Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants

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Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants ∗

Reetam Mondal , Jayati Dey Department of Electrical Engineering (EE), National Institute of Technology(NIT), Durgapur, West Bengal, 713209, India

highlights • • • • •

Explicit design steps for Fractional Order (FO) 2-DOF (Degree of Freedom) Control. Facility of supplementary design parameters with extra degree of freedom in FO 2-DOF Scheme. Distinct design approach of FO pre-filter based on the nature of loop compensator. Stabilization of Cart-Inverted Pendulum System and TRMS System using the proposed scheme. Exposition of MATLAB simulation and Real-Time Experimental results.

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Article history: Received 18 November 2018 Received in revised form 17 May 2019 Accepted 28 May 2019 Available online xxxx Keywords: FO system Fractional Order (FO) controller 2-DOF (Degree of Freedom) controller Fractional Order (FO) pre-filter Robustness

a b s t r a c t The dynamics of Fractional Order (FO) System have recently attracted substantial attention in the field of control. Taking note of this, the Fractional Order (FO) controllers provides additional flexibility in the design by virtue of its non-integer orders. However, all the available control schemes in this domain are mostly featured in 1-DOF (Degree of Freedom) formation which compromises between response and loop goals. Differently, a 2-DOF (Degree of Freedom) topology allows one to shape the transient response ensuring at the same time satisfactory loop margins. It is thus anticipated here that the integration of 2-DOF (Degree of Freedom) control scheme with FO Compensator may accelerate successful attainment of system response and loop robustness. Hence, this paper addresses the development of 2-DOF (Degree of Freedom) FO control system design methodology for integer as well as non-integer order plants. The additional degree of freedom and the non-integer order of the controller together ensure desired output response as well as adequate loop robustness. Novel design procedures for the 2-DOF (Degree of Freedom) control scheme are presented in meticulous manner depending upon the nature of the plant under consideration. A unique approach for FO pre-filter is presented depending upon the nature of the loop compensators in case of non-commensurate order plants. It is observed here that the proposed 2-DOF (Degree of Freedom) scheme bestows an added DOF in addition to the auxiliary design parameter by the virtue of its non-integer orders. The closed loop system response manifests that the proposed approach show cases exclusively surpassing system response and robustness compared to its 1-DOF (Degree of Freedom) as well as integer order 2-DOF (Degree of Freedom) counterparts. The potency of the method put forward is established with MATLAB simulation as well as real-time experimentation. The proposed control schemes are implemented to two highly non-linear real time systems of TRMS system and Cart-Inverted Pendulum System. The experimental outcomes are demonstrated to endorse the benefits of the control scheme advocated. © 2019 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Fractional order (FO) compensators are being put into use at large in diverse fields by several researchers in order to accomplish refined behavior of the various real dynamical models. The state of the art of these various non-integer order controllers ∗ Corresponding author. E-mail address: [email protected] (R. Mondal).

and their different forms has been highlighted in [1]. They are precisely concerned with the employment of fractional-order differentiator or integrator utilizing the theory of fractional calculus. Thus, the potential of the traditional classical PID Controllers is elevated appreciably due to evolution in the arena of fractional calculus trailing to FO-PI λ Dµ (FO-PID) Controller. It is also feasible to comprehend the already established traditional integer order (IO) lead/lag compensator to the FO case, the generalized version of which has been dealt with in [2]. The performance of these non-integer order compensators has been compared and found

https://doi.org/10.1016/j.isatra.2019.05.024 0019-0578/© 2019 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

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to be superior with the conventional IO lead/lag compensators. Closed loop designs that employ these FO controllers are usually restrictive to 1-DOF (Degree of Freedom) unity feedback configuration. Conventional 1-DOF (Degree of Freedom) has exhibited short-comings of satisfying any one of the desired properties of system, i.e. robustness and dynamic response performance of set-point tracking [3]. There always subsists an expedient acceptance to undermine between these response and loop goal performances. 2-DOF (Degree of Freedom) Control, on the contrary, overcomes the above weakness and imperfection thereby attaining both as desired. The amalgamation of the 2-DOF (Degree of Freedom) with Fractional Order thus ameliorates the servo and regulatory problems with augmentation of the design parameters. Such a design retains the loop performance of 1-DOF (Degree of Freedom) Controller but allows adjustments to response and conduct of the closed loop system. Keeping this in view, an elementary design procedure and modulating approach of 2-DOF PI λ Dµ Controller primarily on Internal Model Control (IMC) has attracted curiosity of many individuals of scientific research in the branch of control for the FO systems with dead time [4]. A combination of Fractional Calculus and 2-DOF (Degree of Freedom) PI λ Dµ (FO-PID) Controller is proposed in [5] for competent and scrupulous desired temperature control of a bioreactor, laboratory based temperature control system [6] and real-time Control of Pressure Plant [7]. The 2DOF (Degree of Freedom) PI λ Dµ (FO-PID) Controller integrates a supplementary control loop, whereas the fractional operator offers ancillary malleability for alternation of process properties as desired. Due to this competence, the design and prosecution of 2DOF (Degree of Freedom) PI λ Dµ (FO-PID) Controller for Automatic Generation and Control (AGC) is presented in [8–10] in which the controller tuned by nature inspired Firefly and Cuckoo Search algorithms performance of which is compared with other FOI/FOPI Controllers and conventional integer order PID Controllers. A unity feedback non-integer control strategy for delay dominant fractional order plants has been deliberated in [11]. An adjusting design method of 2-DOF (FO) internal model controller (IMC) for non-integer processes has been addressed in [12] based on the cutoff frequency and desired phase margin to make the system robust with desired dynamic performance. The drawback of the 1-DOF PI λ Dµ Controller in single feedback loop with the remedy of blending a pre-filter in 2-DOF (Degree of Freedom) configuration applied to inverted pendulum system has been discussed in [13]. A brief overview on novel 2-DOF PI λ Dµ (FO-PID) Controller presented and applied on magnetic levitation system, performance of which has been demonstrated experimentally [14–18]. The PI λ Dµ (FO-PID) Controller parameters obtained here are either by pole placement technique with the non-integer orders selected by graphical method or by optimization of multi-objective fitness function framed from the closed loop characteristic equation. The present work highlights the concept of FO differential equation and its extension in the design of the 2-DOF (Degree of Freedom) control for the commensurate and non-commensurate order plants. To this end, two different design methodologies in meticulous manner are portrayed. Commensurate Order compensators are proposed for commensurate plants. The pole placement technique for the integer order (IO) plants discussed in the literature of [3] is extended in the proposed work for commensurate fractional order (FO) plants. The FO pre-filter design is achieved through pole cancellation approach by scraping away the unwanted closed loop poles to achieve improved transient response maintaining the desired stability margins attained. On a different note, the PI λ Dµ Controller, FO-lead and FO-lead/lag compensators [2] are employed to realize the 2-DOF (Degree of Freedom) Controller configuration for the non-commensurate or integer order plants as well. The bibliography in [2] has only

