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A Fractional-Order Repetitive Controller for Periodic Disturbance Rejection Giuseppe Fedele Abstract—A fractional-order repetitive controller is here proposed to reject a periodic disturbance acting on a time-invariant linear stable, possibly nonminimum phase, plant. The controller is designed assuming that the frequency response of the plant, in a particular range of frequencies, belongs to a generic half-plane of the complex plane. No information about the structure of the system, i.e., number and locations of zeros/poles, is required. It is shown that, if this half-plane entirely contains the frequency response of the plant, then the controller can be designed without gain constraints. In the presence of saturation nonlinearities, conditions on the system absolute stability are discussed. Index Terms—Fractional-order systems, periodic disturbance, repetitive control.
I. INTRODUCTION Tracking and rejection of periodic exogenous signals are a very attractive challenge in a wide range of engineering applications [1]– [8]. In the case of unknown disturbance acting on known linear plants, several methods have been recently published [3], [9]. In particular, in [3], the periodic exogenous signal is modeled as a linear combination of finite basis functions with unknown coefficients to be estimated. The method can be applied both for minimum or nonminimum phase systems. As a drawback, this method requires knowledge about the input disturbance frequency. Unlike the previous one, in [9], an adaptive servo compensator is proposed to regulate the output of a known plant affected by a disturbance signal formed by known harmonics with unknown frequency. The semiglobal convergence of this method is also proven. Other approaches are described in [10]–[15]. One of the most commonly approaches used in disturbance cancellation problems is the internal model principle (IMP) [16]. According to the IMP, in order to achieve zero tracking error in the steady state, it is necessary and sufficient to include the generators of the reference signal and/or the disturbance signal in the loop, either in the plant itself or in the controller. When the input frequencies are unknown, an adaptive internal model may be used for disturbance rejection [17]. The IMP and its equivalence with several adaptive algorithms are remarked in [18]. In the family of IMP-based methods, remarkable ones consist in the repetitive control (RC) technique. In the case of a sinusoidal signal, RC is structured as a resonant filter; as a consequence, a periodic signal with infinite harmonics can be tracked or rejected using an infinite number of harmonic oscillators. RC can then be viewed as a Manuscript received June 7, 2017; accepted August 28, 2017. Date of publication September 1, 2017; date of current version April 24, 2018. Recommended by Associate Editor F. Mazenc. The author is with the Department of Informatics, Modeling, Electronics and Systems Engineering, University of Calabria, 87036 Rende, Italy (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2017.2748346
, Member, IEEE periodic waveform generator with infinite poles located in multiple integers of the disturbance frequency [19], [20]. Fortunately, a simple delay line in positive feedback configuration can be used to produce an infinite number of poles and so that to simulate an infinite number of harmonic oscillators. In the devoted literature, the concept of RC has been proven to be a useful control strategy for tracking reference signals or attenuating disturbances with known period [21]. Applications of RC to various industrial fields including power electronics systems such as rectifiers, inverters, and active filters are described in [22]–[24]. The approach discussed in this paper uses some results shown in [25], where the case of unknown periodic disturbance acting on a stable linear plant, as in [3] and [9], is analyzed. Moreover, recent results in [26], about the problem of the minimum information required on the plant to compensate a single tone disturbance with known frequency, are used too. The proposed solution faces the periodic disturbance rejection problem in both the cases of known and unknown fundamental frequency. In the spirit of [25] and [26], the method assumes no information about the structure of the plant, i.e., no information about the number and location of poles and zeros is available. Indeed, the new approach no longer requires a plant estimate, providing a robust controller able to deal with a large class of systems regardless of their order, the number of zeros, the relative degree, the nature of the poles, or the minimum/nonminimum phase properties [5]. The previous results are here reconsidered and extended within the framework of the RC theory. In this case, the proposed solution can deal with a periodic disturbance formed by a theoretically infinite number of harmonics multiple of the fundamental one without using, as in [5] and [25], a bank of filters, each of them representing the internal model of the corresponding harmonic to attenuate at the system output. In particular, a fractional-order repetitive controller is designed assuming that the frequency response of the plant, in the range starting from the fundamental frequency of the disturbance signal to infinity, belongs to a generic half-plane, passing through the origin of the complex plane. Under this assumption, the fractional-order controller ensures closed-loop asymptotic stability and disturbance rejection. It is also shown that, if the frequency response of the plant entirely belongs to a half-plane, a controller can be designed with no restrictions on the control gain. In the case of unknown frequency, a frequency estimator based on a second-order generalized integrator is used to provide the fractional-order repetitive controller with the disturbance fundamental frequency [27], [28]. For the class of systems closed-loop stabilizable under a proportional control, based on some previous results published in [26], the author in [29] discusses the use of a fractional-order repetitive controller with the addition of a proportional gain to extend the frequency range in which the method guarantees closed-loop stability and disturbance cancellation. Unlike [29], the present note also faces with the more realistic scenario where the control input is saturated due to the limited action provided by actuators. In this case, the method makes use of a modified repetitive controller with finite gain at the harmonics frequencies. In order to guarantee the absolute stability of the system, a simple condition on the control gain is provided.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 5, MAY 2018
Fig. 1.
