p} + 20 K_{p} \times (- 1) > 0 \end{matrix}

(23)

Y_{2} = K_{p} K_{i} - 41 K_{i} = K_{i} (K_{p} - 41) < 0

(24)

Obviously, to ensure that $Z$ is positive, $Y_{1}$ must be larger than the absolute value of $Y_{2}$ . By setting $K_{d}$ to 0, and sweeping $K_{i} \in [0, 440 + 400 K_{d}]$ , the variations of $Y_{1}$ and $Y_{2}$ can be shown in Figure 3(a) with $K_{p} \to 21 + 20 K_{d}$ from 0. As can be seen from Figure 3(a), only if $K_{i}$ is smaller than the slope of the tangent line of $- Y_{1}$ , meanwhile $K_{p}$ locates in $[0, 21 + 20 K_{d}]$ , the closed-loop system can remain stable, and its stable region is shown as the blue filled part in Figure 3(a). The tangent equation of $- Y_{1}$ can be described as equation (25); comparing to equation (24), the formula shown in equation (26) should be established

\begin{matrix} Y = {\frac{d (- Y_{1})}{d K_{p}} |}_{K_{p} = K_{p}^{*}} (K_{p} - K_{p}^{*}) + {(- Y_{1}) |}_{K_{p} = K_{p}^{*}} \\ = - 40 (10 - K_{p}^{*} + 10 K_{d}) K_{p} - 20 {(K_{p}^{*})}^{2} - 400 K_{d} - 420 \end{matrix}

(25)

\begin{matrix} - 41 \times (- 40) (10 - K_{p}^{*} + 10 K_{d}) = \\ - 20 {(K_{p}^{*})}^{2} - 400 K_{d} - 420 \\ \Rightarrow {(K_{p}^{*})}^{2} - 82 K_{p}^{*} + 840 K_{d} + 841 = 0 \end{matrix}

(26)

where $K_{p}^{*}$ is the abscissa of the tangent point of $Y_{2}$ on $- Y_{1}$ . Calculating equation (26), the maximum value of $K_{i}$ can be obtained as equation (27)

K_{i . max} = - 40 (10 - K_{p}^{*} + 10 K_{d}) = 40 (31 - 10 K_{d} \pm \sqrt{840 (1 - K_{d})})

(27)

Figure 3.

Stable region of K_p, K_i, K_d in IOPID controller: (a) stable region while K_d = 0 and (b) stable regions while $K_{d} \in [0, 1]$ .

Obviously, only one solution of equation (27) is acceptable for the reason that $K_{p}$ must be bigger than 0 and smaller than $21 + 20 K_{d}$ . By sweeping $K_{d}$ in $[0, 1]$ , a cluster of surfaces can be drawn in Figure 3(b), which indicate the stable regions of $K_{p}$ , $K_{i}$ , and $K_{d}$ of IOPID controller.

Generalization of the method with an example

In this section, the proposed method is generalized with an example; the corresponding flowchart is shown in Figure 4.

Step 1. Select first order plus time delay system $P_{1} (s)$ , for instance, and initialize $ϕ_{m}, θ$ to $50 π / 180$ and $- (π / 6.7)$ , respectively.

Step 2. According to equations (15) and (16) in the “Procedure analysis” section, with the range of gain crossing frequency $ω_{c} \to [0, + \infty)$ , draw a cluster of curves in $(K_{p}, K_{i})$ plane by specifying different fractional order $λ \in (0, 2)$ ; the reference stability surface shown in Figure 5(b) will consist of these curves. Moreover, draw a three-dimensional surface according to equations (10) and (11) of σ-stability boundary and draw the plane $K_{i} = 0$ according to RRB in the “Procedure analysis” section; the feasible stability surface can be obtained as shown in Figure 5(a).

