Classical controller design techniques for fractional order case

ISA Transactions 50 (2011) 461–472

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Classical controller design techniques for fractional order case Celaleddin Yeroglu a,∗ , Nusret Tan b a

Computer Engineering Department, Inonu University, Malatya, 44280, Turkey

b

Electrical and Electronics Engineering Department, Inonu University, Malatya, 44280, Turkey

article

info

Article history: Received 6 December 2010 Received in revised form 21 March 2011 Accepted 22 March 2011 Available online 16 April 2011 Keywords: Controller design Fractional order control system Bode envelope Lag Lag-lead PI PID

abstract This paper presents some classical controller design techniques for the fractional order case. New robust lag, lag-lead, PI controller design methods for control systems with a fractional order interval transfer function (FOITF) are proposed using classical design methods with the Bode envelopes of the FOITF. These controllers satisfy the robust performance specifications of the fractional order interval plant. In order to design a classical PID controller, an optimization technique based on fractional order reference model is used. PID controller parameters are obtained using the least squares optimization method. Different PID controller parameters that satisfy stability have been obtained for the same plant. © 2011 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction The proportional-integral (PI) and proportional integral derivative (PID) controllers are widely used in many industrial control systems for several decades due to their simplicity and reliability [1–3]. The PID control is one of the earlier control strategies with a simple control structure. It has a wide range of applications in industrial control. Several useful PID-type controller design techniques can be found in the literature such as, the well-known empirical Ziegler–Nichols tuning formula, the modified Ziegler–Nichols algorithm, the Åström–Hägglund method, the Chien–Hrones–Reswick formula, the Cohen–Coon formula, refined Ziegler–Nichols tuning, the Wang–Juang–Chan formula and the Zhuang–Atherton optimum PID controller [4]. Controller tuning is the process of obtaining the controller parameters to meet given performance specifications [1]. There are many other type of controllers used for different applications. Thus, many tuning techniques to obtain the parameters of the classical controllers were introduced during last few decades. For example, one of the controllers commonly used for phase compensation is Lag–Lead controllers. The purpose of Lag–Lead compensator design in the frequency domain generally is to satisfy specifications on steady-state accuracy and phase margin. The lead part of the compensator is used to adjust the system’s Bode phase curve

∗

Corresponding author. Tel.: +90 542 4852210; fax: +90 422 3410046. E-mail addresses: [email protected] (C. Yeroglu), [email protected] (N. Tan).

to establish the required phase margin at a specified frequency, without reducing the zero-frequency magnitude value. The lag part of the compensator is used to drop the magnitude curve down to 0 dB at that specified frequency [5,6]. Many applications of the PI, PID Lag, Lead and Lag–Lead controllers can be found for integer order control systems in the literature. Extension of the results obtained for these classical controllers to the fractional order control systems will have a positive contribution to this field. In recent years considerable attention has been given to the fractional order control systems (FOCS) due to the better understanding of fractional calculus. As a result, some important studies dealing with the applications of the fractional calculus to the control systems have been done [7–22]. Robustness analysis of the fractional order control system with a parametric uncertainty structure, is studied in [13–15]. In real time applications, the exact values of the parameters of a control system may not be known due to the tolerance values of the parameters. But these parameters can be estimated in certain intervals. Therefore, modeling this kind of system with a parametric uncertainty structure is a realistic approach. Thus the fractional order control system with a parametric uncertainty structure can be stated as a ‘‘fractional order interval control system (FOICS)’’. Similarly, the fractional order plant with parametric uncertainty structure can be stated as ‘‘fractional order interval plant’’. In this study some classical controller design techniques are extended to the control systems with a fractional order interval plant. Two methods have been used for classical controller design. The classical design techniques for PI, Lag, and

0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.03.004

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C. Yeroglu, N. Tan / ISA Transactions 50 (2011) 461–472

