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Different architectures to develop repetitive controllers
Proceedings of the 20th World Congress Proceedings of the the 20th World World Congress The International Federation of Congress Automatic Control Proceedings of 20th Proceedings of the 20th9-14, World The Federation of Automatic Control Toulouse, France, July 2017 The International International Federation of Congress Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 13408–13413
Different architectures to develop repetitive to develop repetitive Different Different architectures architectures to develop repetitive controllers controllers controllers ∗ ∗∗
V´ıctor Sanz i L´ opez ∗ Ramon Costa-Castell´ o ∗∗ ∗∗∗ V´ıctor ıctor Sanz Sanz German L´ opez pez ∗∗ A. Ramon Costa-Castell´ o ∗∗ V´ ii L´ o Ramon Costa-Castell´ o Ramos ∗∗∗ V´ıctor Sanz German i L´ opez A. Ramon Costa-Castell´ o ∗∗ ∗∗∗ Ramos German A. Ramos ∗∗∗ German A. Ramos ∗ Institut de Rob` otica i Inform` atica Industrial, CSIC-UPC Llorens i ∗ ∗ Institut de Rob` o ii Inform` a Industrial, Llorens Rob` otica tica Barcelona, Inform` atica tica Industrial, CSIC-UPC Llorens ii Artigasde 4-6, 08028 Spain. Email: CSIC-UPC [email protected] ∗ Institut Rob` otica Barcelona, i Inform` atica Industrial, CSIC-UPC Llorens i ∗∗ Institut Artigasde 4-6, 08028 Spain. Email: [email protected] Artigas 4-6, 08028 Barcelona, Spain. Email: [email protected] Universitat Polit` e cnica de Catalunya; Barcelona, Catalunya, Spain. ∗∗ Artigas 4-6, 08028 Barcelona, Spain. Email: [email protected] ∗∗ Universitat Polit` e cnica de Catalunya; Barcelona, Catalunya, Spain. Universitat Polit` e cnica de Catalunya; Barcelona, Catalunya, Spain. Email: [email protected] ∗∗ Universitat Polit`eEmail: cnica [email protected] Catalunya; Barcelona, Catalunya, Spain. ∗∗∗ Email: [email protected] Universidad Nacional de Colombia; Bogot´ a , Colombia. Email: ∗∗∗ Email: [email protected] ∗∗∗ Universidad Nacional de Colombia; Colombia; Bogot´ Bogot´ a,, Colombia. Colombia. Email: Email: de a [email protected] ∗∗∗ Universidad Nacional Universidad Nacional de Colombia; Bogot´ a, Colombia. Email: [email protected] [email protected] [email protected] Abstract: In this work several architectures to implement repetitive controllers are compared. Abstract: In analytical this work work analysis several architectures architectures to implement repetitive controllers are compared. Abstract: In this several repetitive controllers compared. A complete is performed to forimplement these different architectures andare a simulation Abstract: In this work analysis several architectures to implement repetitive controllers are compared. A complete A complete analytical analysis is isisperformed performed for these these different different architectures architectures and and aa simulation simulation example withanalytical a power converter included. for A complete analytical analysis is performed for these different architectures and a simulation example example with with a a power power converter converter is is included. included. example with(International a power converter is included. © 2017, IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Repetitive Control, Disturbance observer, Youla parametrization, inverter control Keywords: Keywords: Repetitive Repetitive Control, Control, Disturbance Disturbance observer, observer, Youla Youla parametrization, parametrization, inverter inverter control control Keywords: Repetitive Control, Disturbance observer, Youla parametrization, inverter control 1. INTRODUCTION example is presented and finally section 8 provides 1. example presented and section 1. INTRODUCTION INTRODUCTION example isabout presented and finally finally section 88 provides provides discussionis the different approaches. 1. INTRODUCTION example isabout presented and finally section 8 provides discussion the different approaches. discussion about the different approaches. discussion about the different approaches. Repetitive Control (RC) (Wang et al., 2009; Longman, Repetitive (RC) (Wang et 2009; Repetitive Control (RC) Ahn (Wang et al., al., 2009; Longman, 2010; ChenControl et al., 2008; et al., 2007) is anLongman, Internal Repetitive Control (RC) (Wang et al., 2009; Longman, 2010; et 2008; et 2007) is 2010; Chen et al., al., (IMP) 2008; Ahn Ahn et al., al.,and 2007) is an an Internal Internal ModelChen Principle (Francis Wonham, 1976; 2010; Chen et al., (IMP) 2008; Ahn et al.,and 2007) is an Internal Model Principle (Francis Model Principle (IMP) (Francis and Wonham, Wonham, 1976; Costa-Castell´ o et al., 2012) based control technique.1976; RC Model Principle al., (IMP) (Francis and Wonham, 1976; Costa-Castell´ based RC Costa-Castell´ et al., 2012) 2012) based control control technique. RC it is speciallyoo et suited for systems subjecttechnique. to periodical Costa-Castell´ o et al., 2012)systems based control technique. RC it is suited to it is specially specially suited for for systems subject to periodical periodical references and disturbances (Ramossubject et al., 2013). During it is specially suited for systems subject to periodical references and disturbances (Ramos et 2013). references andtechnique disturbances (Ramos et al., al., 2013). During During last years this has been applied in different fields references andtechnique disturbances (Ramos et al., different 2013). During last years has applied fields last years this this technique has been been applied inordifferent fields like power electronics (Ramos et al., 2013)in mechatronic last years this technique has been applied in different fields like power electronics (Ramos et 2013) like power electronics (Ramos et al., al.,others. 