Step 3. Take $λ = 0.5$ , for instance, and draw two curves in $(K_{p}, K_{i})$ plane which belong to feasible stability surface and reference stability surface, respectively; $K_{p}$ and $K_{i}$ can be determined from one of the intersection in Figure 6(a) with $ω = 3.926$ and $ω_{c} = 4.1301$ , while the other one with $ω = 0.1771$ and $ω_{c} = 0.1591$ is infeasible because $K_{p}$ is smaller than 0. Moreover, traversing $λ$ in $(0, 2)$ , a cluster of curves can be drawn in Figure 6(b). Connecting all the intersections in Figure 6(b), sub-optimal stability curve can be drawn as the black curve in Figure 7(a), which has appeared on reference stability surface.

Step 4. The point on the sub-optimal stability curve which satisfies the robustness constraints has been presented as the red dot in Figure 7(b). It is the robustness point. The corresponding parameters $(K_{p}, K_{i}, λ)$ of this point will satisfy stability-domain boundaries and frequency-domain specifications simultaneously.

Step 5. With the range of $ϕ_{m} \in [(50 π / 180), (70 π / 180)]$ and $θ \in [- (π / 2), 0]$ , a large number of robustness point can be found by repeating step 4; each point corresponds to a group of gain crossing frequency and the parameters of FOPI controller which satisfy the desired constraints. For comparison, the complete collection of robustness point of the proposed controller $C_{1} (s)$ ; the FOPI controller $C_{2} (s)$ , discussed in Chen et al.;¹⁷ and the IOPID controller $C_{3} (s)$ are all considered. By selecting 50 equidistant points in the interval of $ϕ_{m}$ and $θ$ , and 200 equidistant points in the interval of $λ$ and $K_{d}$ , the complete collection of robustness point of three algorithms is arranged on a two-dimensional plane as shown in the first line of Figure 8.

Step 6. Replace $P_{1} (s)$ with $P_{2} (s)$ and $P_{3} (s)$ , and repeat step 1 to step 5; the complete collection of robustness point of three algorithms is shown in the second line and the third line of Figure 8.

Figure 4.

Flowchart of the control algorithm.

Figure 5.

(a) Feasible stability surface and (b) reference stability surface.

Figure 6.

Intersection point of (K_p, K_i) with (a) λ = 0.5 and (b) $λ \in [0, 2]$ .

Figure 7.

(a) Sub-optimal stability curve and (b) robustness point.

Figure 8.

Complete solution space of $ω_{c}$ using C₁(s), C₂(s), and C₃(s) with $P_{1} (s)$ , $P_{2} (s)$ , and $P_{3} (s)$ .

As can be seen from Figure 8, the complete collection of robustness point in the first line shows the proposed algorithm greatly enlarges the range of the desired parameters by comparison. The second and the third lines show that the robustness point cannot be found in all control systems with different plants $P_{2} (s)$ and $P_{3} (s)$ , except for the proposed algorithm, which indicate the limitation of the FOPI controller discussed in Chen et al.¹⁷ and the IOPID controller. These all show the advantage of the proposed algorithm.

Simulations

In this section, the designed FOPI controller is validated with different control plants by numerical simulation illustrations.

$P_{1} (s)$

In this case, phase margin $ϕ_{m}$ is set to $62.3 π / 180$ , and phase angle $θ$ is set to enable the proposed FOPI controller and the IOPID controller to achieve optimal performance. The designed controllers are shown in equations (28) and (30), respectively. In addition, the FOPI controller discussed in Chen et al.¹⁷ can be designed as (29) with $ϕ_{m}$ = 62.3pi/180 and a fixed $θ = - (π / 2)$ , cause the desired closed-loop poles are placed on the imaginary axis in this case.