Lag–Lead controllers are adapted to the FOICS. In order to design a PID controller, an optimization technique based on a fractional order reference model is introduced. The computation of frequency responses of uncertain transfer functions plays an important role in the application of frequency domain methods for the analysis and design of robust control systems. There are some powerful graphical tools in classical control, such as the Nyquist plot, Bode plots and Nichols charts, which are widely used to evaluate the frequency domain behaviors of the systems. Motivated by the results especially the Kharitonov and the Edge theorems [23,24] obtained in the parametric robust control, there have been several studies on the computation of the frequency responses of control systems under parametric uncertainty [25–27]. The Bode and Nyquist envelopes of a transfer function are important in classical control theory for the analysis and design. For example, the frequency domain specifications such as gain and phase margins can be obtained using the Bode and Nyquist envelopes of a transfer function. The Bode plot of a control system provides a clear indication of how the Bode plot should be modified to meet given specifications. Therefore, controller design based on the Bode plot is simple and straightforward. However, in order to apply classical controller design method to FOICS, it is necessary to compute the Bode and Nyquist envelopes of a given fractional order interval transfer function (FOITF). Computation of the Bode and Nyquist envelopes of the FOICS, have been provided in [28]. In this paper, Bode envelopes of the FOITF are used to design PI, Lag, Lag–Lead controllers for FOICS. Since there have been considerable development in the use of fractional differentiation in various fields during last two decades, fractional differentiation become an important tool in scientific and industrial applications. Thus many authors have started to use fractional order models widely in various system identifications [29]. In recent studies, fractional order models have been used as a reference model for controller tuning [30]. Many optimal control problems can be converted into conventional optimization problems with the powerful tools provided in MATLAB. Since numerical methods are extremely powerful practical techniques for controller design, it is possible to design optimal controller using the optimization toolbox of MATLAB. Some optimal controller design methods are already provided in [4]. The optimization toolbox of MATLAB attempts to solve problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. In this paper, Bode’s ideal transfer function is used for PID controller design. Bode’s ideal transfer function has been chosen as a reference model for optimization. Parameters of the PID controller namely kp , ki and kd have been obtained such that the output of the system is the same as that of the reference model. In this method, desired output response of the system can be achieved by choosing the appropriate parameters for the reference system. The advantages of the proposed methods are clearly presented in the numerical examples. The stability problem of the fractional order systems with parametric uncertainty structure is an important subject. Although there is not a general solution for stability analysis of the fractional order control system yet, some new valuable studies can be found in the literature related to this issue. For example, necessary and sufficient stability condition of fractional order interval linear systems is studied in [31]. Some works related with the robust stability for FOICS can be found in [13,14,32]. In the present study, stability of the FOICS is investigated via step responses and Bode plots. The paper is organized as follows: Mathematical background of fractional representation is given in Section 2. In Section 3, a method is presented for robust parametric classical controller design for fractional order interval systems. An optimal PID controller design based on a fractional order reference model is studied in Section 4. Section 5 includes concluding remarks.

2. Mathematical background Fractional calculus can be considered to be a generalization of integration and differentiation of the integer order expressions to the non-integer order one. The most frequently used integrodifferential definitions are Grünwald–Letnikov, Riemann–Liouville (RL) and Caputo expressions. The Grünwald–Letnikov definition of the fractional order derivative is given by the following equation [33], r a Dt f

(t ) = lim h

−r

h→0

where (−1)j

t −a [− h ]

  r j

(−1)j

j =0

 r j

f (t − jh)

(1) (r )

are the binomial coefficients cj , (j = 0, 1, . . .).

The following expressions can be used to obtain the coefficients [34]. (r )

(r )

c0 = 1,

cj

  1+r (r ) = 1− cj−1 .

(2)

j

Riemann–Liouville definition can be given as, r a Dt f

(t ) =

dn

1

t

∫

0 (n − r ) dt n

a

f (τ ) dτ ( t − τ ) r −n +1

(3)

where n − 1 < r < n and 0 (.) is a Gamma function. A fractional order Caputo expression can be given as, r a Dt f

(t ) =

0 (r − n)

f (n) (τ )

t

∫

1

a

(t − τ )r −n+1

dτ

(4)

where n − 1 < r < n. The Gamma function 0 (m) can be defined for a positive real m as follows [33],

0 (m) =

∞

∫

e−u um−1 du.

(5)

0

Numerical solutions for Grünwald–Letnikov, Riemann–Liouville and Caputo expressions can be obtained using the definitions given in [10,33,34]. Generally, dynamic behaviors of the systems can be analyzed using a transfer function of the control system. Thus, the Laplace transformations of the integro-differential expressions for fractional order control systems are important. Fortunately, there is not big difference between the Laplace transformation of the fractional order expression and that of the integer order. The most general formula for the Laplace transformations of the integrodifferential expressions can be given as [35],

 L

dm f ( t )



dt m

= sm L {f (t )} −

n−1 −

sk

k=0

[

dm−1−k f (t ) dt m−1−k

] (6) t =0

where n is an integer number and m satisfies, n − 1 < m < n. The above expression is simplified as follows if all the derivatives of f (t ) are zero,

 L

dm f ( t )



dt m

= sm L {f (t )} .

(7)

Consider a single input single output control system. Let y(t ) be the output and x(t ) be the input of the system. The relation between input and output of the system can be defined as, an

dαn y(t ) dt αn

= bm

d

+ an − 1

βm

x(t )

dt βm

dαn−1 y(t ) dt αn−1

+ bm−1

d

+ · · · + a0

βm−1

x(t )

dt βm−1

dα0 y(t ) dt α0

+ · · · + b0

dβ0 x(t ) dt β0

.

(8)

C. Yeroglu, N. Tan / ISA Transactions 50 (2011) 461–472

Transfer function of the system can be obtained as follows by taking a Laplace transform of the above equation [4]. G(s) =

Y (s) X ( s)

=

bm sβm + bm−1 sβm−1 + · · · + b0 sβ0 an sαn + an−1 sαn−1 + · · · + a0 sα0

(9)

where αn > αn−1 > · · · > α0 ≥ 0, βm > βm−1 > · · · > β0 ≥ 0, ak , (k = 0, 1, 2, . . . , n) and bl , (l = 0, 1, 2, . . . , m) are constants [4, pp. 285–290]. 3. Robust parametric classical controller design for a fractional order plant The purpose of this section is to present extensions of the classical controller design methods to the fractional order plant with parametric uncertainty structure using Bode envelopes. Since the Bode envelopes of the fractional order interval transfer function are obtained using the frequency domain behavior of the system, it is not necessary to use any integer order approximation of the FOITF. Substituting s = jω in the transfer function of the control system, frequency domain analysis of the fractional order interval control system can be studied. Since (s)µ = (jω)µ , the expression for (jω)µ can be given as [14], µ