2013) or or mechatronic mechatronic systems (Park et al., 2005) among like power electronics (Ramos et al., 2013) or mechatronic systems systems (Park (Park et et al., al., 2005) 2005) among among others. others. systems et al., others. The idea(Park behind RC2005) is toamong include inside the controlThe idea behind RC is to include inside the The ideagenerator behind RC to include inside the iscontrolcontrolloop the of a isperiodical signal, which a high The ideagenerator behind RC isperiodical to include inside the iscontrolloop the of a signal, which loop the generator of a periodical signal, which is aa high high order dynamic systems. Applying conventional stabilizing loop the generator of a periodical signal, whichstabilizing is a high order dynamic systems. Applying order dynamic systems. Applying conventional techniques to this type of systemsconventional would implystabilizing obtaining order dynamic systems. of Applying conventional stabilizing techniques this would imply obtaining techniques to this type type of systems systems would imply obtaining very high to order controllers which would entail huge techniques to this type of systems would imply obtaining very high order controllers which would entail huge very high order controllers which would huge computational resources to implement these entail controllers. very high order controllers which would entail huge computational to implement controllers. computational resources to architectures implement these these controllers. Due to this, RC resources uses specific and anti-windup computational resources to implement these controllers. Due to RC specific architectures and anti-windup Due to this, this,(Ramos RC uses usesand specific architectures and Wang, anti-windup techniques Costa-Castell´ o, 2013; 2016) Due to this,(Ramos RC usesand specific architectures and Wang, anti-windup techniques Costa-Castell´ oo,, 2013; techniques (Ramos andprofit Costa-Castell´ 2013; Wang, 2016) 2016) which allow to take from the steady-state nice techniques (Ramos and Costa-Castell´ o, 2013; Wang, 2016) which allow the nice which allow to take profit from the steady-state steady-state nice properties of to RCtake and profit obtainfrom implementation structures which allow to take profit from the steady-state nice properties RC obtain structures properties of reducing RC and and computational obtain implementation implementation structures which allowof requirements to a properties of RC and obtain implementation structures which allow which allow reducing reducing computational computational requirements requirements to to aa minimum. which allow reducing computational requirements to a minimum. minimum. minimum. Since the seminal work by Inoue et al. (1982), which introSince by al. which Since the seminal work by Inoue Inoue etRC, al. (1982), (1982), which introintroducedthe theseminal plug-in work architecture foret many architectures Since the seminal work by Inoue etRC, al. (1982),architectures which introduced the architecture for duced the plug-in plug-in architecture forinput-output RC, many many architectures have been proposed, both in the (Chen and duced the plug-in architecture for RC, many architectures have been both in (Chen have been proposed, proposed, both in the the input-output input-output (Chen and Tomizuka, 2014, 2015) and state-space (Wu et and al., have been proposed, both in the input-output (Chen and Tomizuka, 2014, state-space (Wu Tomizuka, 2014, 2015) 2015) and the state-space (Wu et et al., al., 2014) formalisms. In thisand workthe most relevant input-output Tomizuka, 2014, 2015) and the state-space (Wu et al., 2014) this work input-output 2014) formalisms. Inare thisreviewed work most most relevant input-output based formalisms. architecturesIn andrelevant compared. The com2014) formalisms. Inare thisreviewed work most relevant input-output based architectures and compared. The comcombased and The parisonarchitectures is illustratedare in reviewed the case of a compared. power inverter. based architectures are reviewed and compared. The comparison is illustrated in the case of a power inverter. parison is illustrated in the case of a power inverter. parison is illustrated in theascase of a power This paper is organized follows: sectioninverter. 2 describes This is as section 22 describes This paper is organized organized as follows: follows: section describes most paper relevant concepts about internal models used in This paper is organized as follows: section 2 describes most about internal used most relevant concepts about internal models models used in in RC, inrelevant section concepts 3 the series architecture is described, most relevant concepts about internal models used in RC, in 33 the series architecture is RC, in 4section section the structure series architecture is described, described, in5 section the plug-in is introduced, in section in RC, in 4section 3 the structure series architecture is described, in section is is introduced, in section section section 4 the the plug-in plug-in structure is introduced, in a disturbance observer approach used, in section 6 the55 section 4 the plug-in structure is is introduced, in section 5 aacontroller disturbance observer in 66 the disturbance observer approach approach used, in 7section section the parametrization is used,isinused, section a numerical a disturbance observer approach is used, in section 6 the controller controller parametrization parametrization is is used, used, in in section section 77 aa numerical numerical controller parametrization is used, in section 7 a numerical
2. 2. 2. 2.