considered lead/lag compensator design for ensuring loop robustness without emphasizing on the response point of view. The design method of generalized lead/lag compensators in [2] is therefore extended in this work from Fractional Order (FO) 1-DOF (Degree of Freedom) Configuration to FO 2-DOF (Degree of Freedom) Structure by virtue of Fractional order (FO) Prefilters for non-commensurate/integer order plants retaining the exact stability margins and attaining significant improvement in plant response. The pre-filter here also shows its effectiveness by appending a zero in the system response transfer function. In addendum to this, a novel systematic strategy to devise Fractional Order (FO) PI λ Dµ Controller with its respective fractional order (FO) pre-filter formulated for integer order (IO) plants as well to enrich the system response maintaining loop robustness in FO 2-DOF (Degree of Freedom) structure. The provided results quantify the performance improvement that can be obtained by using the FO 2-DOF (Degree of Freedom) Controller instead of the integer one. Novel design algorithms are explicitly proposed for the realization of the FO 2-DOF (Degree of Freedom) Controller for both integer order and non-integer order plants. Real-time implementation of the FO 2-DOF (Degree of Freedom) Control is acquired, the working behavior of which is endowed to be fairly passable for both the cart-inverted pendulum system and TRMS (MIMO) system [19–30] which has not been yet observed to be accomplished by FO Controllers in the available literatures. Following are the focal points stand to highlight the novelty and the major contributions of the present research work:

• The present work addresses a FO 2-DOF (Degree of Freedom) •

•

•

•

•

•

•

•

•

linear time-invariant (LTI) Control technique for the integer and non-integer order systems. Auxiliary design parameters of FO Controllers in addition to the supplemental degree of freedom by virtue of the 2DOF (Degree of Freedom) topology are received through this proposed control scheme. Additional degree of freedom and the non-integer order of the compensators allows enhanced flexibility in design which leads to improved system response and loop robustness. The concept of FO differential equation is extended to the design of the 2-DOF (Degree of Freedom) control for the commensurate order plants. Methodical design algorithms are explicitly proposed for the realization of the FO 2-DOF (Degree of Freedom) Controller for both commensurate and non-commensurate order plants. Complete pole placement in the principle sheet of complex (w = sγ ) -plane (γ is the commensurate order) has been achieved in the FO compensator design for the non-integer commensurate order systems leading to stability. Cancellation of the unwanted poles plays an intrinsic pivotal role in FO pre-filter design. The design method of fractional order (FO) lead/lag compensator in [2] has been expanded to non-commensurate/ integer order plants to realize the FO 2-DOF Control plan. A narrative novel design strategy of fractional order (FO) PI λ Dµ Controller has been also appended. A unique approach for FO pre-filter is presented depending upon the nature of the loop compensators for the noncommensurate/integer order plants. The non-integer order topology of 2-DOF (Degree of Freedom) Controller configuration is shown to provide more flexible control scheme with the aid of its supplementary parameters. Ensuring relocation of the closed loop poles in the stable half of s-plane by Riemann Analysis, an exhaustive study of stability analysis of the FO Linear Time In-variant (LTI) systems for all the cases considered are reported.

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

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Phase Margin (PM) are treated as the general benchmarks of loop robustness measurement. The loop robustness will enhanced with the higher values of these stability margins. Sensitivity function in Eq. (4) is a better measure of robust stability. Ensuring minimization of the maximum value of [32,33],

⏐ ⏐ |S(jω)| = ⏐⏐ Fig. 1. Fractional Order (FO) 2-DOF (Degree of Freedom) controller.

• MATLAB simulation and Experimental Results are exposed to endorse the approaches put forward.

• Stabilization and reference tracking of highly non-linear Real-Time Twin Rotor MIMO system and Cart-Inverted Pendulum System using FO 2-DOF (Degree of Freedom) Controller are achieved. The succeeding sections in this present study narrated here subsequently enlightens on the FO 2-DOF (Degree of Freedom) Control framework, the proposed design algorithms, numerical examples, simulations with real-time study and experimental results. Finally, conclusion has been drawn in view of the prevailing outcomes. 2. Structure of fractional order (FO) 2-DOF (degree of freedom) control

⏐ ⏐ ⏐ 1 + L(jω) ⏐ 1

(4)

with L(sγ ) = C (sγ ).G(sγ ), the system will be hardly sensitive to plant parameter variations. The maximum norm of S(s = jω) ∀ω ≥ 0, i.e. maxω |S(jω)| can be utilized to specify the elementary restrictions on both the GM and PM as an estimate of robust stability. To fortify this under plant parameter variations, the highest point on the sensitivity plot, ∥S ∥∞ < 2 in the middle part of the frequency range, effectuated by concurrent fulfillment of GM> 2 or 6 dB and PM > 30◦ is taken care of [32,33]. The primary objective of the design of the controller is model matching which requires a desired closed loop transfer function as in Eq. (5) to be met with unity dc gain is taken as, Y (sγ ) R(sγ )

=

χ B(sγ )

(5)

∧

∆(sγ )

[

∧

γ

]

Here, the deg ∆(s ) = n is the derived pole polynomial and

χ is a scalar. The non-integer order controller polynomials are given by,

The majority of controllers deployed are the conventional controller, driven with servo-mechanism which aches from the frailty that response and loop characteristics cannot be designed independent of each-other. Therefore, it would be worthwhile to design a control strategy that retains the desired time-domain response satisfying the selected frequency domain specifications in the best possible manner. Such a design is viable with a 2-DOF (Degree of Freedom) Controller as launched in Fig. 1. 3. Design steps of fractional order (FO) 2-DOF (degree of freedom) compensator for commensurate order plants Consider an nth order LTI Fractional Order (FO) commensurate strictly proper plant of the form (1) [31], G(sα ) =

γ

B(s ) A(sγ )

=

bm s

γm

+ bm−1 s

γm−1

ak s

+ · · · + b0 s

an sγn + an−1 sγn−1 + · · · + a0 sγ0

∑m ∑m γk kγ k=0 bk s k=0 bk s ∑ = ∑n = n γ kγ k=0

γ0

k

k=0

ak s

(1)

with A(sγ ) a known polynomial of degree n and B(sγ ) another known numerator polynomial. The degree of A(sγ )(= γm ) > degree of B(sγ )(= γn ). ak are the denominator co-efficients and bk are the numerator co-efficients. Here, γk = γ .k. The equation in (1) above can be contemplated as a pseudo-rational function, G(w ) of the variable, w = sγ , in the w -plane as,

∑m

G(w ) = ∑kn=0

k=0

bk w k

(2)

ak w k

Similarly, a single loop compensator of commensurate order can be also defined by rational transfer function,

F (sγ ) = (sγ )p + fp−1 (sγ )p−1 + · · · + f1 sγ + f0 γ

γ p

γ p−1

H(s ) = hp (s ) + hp−1 (s )

γ

+ · · · + h1 s + h0

(6) (7)

which will confirm random closed loop pole placement in the stable half of the s-plane. Thep number of arbitrary coefficients of F (sγ ), f0 , f1 , . . . , fp−1 plus the (p + 1) number of arbitrary coefficients of H(sγ ), that is to say, h0 , h1 , . . . , hp will explicit the (n + p) number of coefficients of closed loop characteristic equation as presented below,