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Control scheme.
This paper is organized as follows. The problem formulation is briefly described in Section II. Section III contains some notes about fractional-order derivatives with the proposed fractional-order repetitive controller. The case of unknown disturbance frequency is faced in Section IV. Section V considers the case of a saturated control signal. Some simulated examples are presented in Section VI. The last section is devoted to conclusions.
Fig. 2.
Stability regions in the W -plane.
As an immediate result, it presents the following Laplace transform: II. PROBLEM FORMULATION Consider the scheme reported in Fig. 1. A periodic disturbance d(t) acts on a linear time-invariant Hurwitz, possibly nonminimum phase, N P (s ) . The problem plant defined by the transfer function P (s) = D P (s ) consists in designing the controller C(s), closed in feedback with the system, to remove the effects of the disturbance on the plant output y(t), i.e., y(t) → 0 for t → ∞. Let d(t) be a Tc -periodic disturbance, where Tc = ω2 πc , expressed in Fourier series as d(t) =
Ac k sin (kωc t + φc k )
(1)
k ≥1
with infinite harmonics and unknown parameters (Ac k , ωc , φc k ). Assumption 1: The frequency response of the plant P (s), for all frequencies jkωc , k = 1, 2, ...., belongs to a half-plane of the complex plane, passing through the origin. III. FRACTIONAL-ORDER REPETITIVE CONTROLLER DESIGN The theory of a fractional-order derivative was defined in the 19th century by Riemann and Liouville [30], [31]. Fractional calculus is a generalization of integration and differentiation to noninteger (fractional) order fundamental operators represented as a Dtμ , where a and t are the limits of the operation and μ ∈ R is its order. The continuous integro-differential operator is defined as
μ a Dt =
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩t a
dμ , dtμ 1,
μ>0 μ=0
(2)
(dτ )−μ , μ < 0.
=
1 Γ(n − μ)
t a
f (n ) (τ ) dτ (t − τ )μ −n + 1
where n − 1 < μ ≤ n, n ∈ N, and Γ(·) is the Gamma function.