INTERNAL INTERNAL INTERNAL INTERNAL
some some some some
MODEL MODEL MODEL MODEL
I(z) I(z) I(z) I(z)
E(z) + E(z) + E(z) + E(z) +
+ + + +
σ · W (z) σ ·· W W (z) (z) σ σ · W (z)
H(z) H(z) H(z) H(z)
Y (z) Y (z) (z) Y Y (z)
Figure 1. Internal model used in Repetitive Control. Figure Figure 1. 1. Internal Internal model model used used in in Repetitive Repetitive Control. Control. Figure 1. Internal model used in Repetitive Control. The internal model is the core element of a RC system The internal model is core aa RC system The internal model by is the the core element element ofcan RC systema and it is composed an element whichof generate The internal model by is the core element ofcan a RC systema and it is composed an element which generate and it is signal. composed by 2anshows element canofgenerate a periodic Figure the which structure a generic and it is composed by an element which can generate a periodic signal.used Figure shows the the structure structure of of a a generic generic periodic signal. Figure 22 shows internal model in RC: periodic signal. Figure 2 shows the structure of a generic internal model model used used in in RC: RC:σH(z)W (z) internal internal model used I(z)in=RC:σH(z)W σH(z)W (z) (z) , σH(z)W I(z) = = 1− σH(z)W (z)(z) ,, I(z) I(z) = 11 − − σH(z)W σH(z)W (z) (z) , where σ = {−1, 1}, W (z) the time(z) delay function, and 1 −is σH(z)W where σ = {−1, 1}, W (z) is the time delay where is σ= 1}, W (z) isAs thean time delay function, function, and H(z) a {−1, low-pass filter. example, for σ =and 1, where is σ= {−1, 1}, W (z) isAs thean time delay function, and −N H(z) a low-pass filter. example, σ H(z) low-pass filter. = As 1ana example, forgenerator, σ = = 1, 1, W (z) is= az −N and H(z) N-periodicfor H(z) is a1z −N low-pass filter. = As an example, forgenerator, σ =− 1, N W (z) and H(z) N-periodic W (z)= = = N z −N and H(z) = 11foraa σN-periodic generator, 2 I(z) , is obtained and = −1, W (z) = z N W (z) =z 11z−1 and H(z) = 1 a N-periodic generator, − −N 2 I(z) = ,, 1,is obtained and = (z) zzand 2 I(z) H(z) = zzN is the obtained and for for σ σgenerator = −1, −1, W W(Gri˜ (z) = = −1 N1−1 −N and = odd-harmonic n o ´ I(z) = zN −1 , is obtained and for σ = −1,−1W (z) = z 2 and H(z) = 1, the odd-harmonic generator (Gri˜ n o ´ and and H(z) = o1,, 2005) the odd-harmonic generator o and Costa-Castell´ is obtained I(z) = N−1 (Gri˜ . n´ and H(z) = 1, the odd-harmonic generator o and 2 +1(Gri˜ −1 Costa-Castell´ .. n´ Costa-Castell´oo,, 2005) 2005) is is obtained obtained I(z) I(z) = = zz N N −1 +1 . Costa-Castell´o, 2005) is obtained I(z) = z N22 +1 For those internal models obtained with H(z) = 1, z 2 +1 For internal those internal internal models obtained infinite with H(z) H(z) = the 1, For those obtained with = 1, the model, models I(z), introduces gain at For internal those internal modelsintroduces obtained with H(z) = 1, the model, gain at the internal model, I(z), infinite gain gain at the the signal frequency andI(z), all itsintroduces harmonics.infinite This high at the internal model, I(z), introduces infinite gain gain at the signal frequency all its signal frequency and and all be its aharmonics. harmonics. This high gain at at high frequencies might problem inThis the high presence of signal frequency and all its harmonics. This high gain at high frequencies frequencies mightthis be gain problem in the the presence of high might be aa problem in presence uncertainty. To reduce a low-pass filter, H(z), of is high frequencies mightthis be a problem in the presence of uncertainty. aa low-pass H(z), is uncertainty. To reduce this gain gain low-pass filter, H(z), is usually used.To A reduce null-phase low pass filter isfilter, usually used uncertainty. To reduce this gain a low-pass filter, H(z), is usually used. A null-phase low pass filter is usually used usually used. A null-phase filter et is al., usually used (Gri˜ n´o and Costa-Castell´ o, low 2005;pass Escobar 2014). usually used. A null-phase low pass filter is usually used (Gri˜ n´o´o and and Costa-Castell´ Costa-Castell´oo,, 2005; 2005; Escobar Escobar et et al., al., 2014). 2014). (Gri˜ n (Gri˜ n´o and Costa-Castell´o, 2005; Escobar et al., 2014).