∆(sγ ) = (sγ )p+n + δp+n−1 (sγ )p+n−1 + · · · + δ1 sγ + δ0 ∧

∧

= A(sγ )F (sγ ) + B(sγ )H(sγ ) = ∆(sγ )q(sγ )

(8)

∧ γ

where, q(s ) is the unwanted closed loop pole algebraic expression apparent from the controller dynamics which are to be canceled out by the pre-filter with the transfer function, Q (sγ ) F (sγ )

∧

=

χ q(sγ )

(9)

F (sγ )

Such a system, framed in Fig. 1 has a reference input–output transfer function stated by, Y (sγ ) R(sγ )

=

B(sγ )Q (sγ )

∆(sγ )

∧

=

χ q(sγ )B(sγ ) ∧

∧

∆(sγ )q(sγ )

=

χ.B(sγ ) ∧

(10)

∆(sγ )

Now, following (1), number of unknown controller parameters ≥ number of closed loop poles to be placed, i.e, (2p + 1) ≥ (p + n) or , p ≥ n − 1

(11)

(3)

The design method for the commensurate order plants is presented as follows, Step 1: H(sγ ) and F (sγ ) can be chosen by solving the Eq. (8). Varia-

which completely determines both the loop and response performance characteristics of the system. Gain Margin (GM) and

tion of q(sγ ) will result in different loop performances. So, one can obtain infinite number of loop characteristics while maintaining same time responses.

∑m γk H(s ) k=0 hk s α ∑ = C (s ) = n γk γk γk

F (s )

k=0 fk s

∧

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

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Step 2: Canceling the unwanted closed loop poles to acquire the ∧ γ

desired transfer function, q(s ) of degree p is determined as, ∧

Q (sγ ) = γ .q(sγ ) = qp (sγ )p + qp−1 (sγ )p−1 + · · · + q1 sγ + q0

(12)

Step 3: The error response transfer function is, E(sγ ) R(sγ )

=1−

Y (sγ )

=

R(sγ )

A(sγ )F (sγ ) + B(sγ ){H(sγ ) − Q (sγ )}

∆(sγ )

ess = Lt sγ E(sγ )

(13)

s→0

For the above Eq. (14) to be robustly zero, the following is postulated to be, (15)

s→0

(15) will be satisfied (i) if h0 = q0 for (ii) automatically for a Type-1 plant and (ii) F (0) = f0 = 0 for a Type-0 plant [32]. 4. Design steps of fractional order 2-DOF (degree of freedom) controller for non-commensurate fractional order and integer order LTI plants Consider an LTI integer order plant as follows, G(s) =

Y (s) U(s)

=

bm s

m−1

+ bm−1 s

m−1

+ · · · + b1 s + b0 B(s) = (16) sn + an−1 sn−1 + · · · + a1 s + a0 A(s)

The nominal plant polynomials B(s) and A(s) are co-prime with the degree of A(s)(= n) > degree of B(s)(= m). Design steps for Fractional Order (FO) 2-DOF control scheme can be narrated as, Step 1. The fractional order compensator or controller to be designed for the plant in (16) as [2], satisfying the desired frequency domain specifications. Here, the FO compensator/controller C (s) adopted from [2] as, C (s) =

H(s) F (s)

=

1 + α Tsβ 1 + Tsβ

(17)

The non-integer order β ∈ (0, 2) [2] consummates accurate and distinctive solution in the design of (17) at a specified gain crossover frequency (ωcg ′ ) with desired robust stability margins intended for the corresponding loop transfer function, L(s) = C (s)G(s) =

H(s)B(s) F (s)A(s)

(18)

fulfilling the defined frequency domain margins and sensitivity peak magnitudes ≤ 2 which in turn, ensures robust stability of the closed loop system. The closed loop characteristic equation is,

∆(s) = A(s)F (s) + H(s)B(s)

Q (s) = q1 sβ + q0

(22)

for compensation with fractional order (FO) lead/lag compensator. At this place, q0 is calculated to meet zero steady-state error requirements as, B(0)Q (0)

(14)

Lt s × [A(sγ )F (sγ ) + B(sγ ){H(sγ ) − Q (sγ )}] = 0

Q (s)

to have F (s) as bi proper. Similarly, the numerator polynomial Q (s) of the non-integer order pre-filter is selected as [13],

∆(0)

= 1 ⇒ q0 =

∆(0) b0

(23)

By proper tuning and variation of q1 of the Q (s)polynomial the distinct place of zeros can be altered to secure a better transient performance with less peak overshoot and settling time [13]. In-case of Type-0 system, compensated with FO lead compensator, a FO-PI λ Controller is needed to be applied to obliterate the steady state error. Therefore, the compensator becomes, H(s) F (s)

=

1 + α Tsβ KP sλ + KI 1 + Tsβ

.

sλ

(24)

The numerator polynomial Q (s) of the non-integer order prefilter here for this instance of lead-PI λ compensator can be formulated as, Q (s) = q2 sλ+β + q1 s

λ+β 2

+ q0

(25)

where, with h0 = q0 , q1 and q2 are calculated by canceling out the unwanted closed loop poles lying in the principal Riemann sheet with less damping ratio which are responsible for high overshoot or the poles with slower dynamic response. 5. Examples and results This section illustrates numerical examples to showcase the benefits of FO 2-DOF controller compared with that of 1-DOF control, exhibiting the expediency of the proposed configuration. Example 1. In this example the dynamic model of an immersed plate of mass M and area S in a Newtonian fluid of infinite extent and connected by a mass less spring of stiffness K to a fixed point as shown in Fig. 2 [34], Assuming, that the plate-fluid system is initially in an equilibrium state, and displacement velocities are initially zero, the dynamics of the system in Fig. 2 is asserted by [34], M .D2 y(t) = f (t) − K .y(t) − 2.S .σ (t , 0)

(26)

with the force f (t) applied to the immersed plate with sufficiently large area S to commence a general transverse motion, y(t) with

(19)

The controller here may be also chosen as PI λ Dµ Controller in the form as in Eq. (20) [1], C (s) =

H(s) F (s)

= KP +

KI sλ

+ KD s µ

(20)

where, KP , KI and KD are the three non-zero controller gains with the additional the fractional order differentiator µ and fractional integrator λ (λ, µ ∈ R+ ). In general, the range of which considered within 0 and 2 [1]. Step 2. As a consequence of this fact, a suitable choice of the Q (s) polynomials of the pre-filter transfer functions, PF (s) = F (s) will assist to improve desired output response. Now, here if the compensator C (s) is a PI λ Dµ Controller, then the numerator polynomial Q (s) of the FO pre-filter will be defined as, Q (s) = q1 sλ + q0

(21)

Fig. 2. Thin rigid plate immersed in a newtonian fluid.