C_{fopi 1} (s) = 3.2876 + \frac{5.301}{s^{0.9825}}

(28)

C_{fopi 2} (s) = 3.3367 + \frac{6.1977}{s^{1.21}}

(29)

C_{iopid} (s) = 0.3582 + \frac{0.7783}{s} + 0.032 s

(30)

The open-loop Bode diagram of the proposed algorithm is presented in Figure 9(a); the step responses and disturbance (magnitude 1 step) rejection responses are presented in Figure 9(b), and the robustness performances are illustrated with $\pm 10 %$ loop gain variations. The contrast simulation of different algorithms is presented in Figure 10. For better illustrating the advantage of the proposed algorithm, the F-MIGO,¹⁹ AMIGO, Ziegler–Nichols methods, and an FOPID controller discussed in Mandić et al.²⁰ are taken into consideration. For a given first order plus time delay system, the designing procedure of F-MIGO and AMIGO methods is shown as equations (31) and (32), respectively

\begin{matrix} λ = {\begin{matrix} 1.1 & if τ \geq 0.6 \\ 1.0 & if 0.4 \leq τ < 0.6 \\ 0.9 & if 0.1 \leq τ < 0.4 \\ 0.7 & if τ < 0.1 \end{matrix} K_{p} = \frac{1}{K} (\frac{0.2978}{τ + 0.00307}) \\ K_{i} = \frac{K_{p} (τ^{2} - 3.4022 τ + 2.405)}{0.8578 T} \end{matrix}

(31)

\begin{matrix} K_{c} = \frac{0.15}{K} + [0.35 - \frac{LT}{{(L + T)}^{2}}] \frac{T}{KL} \\ T_{i} = 0.35 L + \frac{13 L T^{2}}{T^{2} + 12 LT + 7 L^{2}} \end{matrix}

(32)

where $K$ , $T$ , and $L$ represent the open-loop gain, time constant, and time delay of the system; $τ = L / (L + T)$ represents the relative dead time; and $K_{c}$ and $T_{i}$ represent the proportional gain and integral time constant of the designed integer order PI controller, respectively.

Figure 9.

Bode diagram and step response with loop gain variations: (a) open-loop Bode diagram and (b) step response.

Figure 10.

Contrast simulations of different controllers with $P_{1} (s)$ .

The controllers designed by F-MIGO, AMIGO, Ziegler–Nichols methods, and the FOPID controller discussed in Mandić et al.²⁰ are shown in equations (33) to (36)

C_{fopid} (s) = \frac{1.8871 + 2.46 s^{- 0.9} + 2.5184 s^{0.6}}{0.0942 s^{0.6} + 1}

(33)

C_{F - MIGO} (s) = 3.1688 + \frac{7.7723}{s^{0.7}}

(34)

C_{ZN} (s) = 7.4315 + \frac{23.1613}{s}

(35)

C_{AMIGO} (s) = 2.8236 + \frac{4.6464}{s}

(36)

$P_{2} (s)$

In this case, the IOPID controller cannot be obtained because no robustness point exists on the second line and third row of Figure 8. The FOPI controller proposed in this paper and Chen¹⁷ can be designed as equations (37) and (38) by selecting a robustness point on the first and second rows of second line in Figure 8

C_{fopi 1} (s) = 3.7943 + \frac{2.8832}{s^{0.993}}

(37)

C_{fopi 2} (s) = 4.2553 + \frac{2.2216}{s^{1.35}}

(38)

Considering that equations (31) and (32) are applicable for first order plus time delay system, $P_{2} (s)$ should be approximated as equation (39) to ensure that F-MIGO and AMIGO methods could be used. On this basis, the controllers designed by F-MIGO, AMIGO, Ziegler–Nichols methods, and the FOPID controller discussed in Mandić et al.²⁰ are shown in equations (40) to (43), and the step response with loop gain variations and contrast simulations of different controllers are shown in Figure 11

P_{2} (s) = \frac{1}{(s + 1) (s + 2) (s + 3)} e^{- 0.5 s} \approx \frac{0.1291}{s + 0.7745} e^{- 0.5 s}

(39)

C_{fopid} (s) = \frac{5.53 + 4.3679 s^{- 0.9} + 3.3097 s^{0.95}}{0.03213 s^{0.95} + 1}

(40)

C_{F - MIGO} (s) = 6.3304 + \frac{8.7633}{s^{0.9}}

(41)

C_{ZN} (s) = 11.6006 + \frac{5.7971}{s}

(42)

C_{AMIGO} (s) = 1.2793 + \frac{1.6147}{s}

(43)

$P_{3} (s)$

Figure 11.