(3)
n −1
sμ −1 −k f (k ) (0),
k=0
n − 1 < μ ≤ n, n ∈ N
(4)
where only integer-order derivatives of the function f (·) appear. In order to analyze the stability of a fractional-order dynamical system, a new domain, called the fractional-domain (W -domain), has been introduced in [32] by using the transformation w = s1 / q , with q ∈ N + . This transformation can be applied only for rational powers and has the following properties. 1) The ±jω-axes in the s-plane are mapped onto the lines having an angle with respect to the real positive axis equal to |θw | = 2πq . 2) The negative axis of the s-plane is mapped onto a line with θw = πq . 3) The stable and unstable regions transform, respectively, into θw < π and 2πq < |θw | < πq . 2q 4) The region where |θw | > πq is not physical. Therefore, the system will be stable if and only if all roots in the W -plane lie in the region |θw | > 2πq and will oscillate if at least one root lies on the lines |θw | = 2πq , while all other roots are in the stable region. The system will be unstable if there is at least one root in the region |θw | < 2πq (see Fig. 2). The proposed controller r
C(s) = γc
sq 1 − e−s T c
(5)
with q, r ∈ N + , and r ≥ 0, will be designed properly choosing r and q to ensure the closed-loop system stability, as shown in the next theorem. Theorem 1: The transfer function Wd y (s) between the disturbance d(t) and output y(t), i.e., Wd y (s) =
The simplest definition of fractional calculus is represented by the Riemann–Liouville integral [30] even if it suffers for requiring fractional-order initial conditions when solving fractional differential equations. The Caputo definition overcomes such drawbacks, and it is expressed as μ a Dt f (t)
L {0 Dtμ f (t)} = sμ F (s) −
NP (s)(1 − e−s T c ) r
DP (s)(1 − e−s T c ) + γc s q NP (s)
(6)
is stable if the plant P (s) is stable and the following condition holds:
rπ rπ Re[P (jkωc )] cos −Im[P (jkωc )] sin > 0, k = 1, 2, ... 2q 2q (7) with γc sufficiently small. Proof: In order to analyze the stability of the fractional-order sys1 tem Wd y (s), the W -transformation, i.e., w = s q , is applied to (6). In particular, the poles of the system Wd y (s), for s = w q , are obtained
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and
Ξ2 (r, q, P ) = Re[P (jkωc )] cos
Fig. 3.
Stability-preserving movements.
by solving the transformed denominator DP (w q )(1 − e−w
q
Tc
) + γc w r NP (w q ) = 0.
(8)
Since the transfer function P (s) is assumed to be stable, the stable roots of Dp (s) are mapped into the stable region 2πq < θw ≤ πq , i.e., the phase θw is in the stable region, while the imaginary poles s = ±kjωc , for k = 0, 1, ..., are mapped onto the lines |θw | = 2πq , i.e., s = ±jkωc → w = (kωc ) q ej (± 2 q + 1
π
2m π q
)
(9)
with m ∈ N. For γc → 0, the poles are the roots of DP (w q ) = 0 and w = π 2m π 1 (kωc ) q ej (± 2 q + q ) , m = 0, ±1, . . ., as expected. As a consequence, the closed-loop poles will be stable for small γc as long as the poles of the axis |θw | = 2πq move toward the stability region of the complex plane. π 1 The movement of the generic pole at (kωc ) q ej 2 q for small γc is deπ 2m π 1 termined by ∂w/∂γc at γc → 0 and w → w p ≡ (kωc ) q ej ( 2 q + q ) . 1
π
The following analysis holds even for the pole ωcq e−j 2 q due to the symmetry of the generalized fractional root locus of the system. Using (8), the movement ∂w/∂γc is defined as
2πq q −1 q 2 π ∂w qw DP (w q )e−w ω c ωc ∂γc w →w p , γ c →0
+ w r NP (w q ) = 0. (10) w →w p , γ c →0
Therefore, noting that w ¯pq = jkωc for the considered pole, its movement is r+1 ∂w ωc (kωc ) q −1 ξ(r, q, P ) =− (11) ∂γc 2πq w →w p , γ c →0
rπ 2q
− Im[P (jkωc )] sin
rπ . 2q (17)
The proof follows because a necessary and sufficient condition to ensure that the pole moves toward the fractional stability region is the positiveness of the imaginary component of ξ (r, q, P ) [25]. Results stated in Theorem 1 provide a simple way to choose the parameters r and q of the fractional-order controller in order to guarantee the stability of the closed-loop system, even in the case of a rough a priori information on the process. Under these conditions, the attenuation of the input noise is achieved, with a transient depending on the parameter γc . Clearly the previous results hold for small values of γc . Therefore, in all cases where a high value of γc is required to get a faster disturbance cancellation action, the Theorem 1 does not guarantee the closed-loop stability. However, by considering more restrictive hypotheses on the plant, a robust controller can be designed without any restriction on the control gain, as shown in the following result. Theorem 2: Let P (s) be a stable plant satisfying