(jω) = ω

µ



cos

π 2

π  µ + j sin µ . 2

(10)

Bode envelopes of the fractional order interval plant are obtained using the magnitude and phase extremums of the numerator and denominator of the FOITF. The worst case data, which are obtained from the magnitude and phase extremums of the Bode envelopes, are used together with the classical design approach to obtain the parameters of the Lag, Lag–Lead and PI controllers. 3.1. Frequency responses of the fractional order uncertain plant The numerator and denominator polynomials of the fractional order interval plant are a fractional order interval polynomial (FOIP) of the form P (s, q) = q0 sα0 + q1 sα1 + q2 sα2 + · · · + qn sαn

(11)

where α0 < α1 < · · · < αn are generally real numbers, q = [q0 , q1 , q2 , . . . , qn ] is the uncertain parameter vector and the uncertainty box is Q = {q : qi ∈ [qi , qi ], i = 0, 1, 2, . . . , n}. Here qi and qi are specified lower and upper bounds of ith perturbation qi , respectively. A fractional order plant can be represented as, G(s, a, b) =

N (s, b) D(s, a)

=

b0 sα0 + b1 sα1 + · · · + bm sαm a0 s β 0 + a1 s β 1 + · · · + an s β n

(12)

where α0 < α1 < · · · < αm and β0 < β1 < · · · < βn are generally real numbers. The parameters a = [a0 , a1 , . . . , an ] and b = [b0 , b1 , . . . , bm ] might be uncertain parameters of the plant. Exact values of these parameters may not be known. However these parameters can be estimated in a certain intervals. Therefore, modeling of this kind of systems with a parametric uncertainty structure is a realistic approach. Parameters of the plant given in (12) with a parametric uncertainty structure can be defined as, ai ∈ [ai , ai ], i = 0, 1, 2, . . . , n and bi ∈ [bi , bi ], i = 0, 1, 2, . . . , m, where, ai and bi are lower limits, ai and bi are upper limits of the parameters respectively. Since numerator and denominator of the system in (12) are in the form of fractional order interval polynomial, then the uncertainty boxes of the system can be defined as A = {a : ai ∈ [ai , ai ], i = 0, 1, 2, . . . , n} and B = {b : bi ∈ [bi , bi ], i = 0, 1, 2, . . . , m}.

463

The value set of the family of polynomial of (11) can be constructed using the upper and lower values of uncertain parameters [28]. For the FOIP of (11), substituting s = jω gives P (jω, q) = q0 (k0r + jk0i )ωα0 + q1 (k1r + jk1i )ωα1

+ · · · + qn (knr + jkni )ωαn = (q0 k0r ωα0 + q1 k1r ωα1 + · · · + qn knr ωαn ) + j(q0 k0i ωα0 + q1 k1i ωα1 + · · · + qn kni ωαn )

(13)

where klr and kli , l = 1, 2, . . . , n are constant. From (13), it is clear that the uncertain parameters appearing both in the real and imaginary parts are linearly dependent to each other. The value set of such a polynomial in the complex plane is a polygon. Thus the corresponding polytope of a family of (11) in the coefficient space has 2(n+1) vertices and (n + 1)2n exposed edges since the polynomial family has (n + 1) uncertain parameters. Using the upper and lower values of the uncertain parameters all 2n+1 vertex polynomials of P (s, q) can be written in the following pattern,

v1 (s) = q0 sα0 + q1 sα1 + q2 sα2 + · · · + qn sαn v2 (s) = q0 sα0 + q1 sα1 + q2 sα2 + · · · + qn sαn .. . v2(n+1) (s) = q0 sα0 + q1 sα1 + +q2 sα2 + · · · + qn sαn .

(14)

From these vertex polynomials, the exposed edges can be obtained. For example, the vertex polynomial v1 (s) and v2 (s) have the same structure except the parameter (q0 ) is its lower value (q0 ) in v1 (s) and its upper value (q0 ) in v2 (s). One of the exposed edges can be expressed as, e(v1 , v2 ) = (1 − λ)v1 (s) + λ v2 (s)

(15)

where λ ∈ [0, 1]. Similarly, the remaining exposed edges can be constructed. Define the set which contains all the vertex polynomials as, PV = {v1 , v2 , . . . , v2(n+1) }

(16)

and the set of exposed edges as, PE = {e1 , e2 , . . . , e(n+1)2n }.

(17)

Consider the transfer function given in (12), and let n1 , n2 , . . . , n2m+1 and d1 , d2 , . . . , d2n+1 be the vertex polynomials of N (s, b) and D(s, a) polynomials, respectively. Define the sets NV and NE which contains the vertices and edges of N (s, b) as NV = {n1 , n2 , n3 , n4 , . . . , n2m+1 }

(18)

NE = {ne1 , ne2 , ne3 , ne4 , . . . , ne(m+1)2m }

(19)

similarly define DV and DE for the D(s, a) as DV = {d1 , d2 , d3 , d4 , . . . , d2n+1 }

(20)

DE = {de1 , de2 , de3 , de4 , . . . , de(n+1)2n }.