Copyright © 2017, 2017 IFAC 13950 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 2017 IFAC 13950 Copyright ©under 2017 responsibility IFAC 13950 Peer review© of International Federation of Automatic Control. Copyright © 2017 IFAC 13950 10.1016/j.ifacol.2017.08.2282
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Víctor Sanz i López et al. / IFAC PapersOnLine 50-1 (2017) 13408–13413
I(z)
+
R(z) +
σ · W (z)
+
−
G(z)
Gc (z)
H(z)
Y (z)
3. SERIES APPROACH The first approach to RC consists in placing the internal model in series connection with the plant. Additionally a stabilizing controller needs to be introduced to guarantee closed-loop stability (Figure 2). In this case the controller becomes: C(z) = I(z)Gc (z) the complementary sensitivity and sensitivity function become: I(z)Gc (z)G(z) T (z) = 1 + I(z)Gc (z)G(z) 1 . S(z) = 1 + I(z)Gc (z)G(z) Usually, for minimum-phase plants kr Gc (z) = G(z) and :
(1)
kr σW (z)H(z) T (z) = 1 + (kr − 1)σW (z)H(z) 1 − σW (z)H(z) . S(z) = 1 + (kr − 1)σW (z)H(z) As expected, S(z) has in the numerator the denominator of I(z). For non-minimum phase plants it is necessary to use phase cancellation techniques (Ye et al., 2009; Tomizuka, 1987) when designing Gc (z). In case of systems with multiplicative uncertainty: G(z) = anchez-Pe˜ na and Sznaier, 1998), Gn (1 + Wum (z)∆(z)) (S´ the robust stability condition is: m kr σW (z)H(z) Wu (z) . 1 + (kr − 1)σW (z)H(z) ∞
4. PLUG-IN APPROACH
C(z) I(z)
σW (z)
R(z) +
H(z)
Gx (z)
+
−
this internal controller is to guarantee closed-loop stability and robustness to the control system without the internal model. Later, the internal model I(z), and the stabilizing controller, Gx (z), are plugged in to the previous closedloop system. In this case the closed-loop transfer function can be constructed in terms of the closed-loop transfer function without the internal model:
Figure 2. RC : Series approach.
+
13409
+
+ Gc (z)
G(z)
Y (z)
Figure 3. RC : Plug-in approach. The most popular approach in RC is the plug-in architecture (Figure 3). This architecture introduces the internal model to a previously existing control system defined by an internal controller Gc (z) and the plant G(z). The goal of
1 Gc (z)G(z) , To (z) = 1 + Gc (z)G(z) 1 + Gc (z)G(z) and a modifying term: So (z) =
1 − σW (z)H(z) . 1 − σW (z)H(z) (1 − Gx (z)To (z)) So the closed-loop transfer functions are: SM od (z) =
S(z) = So (z)SM od (z) (1 − σW (z)H(z) (1 − Gx (z))) To (z) T (z) = . 1 − σW (z)H(z) (1 − Gx (z)To (z))
For minimum-phase plants 1 the most popular form of for the stabilizing controller is: kr . (2) Gx (z) = To (z) With this selection the closed-loop function becomes: 1 − σW (z)H(z) 1 + (kr − 1) σW (z)H(z) (To (z) − σW (z)H(z) (To (z) − kr )) . T (z) = 1 + (kr − 1) σW (z)H(z) S(z) = So (z)
The following two conditions guarantee closed-loop stability (Ramos et al., 2013): (1) To (z) must be stable (Gc (z) can be designed to fullfil it) (2) W (z)H(z) (1 − kr ) ∞ < 1 (kr can be selected appropriately ) Even thought these conditions are only sufficient it has been proved that they are close to the necessary ones in practice (Songschon and Longman, 2003). It is important to emphasize that in (2) the inversion of To (z) is required while in (1) it is required the inversion of G(z), as To (z) is a closed-loop system its uncertainty should be less than that of G(z). Additionally, it is important to visualize that the sensitivity function in the series approach and the plug-in one are the same except the So (z) term. In case of multiplicative uncertainty the robust stability condition becomes: m (To (z) − σW (z)H(z) (To (z) − kr )) Wu (z) < 1. 1 + (kr − 1) σW (z)H(z) ∞ This condition is quite similar to the one obtained in the series approach but it contains To (z). This term can be shaped using Gc (z), so it is simpler to fulfill this constrain than the one obtained in the series approach. 1 For nonminimum phase plants a phase cancellation approach is usually used
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5. DISTURBANCE REJECTION APPROACH
degrees of freedom they might increase the controller complexity.