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

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Fig. 3. (a) Magnitude and phase diagram of loop transfer function with frequency (b) plot of sensitivity function S(s).

selected as, ∧

Q (s0.5 ) = χ.q(s0.5 ) = 576 × (s0.5 + 7)(s0.5 + 7)(s0.5 + 1)(s0.5 + 1) (30) The pre-filter transfer function in accord to Eq. (12) thus can be written as, PF (s) =

Q (s0.5 ) F (s0.5 )

=

576 × (s2 + 16s1.5 + 78s + 112s0.5 + 49) s2 + 15.5s1.5 + 122.25s + 882.88s0.5 (31)

Fig. 4. Disposition of the poles in the complex w -plane using FO compensator.

velocity v (t , z) and stressesσ (t , z) [31] leading to, M .D2 y(t) + 0.5D1.5 y(t) + K .y(t) = f (t)

(27)

This shows that the system is clearly of commensurate order, with common base power, γ = 0.5. Taking this into account that the maximum order of this differential equation is 2, then the number of states to be considered is, n = 4. The Laplace Transform of Eq. (27) is, G(s0.5 ) =

1 (s0.5 )4

(28)

+ 0.5(s0.5 )3 + 0.5

According to Eq. (5), let the desired overall system transfer function be, Y (s0.5 ) R(s0.5 )

=

χ .B(s0.5 ) ∧

∆(s0.5 )

=

576 (s0.5

(29)

± 4j)(s0.5 ± 6j)

so as to ensure a faster response. The order of the compensator will be, p = n = 4, and f0 = 0 for reference tracking. Therefore, there will be, (n + p) = 8, number of closed loop poles. The other 4 closed loop poles are chosen as -7,-7,-1 and -1 in w -plane so as to γπ satisfy the stability criteria, |φw | > 2 (= 0.7854) [31,35] and the resulting loop transfer function yields satisfactory performance with the desired objectives of stability margins.

The system step response displayed in Fig. 5(a) ensures reduced settling time and % Overshoot with the Fractional Order (FO) 2-DOF(Degree of Freedom) controller with less control signal amplitude as conveyed through Fig. 5(b) compared to that obtained from the 1-DOF(Degree of Freedom) controller configuration. Example 2. Consider the integer order Linear Time invariant (LTI) unstable Magnetic Levitation system as [15], G(s) =

(32)

which has two poles located at ±46.67 making it highly unstable. An attempt has been made in [15] to stabilize the above plant employing PI λ − Dµ controller where the controller gains KP , KI and KD determined by pole placement technique and the noninteger orders λ and µ resolved by graphical tuning approach primarily pivoting on the frequency domain stability margins. The PI λ Dµ controller has been designed on the same system in [16,17] by the optimization of the cost function framed with the desired closed loop characteristic equation composed of dominant poles identified from defined performance specifications. Here, instead a FO lead compensator is formulated and contemplated following [2], to stabilize the plant ensuring adequate robustness. On account of this, to ensure zero steady state error for this Type-0 system a fractional order compensator along with a PI λ Controller in the form of Eq. (33) is introduced in the forward path to maintain satisfactory transient response while eliminating the steady state error. Hence,

B(s0.5 ).H(s0.5 )

The ensuing loop transfer function, L(s0.5 ) = A(s0.5 ).F (s0.5 ) , yields a satisfactory GM of Inf dB and PM of 66◦ at ωcg of 22.9 rad/s. The frequency response of the loop transfer L(s) is made visible in Fig. 3(a). The sensitivity plot in Fig. 3(b) given by Eq. (4) depicts that maxω |S(jω)| reaches to 1.1487. Stability analysis as illustrated in Fig. 4 depicts the absence of poles in the shaded γπ unstable region ensuring, |φw | > 2 (= 0.7854) which ratifies stability of the compensated system [31,35]. To scrap away the unwanted closed loop poles at −7, −7, −1 and −1 along with ensuing unity dc gain at steady state, Q (sα ) is

−3518.85 s2 − 2177.8

C (s) =

KP (sλ + ωc ) sλ

{

1 + α Tsβ 1 + Tsβ

} =

KP (sλ + sλ

KI KP

)

{

1 + α Tsβ

}

1 + Tsβ (33)

The Fractional Order PI λ Controller which contributes a phase lag can be derived as in follows,

(

tan φ =

)

.ωλ sin λπ 2 λπ ) − KP λπ 2 λ 1 + K (ω cos 2 ) I KP KI

(

(34)

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

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Fig. 5. (a) System step response and (b) control signal using FO 1-DOF and 2-DOF compensator.

Fig. 6. (a) Frequency response of loop transfer function using FO lead controller (b) plot for the sensitivity function S(s).

The closed loop characteristic equation using the FO-PI λ Controller and the FO Lead compensator in (33) is, 0.0052522s3.8593 + s2.96 + 174.24s1.8593 + 1341s0.96

+ 742.7s0.89934 + 14075 = 0

(35)

Transmuting Eq. (35) to w -plane by using the notation, s1/γ = w [31], taking γ = 0.01, a polynomial of complex variable w in the following form is obtained as, 0.0052522w 385 + w 296 + 174.24w 185 + 1341w 96 + 742.7w 89

+ 14075 = 0 Fig. 7. Pole positions in complex w -plane using FO lead compensator.

where, φ is the phase lag provided by the FO-PI λ Controller at any frequency ω. With a choice of ω = ωc = 4 rad/s and φ = −5◦ to −6◦ at ωc the solution of λ is obtained as 0.96 which ensures that the Fractional Order (FO) PI λ Controller provides only 5◦ to 6◦ phase lag at ωc . KP of the fractional compensator is } { taken as, -1. The magnitude and phase diagram of KP .G(s)

sλ +ωc sλ

disseminates that the GM = Inf dB and PM = −6.99◦ at ωcg of 37.3 rad/s. A fractional order (FO) lead compensator is designed employing the method in [2] to achieve the objectives Phase Margin (PM) of 45◦ at ωcg ′ = 95 rad/s which yields, β = 0.89934, α = 10.0462 and T = 0.0052522. The magnitude and phase plot of the Loop Transfer function as evinced in Fig. 6(a) confirms that the mentioned objectives of desired stability margins are fulfilled for the unstable system. The sensitivity plot in Fig. 6(b) shows ∥S ∥∞ < 1.5 achieved for this system

(36)

Solving, the above polynomial, the closed loop pole positions are found in the complex s-plane corresponding to the principle Riemann sheet poles 1.0478 ± 0.0252j and 1.0239 ± 0.0260jvisible in the zoomed plot of the complex w -plane (Fig. 7) are −80.8339 ± 73.5286j and −9.0189 ± 6.2444j with the values of the damping ratios 0.7397 and 0.82216 respectively. Non-existence of poles in the shaded region which corresponds to the unstable region is evident from Fig. 7. The unstable γπ region from s-plane mutates to the shaded sector |φw | < 2 [35, 36]. It may be also observed that the argument values |φw1 | = 0.0240 and |φw2 | = 0.0254 of the above roots respectively in γπ the principle sheet satisfies the stability condition |φw | > 2 (= 1 0.0157), where γ = 100 is the total number of sheets in the Riemann surface which confirms the stability [35,36]. Now, with the objective to refine the quality of set-point tracking by reducing the overshoots a fractional order prefilter transfer function is introduced conforming to Eq. (37). According to the method adopted in [15], tuning the FO of the pre-filter, the location of zeros of the fractional PI λ − Dµ controller adjusted to achieve desired transient response. Application of integer order pre-filters with the non-integer order controller has been also conveyed in [14,16,17]. On a distinct note, in this proposed work,

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Table 1 Comparative study of the proposed method with existing ones in available literatures. Sl. No.