(21)

Since at each frequency, the value set of N (s, b) and D(s, a) are similar to the value set of P (s, q), the magnitude and phase extremums of G(s, a, b) can be found from the magnitude and phase extremums of polygons corresponding to N (s, b) and D(s, a). These polygons can be constructed from the sets given in (18)–(21). Bode envelopes of the plant can be obtained using magnitude and phase extremums of the N (s, b) and D(s, a). Detailed explanations and related theorems for obtaining Bode and Nyquist envelopes can be found in [28].

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C. Yeroglu, N. Tan / ISA Transactions 50 (2011) 461–472

3.2. Controller design In this section, the classical Lag, Lag–Lead and PI controller design techniques, which are obtained in [26] for integer order interval control systems, are extended for the FOICS. Robustness of these classical Lag, Lag–Lead and PI controller design techniques for integer order systems are shown in [26]. In case of fractional order interval control systems, the robust design specifications must be satisfied over the entire parameter set. Thus the worst case values must be acceptable. Since these worst case values occur on the boundary of the Bode envelopes, it suffices to verify that the specifications are met over this set. The controller can be designed to satisfy robust stability of the closed loop system under all parameter perturbations and worst case phase margin over the set of uncertain parameters. Bode envelopes of the open loop fractional order interval plant can be constructed using the procedure in [28]. The classical phase Lag, Lag–Lead or PI controllers can be designed using the magnitude and phase extremums of the Bode envelopes of the fractional order interval plant.

1 + a1 T 1 s 1 + T1 s

,

a1 < 1

c

(22)

(23)

The corner frequency 1/a1 T1 can be chosen to be one decade below ωc′ as 1 a1 T 1

ωc ′

=

10

(24)

where the value of a1 can be calculated from (23) and value of T1 obtained from (24). Then the lag compensator C1 (s) can be obtained as in (22). The desired phase margin of the fractional order interval plant can be achieved using C1 (s). 3.2.2. Lag–Lead compensator A lag compensator of the form of (22) can be used to achieve the desired phase margin of the fractional order interval plant. The desired gain margin can be achieved using the Lead compensator. Thus the Lag–Lead compensator can be used to obtain the desired robust phase and gain margins of the fractional order interval plant. In order to achieve the desired gain margin, the phase crossover frequency should be moved to the frequency ω′′ c , where the worst case gain margin is achieved. Thus the following equations should be satisfied

−10 log10 (a2 ) = −(gmd − gml ) dB √ 1 = a2 ω′′ c T2

(25) (26)

where gmd is the desired gain margin and gml is the value of the gain margin obtained by lag compensation. Therefore the cascaded lead compensator can be obtained as C 2 ( s) =

1 1 + a2 T2 s a2 1 + T 2 s

(1 + a1 T1 s)(1 + a2 T2 s) . a2 (1 + T1 s)(1 + T2 s)

(28)

3.2.3. PI controller In order to guarantee the desired phase margin for the entire family of the closed loop systems, the PI compensator given below can be used C3 (s) = KP +

KI s

.

(29)

From the Bode envelope, the gain crossover frequency can be moved to the frequency ωc′ where the desired phase margin is achieved. If the maximum magnitude at the frequency ωc′ is a dB, the value of kp can be obtained from the following equation max G(jωc′ ) = −20 log10 (KP ) = a dB.



ki kp

can be used to bring the magnitude envelope down to 0 dB at the new crossover frequency ωc′ where the worst case phase margin of the system occurs. The controller C1 (s) must provide the attenuation equal to the minimum value of the magnitude envelope at ωc′ . Thus, the following equation should be satisfied

  max G(jω′ ) = −20 log10 (a1 ) dB.

C (s) = C1 (s)C2 (s) =



(30)

In order that the phase lag of the PI compensator does not affect the phase of the compensated system at ωc′ , the corner frequency

3.2.1. Phase lag compensator The classical phase lag compensator of the form C 1 ( s) =

where the values a2 and T2 can be obtained from (25) and (26). Thus, the Lag–Lead compensator can be designed using (22) and (27) as

(27)

can be chosen to be one decade below ωc′ as

ki =

ωc′ 10

kp .

(31)

Then, the values of the kp and ki can be obtained from (30) and (31) respectively. Thus, the PI controller can be designed as in the form of (29). 3.2.4. Illustrative examples Example 1 (Lag Compensator). Consider the following fractional order interval plant G1 (s) =

n0 d3 s3.1 + d2 s2.2 + d1 s0.8 + d0

(32)

where n0 ∈ [10, 20], d3 ∈ [0.06, 0.09], d2 ∈ [0.2, 0.8], d1 ∈ [0.5, 1.5] and d0 = 0. An integrating system is preferred in this example in order to prevent a steady state error which will occur in the step response of the system. The aim of the example is to design a lag compensator for the closed loop system. It is expected that the compensated system will be robustly stable under all parameter perturbations and satisfy the phase margin of at least φpm = 45°. Bode envelopes of the uncompensated and compensated plant of (32) are given in Fig. 1. As seen from phase envelope of the uncompensated system in Fig. 1, the worst case phase margin with some safety factor of φpm = 50° is achieved at the frequency of ωc′ = 0.6548 rad/s. The maximum gain at this frequency is about 40 dB. In order to bring the magnitude envelope of uncompensated plant in Fig. 1, down to 0 dB at the new gain crossover frequency of ωc′ = 0.6548 rad/s, the following equation should be satisfied 40 dB = −20 log10 (a1 ) dB.