C(z) z −m G(z)−1
+
+
6. YOULA PARAMETRIZATION
z −m
C(z)
Q(z)
R(z) +
E(z)
Gc (z)
+
+
+
D(z) + G(z)
R(z) +
Y (z)
−
−
+
F (z)
G(z)
Y (z)
+ G(z)
Figure 4. RC : Disturbance rejection approach. During last years a great effort in the disturbance rejection mechanisms has been made (Chen et al., 2016). In this context a RC has been proposed based on this methodology (Chen and Tomizuka, 2014) and its characteristics are shown in this section. The controller scheme for an m-relative degree minimum phase plant, G(z), is shown in Figure 4. This system uses a baseline controller, Gc (z), plus a disturbance observer composed by the plant model and a filter, Q(z). The sensitivity function for this closed-loop system is: 1 − z −m Q(z) S(z) = . 1 + G(z)Gc (z) In order to transform this in a RC it is necessary to choose Q(z) appropriately, so that the denominator of the internal model, I(z), appears in the numerator of S(z). Consequently, Q(z) must be isolated from the equation: Ns (z) 1 − z −m Q(z) = So (z) (1 − σW (z)H(z)) 1 + G(z)Gc (z) Ds (z) where Ns (z) and Ds (z) are polynomials that must be fixed (this selection requires So (z) to be stable, which can be achieved by selecting and appropriate Gc (z)). So: (1 − σW (z)H(z)) Ns (z) − Ds (z) . Q(z) = z m Ds (z) A simple option is choosing Ns (z) = 1 and Ds (z) = 1 − α · σ · W (z)H(z) with |α| < 1 which generates: (−(1 + α) · σW (z)H(z)) Ns (z) . Q(z) = z m 1 − α · σ · W (z)H(z) Although this is not the conventional shape of Q(z) in disturbance observer based control it allows to reject periodical signals and behave as RC. Finally the controller becomes: Gc (z) + Q(z)z m G(z)−1 C(z) = 1 − z −m Q(z) This controller has a degree of freedom which corresponds to the value of α. This value plays a similar role to kr in the plug-in approach. In case of multiplicative uncertainty the robust stability condition becomes: Wum (z)T (z) ∞ < 1, with 1 − σW (z)H(z) T (z) = 1 − So (z) . 1 − α · σW (z)H(z) Clearly, other selections for Ns (z) and Ds (z) are possible. Although these other options might provide additional
Figure 5. RC : Youla parametrization. It is well-known that all stabilizing controllers for a given stable plant, G(z), can be written in terms of the Youla parametrization (S´anchez-Pe˜ na and Sznaier, 1998) shown in Figure 5. In this case, the sensitivity and complementary sensitivity functions are: S(z) = 1 − F (z)G(z), T (z) = F (z)G(z) where F (z) is a stable system. And the controller is (z) C(z) = 1−FF(z)G(z) . In case of minimum-phase plants it is possible to select F (z) = F (z)G−1 (z), so the closed-loop functions become: S(z) = 1 − F (z), T (z) = F (z). In order to impose that the controller behaves as a RC it is necessary to impose the appropriate shape for F (z). It is necessary that: Ns (z) S(z) = 1 − F (z) = (1 − σW (z)H(z)) Ds (z) with Ns (z) and Ds (z) arbitrary elements. So Ds (z) − (1 − σW (z)H(z)) Ns (z) . F (z) = Ds (z) A simple solution is Ns (z) = 1 and Ds (z) = 1 − α · σ · W (z)H(z) with |α| < 1, which generates: (1 − α) · σ · W (z)H(z) . F (z) = 1 − α · σ · W (z)H(z) This option generates the following closed-loop transfer functions: (1 − α) · σ · W (z)H(z) 1 − α · σ · W (z)H(z) 1 − σ · W (z)H(z) . S(s) = 1 − α · σ · W (z)H(z) Which can be considered a generalized version of the series approach (section 3). T (z) =
Finally, robust stability condition for multiplicative uncertainty is : Wum (z)T (z) ∞ < 1. Clearly, this approach can generate more complex closedloop transfer function, in particular the closed-loop poles
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d
d¯ if
L
io
+ +
Vdc
+
vf _
_
d¯
13411
C L
nonlinear load
vo _
d
PWM
d
voltage controller
vref
Figure 7. Sensitivity function for RC based on plug-in and disturbances observer approaches.