Transient performance indices (proposed method)

Transient performance indices in [15] shown in Table 3

Transient performance indices in [16] shown in Table 10

1. 2.

Settling time(ts ) 0.3 s

Settling time(ts ) 0.71 s

Settling time(ts ) 0.85

% Overshoot 0

% Overshoot 10.72

% Overshoot 2.4

Fig. 8. (a) System step response and (b) control signal using FO 2-DOF (degree of freedom) configuration.

the proposed 2-DOF (Degree of Freedom) configuration compared to the propositions presented in [15,16] as shown in the Table 1. below. It is also well justified from Fig. 8(b) that the amplitude of control signal required is much less in case of FO 2-DOF Controller in disparity to that of FO 1-DOF (Degree of Freedom) Controller. 6. Fractional Order (FO) 2-DOF (Degree of Freedom) Control of cart-inverted pendulum system A novel FO 2-DOF control strategy has been unveiled in this segment of study for the SIMO system with unstable equilibrium [2]. The experimental layout provides two separate outputs, the pendulum angle (θ ) and cart position (x) on the rail as depicted in the control scheme in Fig. 9. The test rig described here has the following input-out transfer functions as [2],

θ (s)

−4.40310s2 U(s) s2 (s + 3.7682)(s − 3.7682) X (s) 3.1800(s + 3.6855)(s − 3.6855) = U(s) s2 (s + 3.7682)(s − 3.7682)

Fig. 9. Overview of the cart-inverted pendulum control set-up.

=

(39) (40)

X (s)

where U(s) stabilized by PD compensator with the resulting inner loop compensated transfer function as [2], the FO pre-filter transfer function has been redesigned newly as, GT (s) = PF (s) =

Q (s) F (s)

=

{q2 s

λ+β

+ q1 s

λ+β 2

+ q0 }

sλ (1 + Tsβ )

(37)

∆(0)

where, q0 is fixed at b = −4 to ensure zero steady state error. 0 The closed loop poles −80.8339 ± 73.5286j with less damping ratio have been selected to coin the Q (s) polynomial of the prefilter as obtained below. Thus, the resulting transfer function is obtained as, PF (s) =

Q (s) F (s)

= −6.4808 × 10−4 .

{

s1.8593 + 96.774s0.92967 + 6170.1

}

0.0052522s1.8593 + s0.96 (38)

The reaction and behavior to step input applied on the FO lead compensator in Fig. 8(a) intimates that it settles with in 0.45 s and the 72.65% Overshoot attained is reduced considerably to 0.3 s and 0% using the Pre-filter transfer function PF(s) in Eq. (38) in

θ (s) U(s)

=

−4.40310s2 s2 (s + 45.19)(s + 3.416)

(41)

which is therefore re-constituted as [2], GF (s) =

X (s) U(s)

=

3.1799(s + 3.6855)(s − 3.6855) s2 (s + 45.19)(s + 3.416)

(42)

A FO lead compensator is thus devised and conceived on Eq. (42) to achieve a PM =45◦ at ωcg ′ = 2 rad/s [2], C (s) =

H(s) F (s)

=

16.6865s0.95711 + 1 0.039679s0.95711 + 1

(43)

The FO compensator is able to achieve the desired loop goals as it can been seen from the magnitude and phase graph plotted against frequency and the sensitivity plot in Fig. 10(a) and (b) respectively. The closed loop pole orientations are found in the complex splane corresponding to the principle Riemann sheet poles 1.0119 ± 0.0244j visible in the zoomed plot of the complex w plane (Fig. 11) are −2.4914 ± 2.2462j with the value of the damping ratio, ξ = 0.7427. Non-appearance of the poles in the

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

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Fig. 10. (a) Magnitude and phase diagram using FO lead controller (b) plot for the sensitivity function.

Fig. 11. Pole positions in complex w -plane using FO lead compensator. Fig. 13. Experimental set-up of twin rotor MIMO system.

shaded region which quadrates to the unstable segment in the complex w -plane [35,36] which reaffirms the stability. Additional zeros are brought in the closed loop transfer function because of the lead compensator in the forward path. In the 2-DOF (Degree of Freedom) configuration, locating the fractional compensator in the feedback path eliminates the zeros from the closed loop transfer function. To supplement this fact, a fractional order (FO) pre-filter has been designed according to methodology narrated in Section 4 as,

PF (s) =

Q (s) F (s)

=

6.1s

0.95711

+1

0.039679s0.95711 + 1

(44)

The numerator polynomial Q (s) of the FO pre-filter is selected ∆(0) conforming to Eq. (22) of step 2. Here, q0 = b is calculated 0 to be −1 to meet the zero steady-state error requirements. By altering the value of q1 of the Q (s) polynomial in the pre-filter, the location of the zeros can be rearranged to get a better transient performance (Fig. 12) leading to competent response goals [15]. The proposed scheme has been administered and applied on the

Cart-Inverted Pendulum System using MATLAB in factual case for stabilization of the system. The experimental results to justify the proposed strategy have been well established in Section 8. 7. Stabilization of twin rotor MIMO system (TRMS) control using fractional order (FO) 2-DOF (degree of freedom) controller A nonlinear laboratory test rig of TRMS system selected to experimentally validate the proposed approach. It is multivariate with two input voltages supplied to the propellers to provide two outputs of vertical and horizontal angles [20] as shown in Fig. 13. To stabilize the TRMS the controllers are designed by decoupling between the two axes of motions. The system is computer controlled and implemented in MATLAB Simulink environment allowing interfacing of measured signals between the TRMS and computer through I/O card in the electrical unit. To compose and express the transfer function of the TRMS, the non-linear model [20,21] has to be linearized. The, numerical

Fig. 12. (a) Transient response and (b) control signal using FO 2-DOF (Degree of Freedom) configuration.

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Fig. 14. Overview on FO 2-DOF control of TRMS system.

Fig. 15. Locus of the roots of the compensated system for (a) main rotor (b) tail rotor.

Fig. 16. Plots of (a) GM (dB) (b) PM (deg) and (c) sensitivity peak variation with variation of FO lambda (λ) and FO Mu (µ) for the main rotor.

Fig. 17. Plots of (a) GM (dB) (b) PM (deg) and (c) sensitivity peak variation with variation of FO Lambda (λ) and FO Mu (µ) for the tail rotor.

Fig. 18. (a) Bode plot of loop transfer functions using FO-PI λ Dµ controller in 1-DOF (Degree of Freedom Configuration) and (b) Plot for the SO (jω) of the Main rotor.