(33)

Then, the value of a1 = 0.01 is obtained from (33). The corner frequency can be chosen to be one decade below ωc′ as defined in (24). The value of T1 = 1527 is obtained from (24). Thus, the lag compensator can be obtained for the fractional order interval plant given in (32) as C1 (s) =

1 + 15.27s 1 + 1527s

.

(34)

The simulation of the fractional order control system can be done using the integer order approximations. A fractional transfer

C. Yeroglu, N. Tan / ISA Transactions 50 (2011) 461–472

465

Fig. 1. Bode envelopes of G1 (s) and C1 (s)G1 (s).

Fig. 3. Bode envelopes of the G2 (s), C1 (s)G2 (s) and C (s)G2 (s).

specifications. As seen from the step responses in Fig. 2, the lag controlled system is stable within the interval of the uncertain parameters of the plant. Example 2 (Lag–Lead Compensator). Consider the following plant for this example G2 (s) =

Fig. 2. Step responses of closed loop system with C1 (s)G1 (s) for 16 different parameters of the G1 (s).

function can be replaced with an integer order transfer function which has almost the same behaviors with the real transfer function. There are several methods for obtaining rational approximations of fractional order systems. For example Carlson’s method, Matsuda’s method, Oustaloup’s method, the Grünwald–Letnikoff approximation, Maclaurin series based approximations, time response based approximations etc. One of the most important approximations for fractional order systems is the Continuous Fractional Expansion–CFE method [36]. Thus, the step response simulation of the fractional order system can be obtained using the toolbox developed by Valério with CFE method [37]. Step responses of the lag controlled system C1 (s)G1 (s) for 16 different parameters of the G1 (s) have been obtained in Fig. 2, by taking two values from the parameters n0 , d3 , d2 and d1 . As seen from compensated gain envelope in Fig. 1, the worst case gain crosses 0 dB at the frequency ωc′ = 0.6548 rad/s and the worst case phase margin is about 45°. Thus one can say that the compensated system robustly meets the required performance

n0 d3 s3.2 + d2 s1.8 + d1 s0.8 + d0

(35)

where n0 ∈ [5, 7], d3 ∈ [0.06, 0.09], d2 ∈ [0.2, 0.8], d1 ∈ [0.5, 1.5] and d0 ∈ [0.1, 0.3]. The objective of this example is to achieve the worst case phase margin to be at least 60° and worst case gain margin to be at least 30 dB. As seen from Bode envelopes of the uncompensated system G2 (s) in Fig. 3, the phase margin of 70°, which is the desired phase margin of 60° plus some safety factor, is achieved at the frequency ωc′ = 0.6472 rad/s. The maximum gain of the system at this frequency is about 20.88 dB. In order to bring magnitude envelope of uncompensated system down to 0 dB at the new gain crossover frequency ωc′ = 0.6472 rad/s, the following equation should be satisfied 20.88 dB = −20 log10 (a1 ) dB.

(36)

Then, the value of a1 = 0.0904 is obtained from (36). The corner frequency can be chosen to be one decade below ωc′ as defined in (24). Thus, the value of, T1 = 171.68 is obtained from (24). Then, the lag compensator is obtained for the fractional order plant with parametric uncertainty structure given in (35) as C1 (s) =

1 + 15.45s 1 + 171.68s

.

(37)

As seen from Fig. 3, the lag compensated system satisfies the required worst case phase margin at the new gain crossover frequency ωc′ = 0.6472 rad/s. Now, the system achieved approximately 60° phase margin and 16.82 dB gain margin. In order to achieve the desired gain margin of 35 dB which is equal to 30 dB plus some safety factors, the worst case phase

466

C. Yeroglu, N. Tan / ISA Transactions 50 (2011) 461–472 Step response 1.8 1.6 1.4 1.2

y(t)

1 0.8 0.6 0.4 0.2 0 0

50

100

150

200

250

300

350

400

450

500

time

Fig. 4. Step responses of closed loop system with C (s)G2 (s) for 32 different parameters of G2 (s).

crossover frequency should be moved to the ω′′ c = 7.4296 rad/s where the desired gain margin is achieved. Thus, the following equation should be satisfied as given in (25).

− 10 log10 (a2 ) = −(35 − 16.82) dB = −18.18 dB.

Fig. 5. Bode envelopes of G3 (s) and C3 (s)G3 (s). Step response

(38)

1.4

The values of a2 = 65.7658 and T2 = 0.0166 can be obtained from (38) and (26) respectively. Then the lead compensator can be obtained using (27) as

1.2

1 65.7658

·

1 + 1.0917s 1 + 0.0166s

1

.