Step response with loop gain variations and contrast simulations of different controllers with $P_{2} (s)$ .

In this case, $P_{3} (s)$ could be approximated as integer order system to better utilize the method proposed in the paper. Compared with the open-loop Bode diagrams and step responses of $P_{3} (s)$ and its approximate model shown in Figure 12, it can be concluded that two models are almost the same, which means the controllers designed by the approximate model can replace the one designed by $P_{3} (s)$

P_{3} (s) = \frac{1}{3 s^{1.05} + 1} e^{- s} \approx \frac{186.1 s^{2} + 3.333 \times 10^{4} s + 3352}{s^{4} + 608.1 s^{3} + 9.432 \times 10^{4} s^{2} + 4.126 \times 10^{4} s + 3352} e^{- s}

(44)

Figure 12.

Comparison of $P_{3} (s)$ and its approximate model.

As can be seen from Figure 8, the FOPI controller discussed in Chen et al.¹⁷ cannot be obtained because no robustness point exists on the third line and the second row. Then the FOPI controller proposed in the paper and IOPID controller can be designed as equations (45) and (46) by selecting a robustness point on the first and third rows of the third line of Figure 8

C_{fopi 1} (s) = 1.6482 + \frac{0.5825}{s^{1.0279}}

(45)

C_{iopid} (s) = 1.868 + \frac{0.688}{s} + 0.4271 s

(46)

Considering that equations (31) and (32) are applicable for first order plus time delay system, $P_{3} (s)$ should be approximated as equation (47) to ensure that F-MIGO and AMIGO methods could be used. On this basis, the controllers designed by F-MIGO, AMIGO, Ziegler–Nichols methods, and the FOPID controller discussed in Mandić et al.²⁰ are shown in equations (48) to (51), and the step response with loop gain variations and contrast simulations of different controllers are shown in Figure 13

P_{3} (s) \approx \frac{0.3878}{s + 0.3878} e^{- s}

(47)

C_{fopid} (s) = \frac{1.8088 + 0.5908 s^{- 0.95} + 0.4277 s^{1.05}}{0.105 s^{1.05} + 1}

(48)

C_{F - MIGO} (s) = 1.0541 + \frac{0.7303}{s^{0.9}}

(49)

C_{ZN} (s) = 2.1406 + \frac{0.7297}{s}

(50)

C_{AMIGO} (s) = 0.5333 + \frac{0.233}{s}

(51)

Figure 13.

Step response with loop gain variations and contrast simulations of different controllers with $P_{3} (s)$ .

For evaluating the advantage of the proposed algorithm compared with other methods, the integrated time absolute error (ITAE) index and the overshoot of step response $δ %$ are applied. The results of numerical comparison are shown in Table 2

ITAE = \int_{0}^{t_{s}} t | e (t) | dt = \int_{0}^{t_{s}} t | r (t) - y (t) | dt

(52)

where $t_{s}$ represents the termination of the simulation, and $r (t)$ , $e (t)$ , and $y (t)$ represent the given input, error, and output of the controlled system, respectively.

Table 2.

Comparisons of different methods.