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

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Fig. 19. (a) Bode plot of loop transfer functions of the tail rotor in 1-DOF (Degree of Freedom Configuration) (b) plot for the SO (jω) of the tail rotor.

Fig. 20. Location of the poles in complex w -plane of the (a) main rotor and (b) tail rotor implemented by FO-PI λ Dµ controller.

Fig. 21. (a) Step response of the main rotor for variation of q1 and (b) step response of the tail rotor for variation of q1 in 2-DOF (Degree of Freedom) arrangement.

Fig. 22. (a) Step response of main (pitch) rotor (b) control signals.

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Fig. 23. (a) Step response of tail (yaw) rotor (b) control signals.

Table 2 Time domain attributes with the FO controllers. Sl. No. Parameters

1.

Settling time(ts ) (s) % Overshoot

2.

2-DOF FO controller

1-DOF FO controller

Integer order PID controller

Main (Pitch)

Tail (Yaw)

Main (Pitch)

Tail (Yaw)

Main (Pitch)

Tail (Yaw)

11

10

13

16

17

26

0

4.77

8.25

35.4

11.5

40.2

parameters adopted from [22]. The TRMS model derived as [21],

[ G(s) =

]

G11 (s) G21 (s)

G12 (s) G22 (s)

⎡

⎤

1.246

⎢ 3 s + 0.9215s2 + 4.77s + 3.918 =⎢ ⎣ 1.482s + 0.4234 s4 + 6.33s3 + 7.07s2 + 2.08s

0 3.6

⎥ ⎥ ⎦

(45)

s3 + 6s2 + 5s

The effects of the undesired cross-couplings are eliminated with de-coupler which transforms the MIMO system into two SISO loops as shown in Fig. 14. The de-coupler matrix elements D11 (s) and D22 (s) are selected as 1 and the remaining elements D12 (s) and D21 (s) as, D12 (s) = −

G21 (s) G22 (s)

and D21 (s) = −

G12 (s) G11 (s)

(46)

The corresponding two SISO transfer functions are found to be,

[ G1 (s) = G11 (s) −

[ G2 (s) = G22 (s) −

G12 (s).G21 (s)

] and

G22 (s) G12 (s).G21 (s)

] (47)

G11 (s)

CM (s) = KP +

where, G1 (s) and G2 (s) are main rotor and tail rotor transfer functions respectively. Decoupled compensation employing two independent FOPI λ Dµ controllers are developed, for each of the rotors of the TRMS system. It is evident from the above linearized model G1 (s) for the positive gain K in Eq. (47) has one real pole near the origin at −0.8341 and two complex poles −0.0437 ± 2.1668j very near to the imaginary axis with the value of damping ratio, ξ = 0.02016 while G2 (s) as one pole at the origin and two real poles on the stable segment of the s-plane at −1 and −5. Since, the FO-PI λ Dµ Controller is a generalization of the traditional integer one with well known transfer function as, CPID (s) = KP +

KI s

+ KD s =

time. The three non-zero controller gains, KP , KI and KD in Eq. (48) can be determined by two finite zeros at −z1 and −z2 placed suitably near the two complex poles of G1 (s) to improve the speed of the response with one pole at the origin. The root loci from the two complex poles will be attracted to the left half of s-plane terminating at infinity with the loci from the poles at the origin and −0.8341 terminating at the two zeros. The zeros are placed at −0.025 ± 1.04851j. Regarding the tail rotor, the controller zeros are suitably placed at −1.8 and −0.2 to which the locus from the two real poles are put to an end. The loci of the roots of the compensated closed loop systems are displayed in Fig. 15. FO-PI λ Dµ Controllers according to Eq. (20) with two more supplementary tuning parameters, fractional order integrator (λ) and fractional differentiator operator (µ) are placed in the feedback path in the outer-loop in accordance with the framework laid down in Fig. 14 for the Main Rotor and Tail Rotor respectively in order to achieve satisfactory robust stabilization A graphical tuning technique based on frequency domain specifications of GM in dB, PM in deg and maximum peak sensitivity is presented in Figs. 16 and 17 for the selection of non integer orders λ and µ of the following FO-PI λ Dµ Controllers according to Eq. (20) which are,

K (s + z1 )(s + z2 ) s

(48)

an acceptable design value of KP , KI and KD are sought which increases the damping with minimum overshoot and settling

KI sλ

+ KD sµ = 0.05 +

1.1 s1.3

+ 1s0.98

(49)

0.3 + 1s0.97 (50) s1.03 for main and tail rotors, respectively. The goal is to adjust and select these non-integer parameters of the FO-PI λ Dµ Controllers in Eq. (49) and (50), so that it is able to achieve desirable positive GM, PM and Sensitivity Margins, variation of which is shown for the plants G1 (s) and G2 (s). The frequency response in Fig. 18(a) of the loop transfer function, LM (s) = CM (s).G1 (s) depicts that the GM achieved is 12.7 dB at ωcp = 0.771rad/s and PM of 35.4◦ at ωcg = 0.39 rad/s. The sensitivity characteristic in Fig. 18(b) portrays that maxω |S(jω)| reaches to 1.965. Further, in case of tail rotor the frequency response in Fig. 19(a) of the loop transfer function, LT (s) = CT (s).G2 (s) depicts that the GM achieved is 65.9 dB at ωcp = 78.9 rad/s and PM of 52.2◦ at a ωcg = 1.05 rad/s. The sensitivity characteristic shown in Fig. 19(b) yields maxω |S(jω)| = 1.3289. CT (s) = 2 +

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Fig. 24. (a) x, θ and u diagrams with coveted PM = 45◦ at ωcg ′ = 2 rad/s (b) x, θ and u diagrams with Square Reference signal of amplitude 0.2 m.

Fig. 25. (a) Experimental results of main (pitch) rotor (b) Control signal applied to main (pitch) rotor (c) experimental results of tail (yaw) rotor (d) control signal applied to tail (yaw) rotor.

The closed loop transfer function using G1 (s) and CM (s) has a

and ξ = 0.3259 in s-plane corresponding to the pair of poles

pair of complex conjugate poles −0.1232 ± 2.3779j and

1.0086 ± 0.0164j and 0.9918 ± 0.0189j visible in the zoomed

−0.1463 ± 0.4244j with respective damping ratios of ξ = 0.0516

plot (Fig. 20(a)) of the principal sheet of Riemann surfaces in

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Fig. 26. (a) Experimental results of main (pitch) rotor to a square reference pitch input of +1 rad with yaw reference input kept constant to +0.5 rad (b) experimental Results of the tail (yaw) rotor to a square reference pitch input of +1 rad with main (pitch) rotor reference input kept constant to +0.5 rad.

Fig. 27. Experimental results of (a) main (pitch) rotor and tail (yaw) rotor with δ = 0.22 (b) main (pitch) rotor and tail (yaw) rotor with δ = −0.44.