(39) 0.8

Therefore the Lag–Lead compensator for the fractional order interval system G2 (s) can be constructed using (37) and (39) in (28) as

(1 + 15.45s)(1 + 1.0917s) . C ( s) = 65.7658(1 + 171.68s)(1 + 0.0166s)

(40)

As seen from Fig. 3, the Lag–Lead compensated system satisfies the desired robust performances. Since the plant given in (35) is not an integrating system, there will be steady state error in the step response of the Lag–Lead controlled system. Thus, an integrator is added to the system for simulation. The step responses of C (s)G2 (s) for 32 different transfer functions of G2 (s), which are obtained by taking limit values of the each parameters, can be obtained as shown in Fig. 4. As seen from Fig. 4, the Lag–Lead controller constructed in (40) satisfies the robust performance of the system for the interval of the uncertain parameters. Example 3 (PI Compensator). Consider the following fractional order interval plant G3 (s) =

n1 s + n0 d4

s3.9

+ d3

s3.2

+ d2 s2.3 + d1 s0.8 + d0

(41)

where n1 ∈ [0.35, 0.45], n0 = 1, d4 = 0.02, d3 ∈ [0.25, 0.35], d2 ∈ [0.9, 1.1], d1 = 0.04 and d0 = 0. In this example, the desired phase margin of the entire family of closed loop system is at least 45°. As seen from the Bode envelope of G3 (s) in Fig. 5, a phase margin of 50°, which is 45° plus some safety factor, is achieved at the frequency of ωc′ = 0.1 rad/s. The maximum gain at this frequency can be calculated from Fig. 5 as 47.92 dB. Thus, the following equation should be satisfied as given in (30)

− 20 log10 (KP ) = 47.92 dB.

(42)

y(t)

C 2 ( s) =

0.6

0.4

0.2

0

0

100

200

300

400

500

600

700

800

900

1000

time

Fig. 6. Step responses of closed loop system with C3 (s)G3 (s) for 8 different parameters of G3 (s).

The value of proportional constant kp = 0.004 can be calculated from (42). The value of integral term ki = 0.00004 can be calculated using (31). Thus the PI compensator for the fractional order plant G3 (s) can be constructed as in (43) using the values of kp and ki in (29). As seen from Fig. 5, the PI-compensated system satisfies the desired phase margin robustly. C3 (s) = 0.004 +

0.00004 s

.

(43)

Step responses of C3 (s)G3 (s) for 8 different transfer functions of G3 (s), which are obtained by taking limit values of the each parameters, can be obtained as shown in Fig. 6. As seen from Fig. 6, the PI controller in (43) satisfies the robust performance of the system for the interval of the uncertain parameters. As seen from the examples, the controllers, which are designed by the proposed method in this section, satisfy robust stability and required performance specifications of the fractional order plant with a parametric uncertainty structure. The results of this section

C. Yeroglu, N. Tan / ISA Transactions 50 (2011) 461–472

467

will be important for the robust stability analysis and design of the fractional order control systems with parametric uncertainty structure. 4. Optimal PID controller design based on fractional order reference model

Fig. 7. Fractional order control loop with Bode’s ideal transfer function.

This section presents the tuning of a PID controller based on fractional order reference model using optimization techniques. PID controller parameters are obtained using least squares optimization method according to the fractional order reference model. It has been shown that the step responses of the plant, which is controlled with the proposed method, are almost the same as the step responses of the reference model. Consequently, the overshoots and other time characteristics of the step responses of the plant can be adjusted using the reference model with the suitable parameters. In recent studies, fractional order models have been used as a reference model for controller tuning [30]. Many optimal control problems can be converted into conventional optimization problems with the powerful tools provided in MATLAB. Since numerical methods are extremely powerful practical techniques for controller design, it is possible to design an optimal controller using the optimization toolbox of MATLAB. The optimization toolbox of MATLAB attempts to solve problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. Optimization algorithms solve constrained and unconstrained continuous and discrete problems. The toolbox includes functions for linear programming, quadratic programming, binary integer programming, nonlinear optimization, nonlinear least squares, systems of nonlinear equations, and multi-objective optimization. One can use them to find optimal solutions. Unconstrained optimization solves the problems of the form min F (x).

(44)

x

Optimization finds the vector x such that the objective function F (x) is minimized. In this section, tuning of a PID controller has been done using a least squares optimization method based on a fractional order reference model. Bode’s ideal transfer function has been chosen as a reference model for optimization. Parameters of the PID controller have been obtained such that the output of the system is the same as the reference model. The desired output response of the system can be achieved by choosing the appropriate parameters for the reference system. 4.1. Bode’s ideal control loop Bode [38] introduced new analysis tools for the specification and design of feedback systems in his well-known book [38]. Bode’s ideal control loop is summarized from the paper in the Ref. [30] as follows. Bode has suggested an ideal shape of the openloop transfer function of the form L(s) =

Fig. 8. Controller tuning setup.

 ω γ c

s

,

γ ∈R

Fig. 9. Step responses of the reference model for γ = 1.05.

The closed loop transfer function of the system of Fig. 7 can be given as T (s) =

L(s) 1 + L(s)

= 

1 s

ωc

γ

,

1 < γ < 2.

(46)

+1

So, the desired time response specifications of the reference system such as maximum overshoot, peak time, rise time, settling time, can be adjusted using the appropriate values of the parameters γ and ωc of the closed loop transfer function of the reference system. Thus, the Bode’s ideal control loop can be used as a reference model in the controller tuning algorithm. 4.2. Controller tuning algorithm

(45)

where ωc is the gain crossover frequency, that is |L(ωc )| = 1. The γ is the slope of the magnitude curve, on a log–log scale, and may assume integer as well as noninteger values. The transfer function L(s) can be considered as a fractional order differentiator for γ < 0 and fractional order integrator for γ > 0. As explained in [30], the L(s) gives a closed loop system with a desirable property of being insensitive to gain changes. Thus, the closed loop system with the transfer function L(s) given in the figure below can be used as a reference model.