Method	$P_{1} (s)$		$P_{2} (s)$		$P_{3} (s)$
	ITAE	$δ %$	ITAE	$δ %$	ITAE	$δ %$
C_fopi ₁(s)	2.314	8.69	14.55	7.6	38.26	10.5
C_fopi ₂(s)	3.482	14.65	39.85	31.51	–	–
C_iopid(s)	18.62	3.89	–	–	43.16	16.28
C_fopid(s)	8.006	–	18.71	–	59.65	7.764
C_F-MIGO(s)	5.124	21.24	242.5	84.36	64.39	23.0476
C_ZN(s)	0.5497	57.01	60.28	62.89	38.52	29.27
C_AMIGO(s)	2.502	8.77	24.9	1.11	126.5	0.31

As can be seen from Table 2, the best ITAE and overshoot for each systems are shown in bold, which represent for they can not reach optimal result at the same time. Therefore, a compromise must be sought. For the first order plus time delay system $P_{1} (s)$ , the best ITAE value and $δ %$ belong to the IOPI controller designed by Ziegler–Nichols method and IOPID controller, but the former one has a too large $δ %$ and the latter one has a too large ITAE value; therefore, the FOPI controller designed by the proposed algorithm is the best through comprehensive comparison. For the third order and fractional order plus time delay system $P_{2} (s)$ and $P_{3} (s)$ , the ITAE value simulated by the proposed algorithm is the best. By comparison, the algorithm presented in this paper has the best comprehensive performance.

Experimental validation

Hardware-in-the-loop simulation platform introduction

In this section, the proposed algorithm is validated in permanent magnet synchronous motor (PMSM) servo control system. Figure 14 shows the hardware-in-the-loop simulation platform. The MATLAB/Simulink software in personal computer is used to edit the proposed algorithm, and cSPACE module is used to download the edited algorithm to the control board. The main function of the control board is to drive the PMSM; its specific model is TMS320F28335. Parameters of PMSM are presented in Table 3. A coaxial DC motor is used as the load.

Figure 14.

Hardware-in-the-loop simulation platform.

Table 3.

Parameters of PMSM.

Parameter	Value	Units	Definition
R	0.33	Ω	Stator resistor
L	0.9	mH	Stator inductor
J	0.189	kg m² 10–4	Moment of inertia
C_E	5.4	mV/r/min	Back-EMF coefficient
C_M	0.087	N m/A	Torque coefficient
B_m	8 × 10–4	N m s	Friction factor
p	4	–	Pole pairs

PMSM: permanent magnet synchronous motor; back-EMF: back electromagnetic force.

Experimental results

According to Table 3, the object model of the system can be obtained as follows

P_{4} (s) = \frac{153.4392 s^{2} + 30.6878 s + 1.5344}{s^{4} + 700.1042 s^{3} + 109.0586 s^{2} + 4.0469 s + 0.0141}

(53)

In order to satisfy the experimental conditions, the fractional operator in fractional order controller is discretized by impulse-invariant discretization method.²¹ Based on equation (53), four fractional order controllers are designed by the algorithms proposed in this paper;^17,19,20 then setting sampling period to 0.000625 s, the discrete transfer function of those controllers can be described as follows

C_{f o p i 1} (s) = 0.0019 + \frac{0.0011}{s^{0.4649}} \underset{\to}{d i s c r e t e} \frac{0.001933 z^{5} - 0.006067 z^{4} + 0.007068 z^{3} - 0.003693 z^{2} + 0.0008086 z - 4.949 \times 10^{- 5}}{z^{5} - 3.148 z^{4} + 3.679 z^{3} - 1.93 z^{2} + 0.4246 z - 0.02618}

(54)

C_{f o p i 2} (s) = 0.0031 + \frac{0.0015}{s^{0.3768}} \underset{\to}{d i s c r e t e} \frac{0.003188 z^{5} - 0.009871 z^{4} + 0.01131 z^{3} - 0.005792 z^{2} + 0.001233 z - 7.212 \times 10^{- 5}}{z^{5} - 3.108 z^{4} + 3.579 z^{3} - 1.842 z^{2} + 0.3951 z - 0.02337}