Fig. 28. (a) Experimental results of main (pitch) rotor with τd = 0.15 s (b) experimental results of tail (yaw) rotor with τd = 0.15 s.

w-plane which develops unwanted dynamics of oscillations in the system [31]. Likewise, the principal sheet of the Riemann surface in w -plane employing G2 (s) and CT (s) for the Tail Rotor is displayed in Fig. 20(b). To reduce the effects of overshoots and oscillations and achieve an over-damped response with reduced settling time, the prefilter transfer function for the Main Rotor is ascertained by tuning q1 of the pre-filter transfer function in Eq. (51) as portrayed in

Fig. 21(a). q0 is already set as

PFM (s) =

QM (s) FM (s)

=

q1 sλ + q0 sλ

∆(0)

=

b0

= 1.1.

0.05s1.3 + 1.1

(51)

s1.3

Applying, the same method discussed above, the pre-filter transfer function for the Tail Rotor is taken as Eq. (52) by proper ∆(0) tuning and variation of q1 as in Fig. 21(b). q0 is set to b = 0.3 0

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in this case. PFT (s) =

QT (s) FT (s)

=

q1 sλ + q0 sλ

=

0.1s1.03 + 0.3 s1.03

(52)

The system step response of the rotors are exhibited as in Figs. 22(a) and 23(a) respectively which elucidates that the Fractional Order (FO) 2-DOF (Degree of Freedom) Controller has reduced settling time and less percentage overshoot compared to integer order PID Controller and Fractional Order (FO) 1-DOF (Degree of Freedom) Controller as tabulated in Table 2. It has been also verified through simulation as demonstrated in Figs. 22(b) and 23(b) that with the above choice of the controller parameters, the decoupled controller outputs for both the cases of main and tail rotors using Fractional Order (FO) 2-DOF (Degree of Freedom) Controller lies well between ±2.5 V. The magnitude of control signal applied to the DC Motor actuator is much less in each of the cases studied compared to the 1-DOF (Degree of Freedom) integer order PID Controller. Thus, it can be executed and administered on the real time system as vindicated in the next Section 8. 8. Real-time experimental results and discussions 8.1. Stabilization and position tracking of the Cart-inverted pendulum system The non-integer order compensator presented in Eq. (43) and the fractional order pre-filter in Eq. (44) methodized in 2-DOF (Degree of Freedom) framework is implemented on a Real time workshop of the system under control. It is observed from the nominal response in Fig. 24(a) that it remains upright maintaining the pendulum in equilibrium position following the desired input coveted. The initial pendulum is, ±0.05 rad (2.86◦ ). The cart position is skillfully sustained by the non-integer order compensator within the restricted rail limit of ±0.4 m with the control voltage to the DC Motor actuator well confined to the rated safe limit of, ±2.5 V [2]. Additionally, to accredit and asses the potency of the compensator designed on the actual physical system, a square reference signal varying from 0 to 0.2 m in the course of the Cart travel has been infringed on the system under test. The position tracking performance of the test rig revealed here in Fig. 24(b) has been gratified acceptably in real time in aid of the claim asserted. It is perceived here that the compensator in the proposed configuration is certainly capable to obey the desired signal quite conveniently. 8.2. Implementation of fractional order (FO) 2-DOF (degree of freedom) controller on twin rotor MIMO system The tracking performance of the compensated system to a step reference input for both the output channels taken making use of the FO 2-DOF controller investigated through experimentation on a TRMS set-up [22]. The magnitude of the step signal administered here is 1 rad. The real-time response obtained in this regard has been given in Fig. 25 proving satisfactory reference tracking. Again, to verify the effectiveness of the controller, a square reference pitch +1 rad and time period T = 60 secs is fed as desired input while the reference input to the yaw is kept constant at + 0.5 rad. Under these circumstances the experimental outputs obtained are presented in Fig. 26(a). Similarly, a square wave of same amplitude and period is applied to the yaw while the reference input to the pitch is kept constant to a magnitude of +0.5 rad. The corresponding experimental response obtained is depicted in Fig. 26(b). This particular experiment is comprehended to uncertainty analysis as well with multi-channel input–output perturbations

of the form [21,23] of diag. [1+δ, 1−δ] and diag.[1−δ, 1+δ] is applied and verified through both simulation and experiment. Unit step pitch and yaw reference inputs are administered and the value of δ (positive and negative) is recorded for which the closed loop feedback system using the Fractional Order 2-DOF (Degree of Freedom) Controller remains stable. In the present research, in simulation the range of δ for which the closed loop system remains stable is [−0.55, 0.25] which is found experimentally on real time system as [−0.44, 0.22]. Fig. 27(a) demonstrates the experimental step response for main and tail rotor with δ = 0.22 and Fig. 27(b) for δ = −0.44. In simulation the delay margin is obtained as 0.2 s for main and tail rotor which is verified and found to be 0.15 s experimentally, the response of which has been displayed in Fig. 28. 9. Conclusions The present work proposes a 2-DOF FO control design methodology for commensurate FO Plants. Further, a lucid design propositions on FO 2-DOF Controller based on non-integer order lead/ lag compensator and PI λ Dµ Controller for linear time-invariant (LTI) integer order and non-commensurate order plants have been also proposed. Methodical design of FO pre-filters is also demonstrated in detail considering different cases for the methods postulated above. The design algorithm shows that the robustness and dynamic performance of the closed loop control system can be decoupled from each-other in which set-point tracking and robustness can be regulated within desirable values independently. The robustness of the designed 2-DOF controllers is scrutinized with respect to sensitivity function, GM and PM fulfilling the possible robustness criteria. The simulation results indicate that the design put forward is not only convenient and simple in parameter tuning of the compensators, but also provides better performance in both reference tracking and loop robustness. The distinct attributes to be relevant or appropriate and potency of the design schemes asserted and substantiated with both numerical cases and experimentations. As an extension of this work, the methodology employed can be extended to other time delay non-linear plants and multi-variable control systems, to study the attributes of the proposed scheme. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Shah Pritesh, Agashe Sudhir. Review of fractional PID controller. Mechatronics 2016;38:29–41, Science Direct, Elsevier. [2] Dey Jayati, Mondal Reetam, Halder Suman. Generalized phase compensator of continuous time plants. ISA Trans 2018;81:141–54, Elsevier. [3] Dey Jayati, Saha Tapas Kumar. Design and performance analysis of two degree of freedom (2-DOF) control of DC-DC boost converter. In: IEEE international conference on industrial technology (ICIT). 2013, p. 493–8. [4] Li Mingjie, Zhou Ping, Zhao Zhicheng, Zhang Jinggang. Two degree of freedom fractional order PID controller design for fractional order processes with dead time. ISA Trans 2016;61:147–54, Elsevier. [5] Pachauri Nikhil, Singh Vijander, Rani Asha. Two degree of freedom fractional order proportional-integral-derivative based temperature control of fermentation process. J. Dynam. Syst. Meas. Control (ASME) 2018;140:1–10. [6] Deshmukh GL, Kadu CB. Design of two degree of freedom PID controller for temperature control system. In: IEEE international conference on automatic control and dynamic optimization techniques (ICACDOT). 2016, p. 586–9. [7] Bingi Kishore, Ibrahim Rosdiazli, Karsiti Mohd Noh, Miya Hassan Sabo, Rajah Harindran Vivekananda. Real-time control of pressure plant using 2-DOF fractional order PID controller. Arab J Sci Eng 2018;1–12, Springer.