Figure 6. Circuit diagram. could be arbitrarily placed by choosing and appropriate value for Ds (z). Alternatively, these degrees of freedom could be used to optimize any criteria such as robustness. This increase in the controller complexity would increase the computational resources required to implement the controller. 7. NUMERICAL EXAMPLE The numerical example is based on the system depicted in Figure 6. It consists of a Voltage Source Inverter (VSI) which transforms a DC supply, Vdc , in an AC power source, vo . The duty cycle, d, of a Pulse Width Modulation (PWM) signal is the control action used to switch the power transistors and the produced signal is filtered by an LC network to obtain the output AC voltage vo (t). The control objective is providing a sinusoidal waveform voltage signal with specific amplitude and frequency (vref (t)). At the same time, it is needed to reject disturbances caused by the load. Nonlinear loads, as the one produced by full-bridge diode rectifiers, are of special interest since they produce disturbances with high harmonic content. A way of measuring the system performance is through the Total Harmonic Distortion (THD), thus a low THD in voltage waveform (vo (t)) is desirable.
Additionally, the control action is Vf (s) = (2d − 1)Vdc , where d ∈ [0, 1] is the duty cycle of the PWM signal and Vdc is the DC input source. The PWM signals has a frequency of 10KHz, and the duty cycle is updated each period, so a sampling time of Ts = 0.0001s is used.
The system transfer function is based on the LC filter circuit equations to obtain: (3) Vo (s) = Gp (s)Vf (s) + Gd (s)Io (s), with RC , Gp (s) = 2LCRC s2 + (2CRL RC + 2L)s + (2RL + RC ) (4) and 2LRC + 2RL + RC , Gd (s) = − 2 2LCRC s + (2CRL RC + 2L)s + (2RL + RC ) (5) where Gp (s) is the plant transfer function, Gd (s)Io (s) is the disturbance signal caused by the load, L = 300 µH and C = 80 µF are the inductive and capacitive part of the filter, RL = 0.1 Ω and RC = 8200 Ω are the parasitic resistance of inductance and capacitance respectively.
Although the inverter is designed to feed generic loads, it is difficult to design a control system without some assumptions about the concrete load being used. The load has been modeled by means of an inductance and a resistor in series placed in parallel with the capacitor. For this reason, an impedance Z(s) to relate the current introduced to the load Io (s) and the voltage supplied to it Vo (s). This impedance is as can be seen in equations (6) and (7): (6) Io (s) = Z(s)Vo (s) 1 (7) Z(s) = L · s + RL Taking this into account the nominal model is constructed as: Gp (s) Gn (s) = . 1 + Gd (s)Z(s) This model will be the one used to design the controller.
Figure 8. Sensitivity function for RC based on series and Youla parametrization approaches.
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closed-loop poles. In fact, a relation between kr and α can be established, α = 1 − kr , so that the closed-loop poles from different architectures are placed in the same location. For consistency with previous cases, α = 0.3 is choosen. An element present in all these architectures is the lowpass filter. This element’s most relevant goal is introducing robustness in the high frequency range. In this work the following filter has been used: 0.25z 2 + 0.5z + 0.25 , (8) H(z) = z it has null-phase, a gain close to 1 in low and medium frequency range and a important attenuation in the high frequency range. Figure 9. Bode diagram of 1/T (z) for all four approaches.
Figure 10. Bode diagram of 1/T (z) for all four approaches.
Finally, the plug-in approach (section 4) and disturbance rejection one (section 5) contain an internal controller in charge of providing robustness. To prevent increasing the complexity and allowing a better comparison a proportional controller like the following one has been selected: Gc (z) = 0.1. (9) Taking this into consideration and noting that the discretetime plant obtained when applying the z-transform to Gn (s) is minimum-phase plant, all the controllers introduced in this work are completely defined. Figure 7 shows the frequency response of the sensitivity function for the plug-in approach and the disturbance observer one. As it can be seen, both responses are similar, have small values for the working frequency and all the harmonics. The effect of the filter can also be seen, the small values at harmonics increase as frequency increases. Additionally, as the sensitivity function does not take values over 6dB it can be stated that both of them are quite robust. Figure 8 shows the sensitivity function for the series and Youla architectures. As it can be seen, both schemes provide the same result and it is quite similar to the one observed in Figure 7.