(55)

\begin{matrix} C_{fopid} (s) = \frac{3.4447 + 0.1862 s^{- 0.88} + 0.2116 s^{0.9}}{0.37801 s^{0.9} + 1} \\ \underset{\to}{d i s c r e t e} \frac{0.5168 z^{9} - 2.848 z^{8} + 6.64 z^{7} - 8.47 z^{6} + 6.344 z^{5} - 2.739 z^{4} + 0.5767 z^{3} - 0.0041 z^{2} - 0.0204 z + 0.0026}{z^{9} - 5.53 z^{8} + 12.94 z^{7} - 16.564 z^{6} + 12.45 z^{5} - 5.398 z^{4} + 1.146 z^{3} - 0.0112 z^{2} - 0.0399 z + 0.0051} \end{matrix}

(56)

C_{F - M I G O} (s) = 0.0817 + \frac{0.0008}{s^{0.7}} \underset{\to}{d i s c r e t e} \frac{0.08174 z^{5} - 0.2659 z^{4} + 0.3233 z^{3} - 0.1782 z^{2} + 0.04191 z - 0.00288}{z^{5} - 3.253 z^{4} + 3.955 z^{3} - 2.18 z^{2} + 0.5128 z - 0.03524}

(57)

Besides, other three integer order controllers based on Ziegler–Nichols, AMIGO, and the proposed method are designed as follows

C_{ZN} (s) = 0.0737 + \frac{8.5445 \times 10^{- 4}}{s}

(58)

C_{AMIGO} (s) = 0.0953 + \frac{0.0012}{s}

(59)

C_{iopid} (s) = 0.1337 + \frac{0.0027}{s} + 0.5385 s

(60)

Using the proposed FOPI controller, the open-loop Bode diagram can be shown in Figure 15(a). The gain crossing frequency and phase margin are 0.00601 rad/s and 50°, respectively. The step responses are presented in Figure 15(b), and the robustness performances are illustrated with $\pm 20 %$ loop gain variations. For the superiority verification of the proposed FOPI controller, the experimental comparisons with other methods are shown in Figure 16 and Table 4. It is obvious that the proposed method can obtain the minimum ITAE value and overshoot, which clearly shows the advantage of the proposed method over other methods.

Figure 15.

Bode diagram and step response with loop gain variations: (a) open-loop Bode diagram and (b) step response.

Figure 16.

Contrast experiments of different controllers with $P_{4} (s)$ .

Table 4.

Comparisons of different methods.

	Proposed	Chen et al.¹⁷	F-MIGO	FOPID	IOPID	ZN	AMIGO
ITAE	23.13	41.95	40.87	37.64	40.54	40.63	86.07
δ %	4.4	36.8	26.8	28.8	49.8	30.8	24.8

FOPID: fractional order proportional integral derivative; IOPID: integral order proportional integral derivative.

Conclusion

According to the requirements for the stability and robustness of the arbitrary integer order plus time delay system, this paper provides a method to construct the complete solution space of the parameters of the FOPI controller with specific stability-domain boundaries and frequency-domain specifications. The detailed designing procedures of the proposed FOPI controller are generalized with an example. By comparing the proposed FOPI controller with other controllers, the proposed one has the largest complete solution space and can be applied in arbitrary integer order plus time delay system. In the end, the comparisons of simulation and experimental results between the proposed method and some other methods are presented, which all shows the performance and benefit of the proposed FOPI controller.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

SiYi Chen

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Design of fractional order proportional integral controller using stability and robustness criteria in time delay system

Abstract

Keywords

Introduction

Procedure analysis

The structure of the system

Determine stable solution space by stability-domain boundaries

Determine optimal solution by frequency-domain specifications

Complete solution space of gain crossing frequency ω c

Generalization of the method with an example

Simulations

Experimental validation

Hardware-in-the-loop simulation platform introduction

Experimental results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Complete solution space of gain crossing frequency $ω_{c}$