rπ rπ − Im[P (jω)] sin > 0 ∀ω ≥ 0. (18) Re[P (jω)] cos 2q 2q Then, the fractional-order repetitive controller provides the input u(t) able to cancel the effects of the periodic disturbance on the plant output for any finite γc > 0. Proof: From Theorem 1, Wd y (s) is stable for sufficiently small values of γc , since the poles on the axis |θw | = (π/2q) move toward the stability region of the complex plane. Let w ˆ = ρej (π / 2 q ) be a point with phase equal to (π/2q) and ρ = q ˆ to belong to the positive (kωc ) , k = 0, 1, ... The phase condition for w root locus is
rπ q 2π − arg 1 − ej ρ ω c = (2ν + 1)π, ν = 0, 1, .... arg [P (jρq )]+ 2q (19) Noting that
π π q 2π − < arg 1 − ej ρ ω c < (20) 2 2 and −
π 2
1+
r q
< arg [P (jω)]
0, has an equilibrium point in a neighborhood O(γ) of the input frequency ωc . Moreover, the equilibrium point is asymptotically stable for sufficiently small values of γ. The fractional-order repetitive controller is then used to refine the fundamental frequency estimation reducing the harmonic components at the system output. As a consequence, the input to the OSG-SOGI becomes a sinusoid with the same period of the disturbance. To investigate the signal v(t) sensitivity w.r.t. the inaccurate estimate of the fundamental frequency ωc , consider the transfer function from d(t) to v(t) [19], [20]: ˆ
Wd v (s) =
(1 − e−s T c )DF (s)NP (s) (s2 + ωs2 )DW (s)
(25)
where DW (s) is the denominator of transfer function Wd y (s) and DF (s) = (s2 + Ks ωs s + ωs2 ). From Theorem 3, it follows that, after the transient, the OSG-SOGI resonant frequency ωs converges to a neighborhood of the fundamental one, i.e., ωs = ωc + δω, where δω is O(γ). Therefore, δω can be made reduced by means of an opportune choice of γ. Aiming at ensuring that Wd v (s) behaves as a filter for all frequencies kωc with k ≥ 2, the filter response when Tˆc = 2π/(ωc + δω) and s = jkωc has to be investigated. Substituting these values into (25) and using Taylor expansion with δω → 0, it follows that |Wd v (jkωc )| = c0 + c1 δω + O(δω 2 )
(26)
c0 =
2 |sin(kπ)| (k 2 − 1)2 + k 2 Ks2 r 2 k − 1 γc (kωc ) q
(27)
c1 =
r −1 − r 2k 1 − q πωc q (k 2 − 1)2 + k 2 Ks2 . 2 γc (k − 1)
(28)
Remark 1: It is straightforward to note that other terms in the Taylor expansion, as c0 and c1 , contain k 2 − 1 in the denominator. This means that |Wd v (jkωc )| is infinite only for k = 1 and the OSG-SOGI input v(t) is formed by a single sinusoid at the periodic signal fundamental frequency. It follows that the adaptive regulation makes the resonant frequency converging to the correct value. From Fig. 4, the signals v(t) and y(t) are related by y(s) = (1 − F1 (s)) v(s).
(29)
Taking into account (29), y(t) goes to zero, since v(t) is actually a pure sinusoid at the fundamental frequency ωc and |F1 (jωc )| ≈ 1 for ωs = ωc + δω. V. CASE OF SATURATED INPUT: A QUASI-REPETITIVE CONTROLLER Designing a control system when the actuators are subject to hard constraints is a fundamental problem. As it is well known, actuator limitations are one of the main sources of performance degradation, and they may even cause fatal consequences in several situations. This problem can be faced within the framework of systems consisting of a forward path with a linear time-invariant element and of a feedback path with a sector-bounded nonlinearity. With origins in the work [34], these interconnections are referred to as Lur’e systems, while their stability properties can be studied through the absolute stability theory. Absolute stability theory investigates stability through the interplay of linear component frequency-domain properties and sector data for the nonlinearity [35]. In presence of actuator saturation, the scheme in Fig. 1 can be modified as in Fig. 5. Clearly, if the saturation level is less than the |d(t)|∞ , then the disturbance cancellation cannot be exactly accomplished. In the new control scheme, the stability of the fractional-order repetitive controller in the presence of actuator saturation, where u ˆ(t), |ˆ u(t)| ≤ M (30) u(t) = sat[ˆ u(t)] = sign[ˆ u(t)] M, otherwise has to be analyzed. The considered nonlinearity sat[ˆ u(t)] belongs to the sector [0, 1], since 0 ≤ sat[ˆ u(t)] ≤ u ˆ(t). The fractional-order repetitive controller transfer function has an infinite number of poles on the axis of phase π/(2q) for frequencies ω = k 2Tˆπ , k = 0, 1, .... In these frequencies, the magnitude of the c denominator of C(s) is zero, making the gain of the transfer function infinite. Instability problems may then arise. To overcome this issue ensuring the system bounded-input bounded-output (BIBO) stability, the previously proposed repetitive controller can be modified by adding
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ˆ The poles of C(w) can be obtained by solving wq = −
log(1 + ) . Tc
(35)
Let w = ρej θ ; then, it follows that
1 log(1 + ) q ρ= Tc
(36)
and Fig. 6.