The goal in this type of system is feeding the load with a sinusoidal voltage of frequency f = 50 Hz and amplitude √ 220 2 V. Consequently, the period signal of to be tracked 1 is Tp = 50 = 0.02s. Consequently, the discrete time period, N , can be computed as N = Tp /Ts = 200. This allows to obtain a good reconstruction of the continuous time signal.
Figure 9 shows the closed-loop time response for all described architectures when a sinusoidal reference of the working frequency is introduced in the system. As it can be seen, after a small transient all closed-loop schemes converge to the reference with null-steady state error. No relevant differences can be observed.
In the following, 4 different RC systems, corresponding to the previously introduced architectures will be designed. In each approach different parameters have been tuned.
Finally, the attention is turned to robustness eases. In this type of application, the real load is not know. A way to analyze the robustness of the control system against changes in the load characteristics is to model the variation in the load as a multiplicative uncertainty Wum (z) in the plant and perform a robust stability analysis. The robust stability condition is Wum (z)T (z) ∞ < 1, which can be analyzed frequency by frequency as : 1 |Wum (ejωTs )| < . (10) |T (ejωTs )| Figure 10, shows the frequency response of T (z)−1 . As it can be seen it is always over 0dB in all the frequency range, and specially in the high frequency range. Consequently, a 100% change in the impedance at each frequency can be handled. It can be stated that the closed-loop system is very robust. Additionally, it is possible to see that all architectures provide a similar robustness.
In the series (section 3) and the plug-in (section 4) approaches, kr is the most relevant degree of freedom. It is related with the most relevant closed-loop poles. Approximately, these are the solutions of z N = 1 − kr . These poles are homogeneously distributed over a circle of radius |1 − kr |, which directly defines the closed-loop settling time (Yeol et al., 2008; Garimella and Srinivasan, 1994). Due to this, the selection of kr is a trade-off between settling time and robustness. A reasonable value is kr = 0.7. In the disturbance rejection (section 5) and the Youla parametrization (section 6) a parameter called α is introduced. Similarly to kr , α is directly related with the
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8. CONCLUSIONS In this paper most relevant architectures used to implement RC in input-output form have been reviewed and analyzed for the case of minimum-phase plants and its applications to the case of an inverter has been illustrated. As it can be shown, most architectures achieve similar results. Obtaining good steady-state results and reasonably robustness conditions. Although it has not been shown in this work, the plug-in approach and the disturbance observer contains an additional degree of freedom which can be used to slightly improve robustness and transient behavior. Youla parametrization is the most generic by far because it can be used to reduce all the other approaches. Finding new approaches based on Youla parametrization which improve transient and robustness with a limited complexity increase is a current research area. ACKNOWLEDGEMENTS This work was partially supported by the spanish Ministerio de Educaci´ on project DPI2015-69286-C3-2-R (MINECO/FEDER) and the catalan AGAUR project 2014 SGR 267. REFERENCES Ahn, H.S., Chen, Y.Q., and Moore, K.L. (2007). Iterative Learning Control: Brief Survey and Categorization. IEEE Transactions on Systems Man & Cybernetics Part C, 37(6), 1099–1121. Chen, W.H., Yang, J., Guo, L., and Li, S. (2016). Disturbance-observer-based control and related methods : An overview. IEEE Transactions on Industrial Electronics, 63(2), 1083–1095. doi: 10.1109/TIE.2015.2478397. Chen, X. and Tomizuka, M. (2014). New repetitive control with improved steady-state performance and accelerated transient. IEEE Transactions on Control Systems Technology, 22(2), 664–675. doi: 10.1109/TCST.2013.2253102. Chen, X. and Tomizuka, M. (2015). Overview and new results in disturbance observer based adaptive vibration rejection with application to advanced manufacturing. International Journal of Adaptive Control and Signal Processing, 29(11), 1459–1474. doi:10.1002/acs.2546. Acs.2546. Chen, Y., Moore, K.L., Yu, J., and Zhang, T. (2008). Iterative learning control and repetitive control in hard disk drive industry—a tutorial. International Journal of Adaptive Control and Signal Processing, 22(4), 325–343. doi:10.1002/acs.1003. Costa-Castell´ o, R., Olm, J.M., Vargas, H., and Ramos., G.A. (2012). An educational approach to the internal model principle for periodic signals. International Journal of Innovative Computing, Information and Control, 8(8), 5591–5606. Escobar, G., Mattavelli, P., Hernandez-Gomez, M., and Martinez-Rodriguez, P.R. (2014). Filters with linearphase properties for repetitive feedback. IEEE Transactions on Industrial Electronics, 61(1), 405–413. doi: 10.1109/TIE.2013.2240634. Francis, B. and Wonham, W. (1976). The internal model principle of control theory. Automatica, 12(5), 457 – 465. doi:10.1016/0005-1098(76)90006-6.