Specifications of the Bode’s ideal transfer function are given in the previous section depending on the parameters γ and ωc . In this section a closed loop system with Bode’s ideal transfer function in Fig. 7, is taken as a reference system for tuning of the controller. The objective of the setup given in Fig. 8 is to minimize the output of the system Y (s). Parameters of the controllers are computed such as the output of the plant YP (s) will be equal to the output of the reference system YR (s). Least squares optimization algorithm ‘‘lsqnonlin’’ of MATLAB has been used to obtain the optimum values of the parameters of

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time, peak time. Figs. 9–12, shows the step responses of the Bode’s ideal control loop given in Fig. 7 for different values of the γ and ωc . As seen from Figs. 9–12, overshoots of the step responses are almost the same for different values of ωc at a fixed value of the γ . But time characteristics are different for different values of ωc . As seen from Fig. 13, time characteristics of the step responses are almost the same for different values of γ at a fixed value of the ωc . But the overshoots are different for different values of γ . Looking at Figs. 9–13, one can conclude that the desired step response can be achieved for Bode’s ideal control loop using appropriate values of the γ and ωc . Time characteristics of the reference model have been studied and some equations are given for computation of the step response parameters such as maximum overshoot, peak time, rise time, and settling time in [30]. One can conclude form the results given in [30] and from the Figs. 9–13 that the value of γ affects the overshoot of the step response while the value of ωc has an influence on the other time characteristics. One can also conclude that the value of ωc can be around the gain crossover frequency of the uncontrolled system. As stated in [30], the value of γ affects the slope of the magnitude curve of the reference system and reference system gives a satisfactory result for 1 < γ < 2. But the

Fig. 10. Step responses of the reference model for γ = 1.1.

Fig. 11. Step responses of the reference model for γ = 1.2.

Fig. 12. Step responses of the reference model for γ = 1.3.

the controller C (s). This optimization algorithm attempts to solve the problems of the form min

−  (Y (s))2 .

(47)

Since Y (s) = YR (s)− YP (s), the following equation can be solved using the same optimization algorithm, min

−  (YR (s) − YP (s))2 .

(48)

The optimization algorithm starts with the initial values of the controller parameters and finds the optimum values to satisfy the Eq. (48). The desired output response of the system can be achieved by choosing the appropriate values of the parameters γ and ωc of the reference system. Then the appropriate values of the PID controller parameters, kp , ki and kd , can be computed using the controller’s tuning setup in Fig. 8. Maximum overshoot and time characteristics of the step response can be adjusted by taking the appropriate value of γ and ωc in Bode’s ideal control loop. The value of the γ considerably effects the overshoots of the step response. Similarly, the value of the ωc considerably effects the time characteristics of the step response, such as rise time, settling

Fig. 13. Step responses of the reference model for ωc = 0.5.

C. Yeroglu, N. Tan / ISA Transactions 50 (2011) 461–472 Table 1 Values of the PID Parameters for different values of γ and ωc .

469 Step Response

1.4

ωc

kp

ki

kd

2 1 0.5 1.5 1 0.5 2 1 0.5

14.3577 6.7755 3.5479 6.7908 9.5998 3.3855 12.6904 6.1684 2.8947

2.4945 1.0600 0.5426 1.5649 1.5240 0.6373 4.6577 1.8054 0.8241

7.8944 6.6891 7.0386 4.4440 9.6580 6.9631 6.8623 6.2886 6.5303

1.2

1

0.8 y(t)

γ 1.05 1.05 1.05 1.1 1.1 1.1 1.2 1.2 1.2

0.6

best parameter for γ and ωc can be obtained by a trial-and-error method considering the constraints given above. Controller Tuning Procedure can be given as follows,

0.4

0.2

• Select appropriate values of the parameters γ and ωc of

Applications of the classical PID tuning based on the fractional order reference model are illustrated with three examples as given below.

0

0

5

10

15

20

25

30

35

40

time

Fig. 14. Step responses of the C41 (s)G4 (s), C42 (s)G4 (s), CA (s)G4 (s) and CISTE (s)G4 (s).

Bode Diagram 60 40 Magnitude (dB)

the reference model to obtain the desired step response specification for the reference system. • Specify initial values for the PID controller parameters kp , ki and kd . • In order to achieve desired step response of the given plant, use the optimization algorithm ‘‘lsqnonlin’’ of MATLAB with the controller tuning setup given in Fig. 8. • Obtain new values for kp , ki and kd to satisfy the desired step response of the given plant.

20 0 -20 -40 450

Example 4. Consider the following plant, G4 (s) =

s2 + 0.02s + 1

e

−0.1s

.

(49)

The purpose of the study is to find a PID controller for the plant such that the step response characteristics will be the same as the desired step response characteristics of the reference system. The PID controller can be represented as, C4 (s) = kp +

ki s

+ kd s.

2.3595 s

+ 5.8399s.

(51)

Another controller have been obtained for the values γ = 1.1 and ωc = 1.5 rad/s as, C42 (s) = 6.7908 +

270 180 90 0 -90 10-2

10-1

1.5649 s

+ 4.444s.