θ=
Fractional-order quasi-repetitive controller.
Fig. 7. Magnitude frequency response of the FO-QRC for T c = 1 and r/q = 0.
π + 2kπ , k = 0, 1, .... q
(37)
ˆ The argument of poles is greater than 2πq ; as a consequence, C(s) is BIBO stable, and then, it belongs to L1 . ˆ is bounded, Moreover, the L2 -norm of the impulse response of C(w) since the difference between the denominator degree and numerator degree is greater than 12 [36]. The global stability properties of the system in Fig. 5 can be analyzed, for example, by considering frequency criteria for forced systems. In particular, if the impulse response of W (s) satisfies the necessary conditions of being in L1 and L2 , then the traditional absolute stability criteria in the frequency domain [37] can be generalized to the related fractional-order Lur’e system. Therefore, the following result holds. Theorem 4: If the system in Fig. 5 satisfies the conditions: ˆ 1) the system W (s) = P (s)C(s) is in L1 and L2 ; 2) the nonlinearity belongs to the sector [0, 1]; 3) the Nyquist plot of W (jω) stays in the half-plane Re[s] > −1; then the equilibrium point 0 of the system is globally asymptotically stable. In particular, since the forcing signal d(t) is Tc -periodic, then the signal u(t) is also periodic with the same period [38]. Theorem 5: Consider the system in Fig. 5 and assume that the following conditions are satisfied: r
the gain as in Fig. 6. In this case, the transfer function from y(t) to u ˆ(t) is ˆ C(s) = γc
s 1−
r q
e−s T c
+
.
|P (jω)|ω q < ∞ ∀ω ≥ 0 and γc < min
(31)
ˆ In ω = k 2Tπc , the magnitude of the denominator of C(s) is equal to . Clearly, when goes to zero, the transfer function gain is infinite, obtaining the previous repetitive controller, the principle of which is to have infinite loop gain at the harmonics of the disturbance. For = 0, it is possible to choose the (finite) gain at the harmonics frequencies, from which the term fractional-order quasi-repetitive controller (FOQRC) (see Fig. 7, where the resonant frequency is set with Tc = 1 and r/q = 0). Definition 1: L1 is the set of absolutely integrable functions on R: ∞ + |u(t)|dt < ∞ . (32) L1 = u(t) : R → R,
ω ≥0
wr . 1 + − e−w q T c
.
(39)
r
2 + (2 + ) − 2(1 + ) cos Since
2 + (2 + ) − 2(1 + ) cos
2πω ωc
2π ω ωc
.
(40)
≥ 2
(41)
then r
γc |P (jω)|ω q .
By using Theorem 4, the proof follows if |W (jω)| < 1.
(42)
VI. SIMULATION RESULTS
Proposition 1: The FO-QRC transfer function is in both L1 and L2 . 1 Proof: Consider the W -transformation with w = s q ; then ˆ C(w) =
r
|P (jω)|ω q
|W (jω)| = γc
|W (jω)| ≤ (33)
|P (jω)|ω q
Then, the system is globally asymptotically stable. Proof: The magnitude of W (jω) is
0
L2 is the set of square-integrable functions on R:
12 ∞ L2 = u(t) : R+ → R, u(t)2 dt