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.

R. Mondal and J. Dey / ISA Transactions xxx (xxxx) xxx [8] Debbarma Sanjoy, Saikia Lalit Chandra, Sinha Nidul. Automatic generation control of multi-area system using two degree of freedom fractional order PID controller: A preliminary study. In: IEEE (PES) Asia pacific power and energy engineering conference (APPEEC). 2014, p. 1–6. [9] Debbarma Sanjoy, Saikia Lalit Chandra, Sinha Nidul, Kar Biswajit, Datta Asim. Fractional order two degree of freedom control for AGC of an interconnected multi-source power system. In: IEEE International conference on industrial technology (ICIT). 2016, p. 505–10. [10] Debbarma Sanjoy, Saikia Lalit Chandra, Sinha Nidul. Automatic generation control using two degree of freedom fractional order PID controller. Electr Power Energy Syst 2014;58:120–9, Elsevier. [11] Padula Fabrizio, AntonioVisioli. Set-point filter design for a two degree of freedom fractional control system. IEEE J Autom 2016;3(4):451–61. [12] Jia Dong, Yanjun Cui, Zhiqiang Wang. Two degree of freedom fractional order internal model controller design for non-integer processes with time-delay. In: Proceedings of the 36th chinese control conference. China; 2017, p. 4380–5. [13] Dwivedi Prakash, Pandey Sandeep, Junghare AS. Robust and novel two degree of freedom fractional controller based on two-loop topology for inverted pendulum. ISA Trans 2018;75:189–206, Elsevier. [14] Acharya DS, Mishra SK, Ranjan PK, Misra S, Pallavi S. Design of optimally tuned two degree of freedom fractional order PID controller for magnetic levitation plant. In: IEEE international conference on electrical, electronics and computer engineering (UPCON). 2018, p. 1–6. [15] Pandey Sandeep, Dwivedi Prakash, Junghare AS. A novel 2-DOF fractional order PI λ Dµ controller with inherent anti-wind up capability for magnetic levitation system. Int J Electron Commun (AEU) 2017;79:158–71, Elsevier. [16] Swain Subrat Kumar, Sain Debdoot, Mishra Sudhansu Kumar, Ghosh Subhojit. Real time implementation of fractional order PID controllers for magnetic levitation plant. Int J Electron Commun (AEU) 2017;78:141–56, Elsevier. [17] Sain Debdoot, Swain Subrat Kumar, Kumar Tapas, Mishra Sudhansu Kumar. Robust 2-DOF FOPID controller design for maglev system using jaya algorithm. IETE J Res 2018. http://dx.doi.org/10.1080/03772063.2018.1496800, Taylor and Francis. [18] Gaidhane Prashant J, Kumar Anupam, Nigam Madhav J. Tuning of two DOF FOPID controller for magnetic levitation system: A multi objective optimization approach. In: IEEE 6th international conference on computer applications in electrical engineering recent advances (CERA). 2017, p. 479–84. [19] Ates A, Yeroglu C. Online tuning of two degree of freedom fractional order control loops. Balkan J Electr Comput Eng 2016;4(1):5–11. [20] Pandey Sumit Kumar, Dey Jayati, Banerjee Subrata. Design and real time implementation of robust PID controller for twin rotor MIMO system (TRMS) based on Kharitonov’s theorem. In: IEEE international conference on power electronics, intelligent control and energy systems (ICPEICES). 2016, p. 1–6.

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[21] Pradhan Jatin Kumar, Ghosh Arun. Design and implementation of decoupled compensation for a twin rotor multiple-input and multiple-output system. IET Control Theory Appl 2013;7(2):282–9. [22] Twin Rotor MIMO system Control Experiments User Manual (33-949S), Feedback Instruments Ltd, East Sussex, UK. [23] Pandey Sumit Kumar, Dey Jayati, Banerjee Subrata. Design of robust proportional–integral–derivative controller for generalized decoupled twin rotor multi-input multi-output system with actuator non-linearity. J Syst Control Eng 2018;232(8):971–82. [24] Abdulwahhab Omar Waleed, Abbas Nizar Hadi. A new method to tune a fractional order PID controller for a twin rotor aerodynamic system. Arab J Sci Eng 2017;42(12):5179–89, Springer. [25] Ramalakshmi APS, Manoharan PS, Deepamangai P. PID tuning and control for 2DOF helicopter using particle swarm optimization. Lecture Notes in Computer Science, vol. 8297, Springer; 2013, p. 662–72. [26] Belmonte Lidia Maria, Morales Rafael, Caballero Antonio Fernandez, Somolinos Jose Andres. Robust decentralized nonlinear control for a twin rotor MIMO system. Sensors 2016;16(8):1–22. [27] Juang Jih-Gau, Liu Wen-Kai, Lin Ren-Wei. A hybrid intelligent controller for a twin rotor MIMO system and its hardware implementation. ISA Trans 2011;50:609–19, Elsevier. [28] Yeroglu Celaleddin, Ates Abdullah. A stochastic multi-parameters divergence method for online auto-tuning of fractional order PID controllers. J Franklin Inst B 2014;351:2411–29, Elsevier. [29] Juang Jih-Gau, Lin Ren-Wei, Liu Wen-Kai. Comparison of classical control and intelligent control for a MIMO system. Appl Math Comput 2008;205:778–91, Elsevier. [30] Yang Xiaoyan, Cui Jianwei, Lao Dazhong, Li Donghai, Chen Junhui. Input shaping enhanced active disturbance rejection control for twin rotor multi-input multi-output system (TRMS). ISA Trans 2016;62:287–98, Elsevier. [31] Monje Concepcion A, Chen Yang Quan, Vinagre Blas M, Xue Dingyu, Feliu Vicente. Fractional order systems and controls- fundamentals and applications. In: Advances in industrial control (AIC). Springer; 2010. [32] Wolovich William A. Automatic control systems. Saunders College Publishing; 1994. [33] Doyle JC, Francis BA, Tannenbaum AR. Feedback control theory. New York: Macmillan; 1993. [34] Torvik PJ, Bagley RL. On the appearance of the fractional derivative in the behavior of real materials. Trans ASME J Appl Mech 1984;51:294–8. [35] Tepljakov Aleksei, Petlenkov Eduard, Belikov Juri. FOMCON: a MATLAB toolbox for fractional-order system identification and control. Int J Microelectron Comput Sci 2011;2(2):51–62. [36] Chen Yang Quan, Petras Ivo, Xue Dingyu. Fractional order control - a tutorial. In: American control conference (AACC) hyatt regency riverfront. St. Louis, MO, USA; 2009, p. 1397–411.

Please cite this article as: R. Mondal and J. Dey, Fractional Order (FO) Two Degree of Freedom (2-DOF) control of Linear Time Invariant (LTI) plants. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.05.024.