(52)

100

101

102

Frequency (rad/sec)

Fig. 15. Bode plots of C41 (s)G4 (s).

(50)

As explained in Section 4.2, the appropriate controller parameters can be obtained using a suitable reference model in the controller tuning setup given in Fig. 8. Table 1 shows the values of the PID controller parameters obtained using different reference models for different values of the γ and ωc . As seen from Table 1, different values for the parameters of the PID controller kp , ki and kd can be obtained using the least squares optimization algorithm for different values of the parameters γ and ωc of the reference system. Two different controllers, which are obtained for different values of γ and ωc , have been used for step response simulation for the plant in (49). One of the controllers has been obtained for the values γ = 1.3 and ωc = 1 rad/s as, C41 (s) = 5.4 +

Phase (deg)

360

1

The PID controllers obtained by the classical Åström–Hägglund method [39] and the ISTE optimization method which is studied in [40] for the same plant can be given respectively as follows, CA (s) = 0.142 +

0.0323

CISTE (s) = 0.144 +

s 0.387 s

+ 0.1562s

(53)

+ 0.808s.

(54)

Step responses of the PID controller C41 (s) and C42 (s) obtained by the proposed method are compared with PID controllers CA (s) and CISTE (s) as shown in Fig. 14. As seen from Fig. 14, the proposed method gives a much better result than the other methods for suitable values of the reference system. Bode plots of C41 (s)G4 (s), are given in Fig. 15. As seen from the Bode plots of the system, the controller satisfies the stability of the plant. Step responses of the system C4 (s)G4 (s) for different controllers in Table 1 are given in Figs. 16 and 17. Fig. 16 gives step responses of the system such that the controllers are obtained for different values of ωc at the fixed value of γ = 1.2. Fig. 17 gives the step responses of C4 (s)G4 (s) where controller parameters are obtained

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for different values of γ at ωc = 1 rad/s. As seen from Figs. 16 and 17, controllers given in Table 1 satisfy the stability of the control system with the desired step response characteristics. Example 5. Now, consider the following higher order control system, G5 (s) =

1

(s + 1)3

.

(55)

A PID controller is obtained using the least squares optimization algorithm as follows, C5 (s) = 11.7101 +

0.7095 s

+ 21.3256 s.

(56)

Parameters of the reference systems are γ = 1.05 and = 0.5 rad/s for this case. Step response of the control system C5 (s)G5 (s), where the controller parameters are obtained by proposed method, is given in Fig. 18. Bode plots of, C5 (s)G5 (s) are given in Fig. 19.

ωc

Fig. 18. Step responses of the C5 (s)G5 (s). Bode Diagram

Magnitude (dB)

40 20 0 -20 -40

Phase (deg)

-60 0 -45 -90 -135 -180 10-2

10-1

100

101

102

Frequency (rad/sec)

Fig. 19. Bode plots of C5 (s)G5 (s). Fig. 16. Step responses of the controlled plant C4 (s)G4 (s) where the controller parameters obtained for γ = 1.2.

Example 6. Consider the following integrating system, G6 (s) =

1 s(s + 1)3

.

(57)

A PID controller is obtained using least squares optimization algorithm as follows, C6 (s) = 0.4315 +

0.0070 s

+ 3.7451 s.

(58)

Parameters of the reference systems are γ = 1.1 and ωc = 0.1 rad/s for this case. The step response of the control system C6 (s)G6 (s), where the controller parameters are obtained by the proposed method, is given in Fig. 20. Bode plots of C6 (s)G6 (s) are given in Fig. 21. As seen from the step responses and the plots of the systems given in Figs. 18–21, the controllers C5 (s) and C6 (s) satisfy the stability of the systems G5 (s) and G6 (s) respectively. 5. Conclusion

Fig. 17. Step responses of the controlled plant C4 (s)G4 (s) where the controller parameters obtained for ωc = 1 rad/s.

In this paper, two methods are presented for controller design in the presence of a fractional case. The first method presents a robust parametric classical controller design to satisfy robust

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471

References

Fig. 20. Step responses of the C6 (s)G6 (s). Bode Diagram

Magnitude (dB)

50 0 -50 -100 -150 -45

Phase (deg)

-90 -135 -180 -225 -270 10-2

10-1

100 Frequency (rad/sec)

101

102

Fig. 21. Bode plots of C6 (s)G6 (s).

stability and the required performance specifications of the fractional order plant with a parametric uncertainty structure. Magnitude and phase envelopes of the fractional order interval plant are used to obtain the parameters of the Lag, Lag–Lead and PI controllers. It has been shown that the controllers, which are obtained by the proposed method, satisfy the robust stability of the fractional order plants and the system meets the required performance specifications. Numerical examples have been given for each type of controller to illustrate the application of the method presented. The results of the paper will be important for the robust stability analysis and design of the fractional order control systems with a parametric uncertainty structure. In the second method, PID controller parameters are obtained using a least squares optimization method. Bode’s ideal control loop has been used as a reference model. The desired step response of the plant can be achieved by choosing the appropriate parameters of the reference model. Different PID controller parameters have been obtained for the same plant using different values of the parameters of the reference model. Each of the controllers satisfies the stability of the system. Desired overshoots and time characteristics of the plant have been achieved with different parameters of the PID controllers.

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