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Garimella, S. and Srinivasan, K. (1994). Transient response of repetitive control systems. In Proceedings of the American Control Conference, 2909–2913. Baltimore. Gri˜ n´o, R. and Costa-Castell´o, R. (2005). Digital repetitive plug-in controller for odd-harmonic periodic references and disturbances. Automatica, 41(1), 153 – 157. doi: http://dx.doi.org/10.1016/j.automatica.2004.08.006. Inoue, T., Nakano, M., Kubo, T., Matsumoto, S., and Baba, H. (1982). High Accuracy control of a proton syncrotron magnet powe supply, 3137–3142. IFAC by Pergamon Press. Longman, R.W. (2010). Iterative learning control and repetitive control for engineering practice. International Journal of Control, 73(10), 930–954. doi: 10.1080/002071700405905. Park, S.w., Jeong, J., Yang, H.S., Park, Y.p., and Park, N.c. (2005). Repetitive controller design for minimum track misregistration in hard disk drives. IEEE Transactions on Magnetics, 41(9), 2522–2528. doi: 10.1109/TMAG.2005.854338. Ramos, G.A. and Costa-Castell´o, R. (2013). Optimal anti-windup synthesis for repetitive controllers. Journal of Process Control, 23(8), 1149–1158. doi: 10.1016/j.jprocont.2013.07.004. Ramos, G., Costa-Castell´o, R., and Olm, J.M. (2013). Digital Repetitive Control under Varying Frequency Conditions, volume 446 of Lecture Notes in Control and Information Sciences. Springer. ISBN: 978-3-642-377785. S´anchez-Pe˜ na, R.S. and Sznaier, M. (1998). Robust Systems Theory and Applications. Adaptive and Learning Systems for Signal Processing, Communications and Control Series. Wiley-Interscience. Songschon, S. and Longman, R.W. (2003). Comparison of the stability boundary and the frequency response stability condition in learning and repetitive control. International Journal of Applied Mathematics and Computer Science, 13(2), 169–177. Tomizuka, M. (1987). Zero phase error tracking algorithm for digital control. Journal of Dynamic Systems, Measurement, and Control, 109(1), 65–68. doi: 10.1115/1.3143822. Wang, L. (2016). Tutorial review on repetitive control with anti-windup mechanisms. Annual Reviews in Control, 0, 1–14. doi:10.1016/j.arcontrol.2016.09.016. Wang, Y., Gao, F., and Doyle, F.J. (2009). Survey on iterative learning control, repetitive control, and run-torun control. Journal of Process Control, 19(10), 1589– 1600. doi:10.1016/j.jprocont.2009.09.006. Wu, M., Xu, B., Cao, W., and She, J. (2014). Aperiodic disturbance rejection in repetitive-control systems. IEEE Transactions on Control Systems Technology, 22(3), 1044–1051. doi:10.1109/TCST.2013.2272637. Ye, Y., Tayebi, A., and Liu, X. (2009). All-pass filtering in iterative learning control. Automatica, 45(1), 257 – 264. doi:10.1016/j.automatica.2008.07.011. Yeol, J.W., Longman, R.W., and Ryu, Y.S. (2008). On the settling time in repetitive control systems. In Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, 12460–12467. IFAC, Korea.
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Different architectures to develop repetitive to develop repetitive Different Different architectures architectures to develop repetitive controllers controllers controllers ∗ ∗∗
V´ıctor Sanz i L´ opez ∗ Ramon Costa-Castell´ o ∗∗ ∗∗∗ V´ıctor ıctor Sanz Sanz German L´ opez pez ∗∗ A. Ramon Costa-Castell´ o ∗∗ V´ ii L´ o Ramon Costa-Castell´ o Ramos ∗∗∗ V´ıctor Sanz German i L´ opez A. Ramon Costa-Castell´ o ∗∗ ∗∗∗ Ramos German A. Ramos ∗∗∗ German A. Ramos ∗ Institut de Rob` otica i Inform` atica Industrial, CSIC-UPC Llorens i ∗ ∗ Institut de Rob` o ii Inform` a Industrial, Llorens Rob` otica tica Barcelona, Inform` atica tica Industrial, CSIC-UPC Llorens ii Artigasde 4-6, 08028 Spain. Email: CSIC-UPC [email protected] ∗ Institut Rob` otica Barcelona, i Inform` atica Industrial, CSIC-UPC Llorens i ∗∗ Institut Artigasde 4-6, 08028 Spain. Email: [email protected] Artigas 4-6, 08028 Barcelona, Spain. Email: [email protected] Universitat Polit` e cnica de Catalunya; Barcelona, Catalunya, Spain. ∗∗ Artigas 4-6, 08028 Barcelona, Spain. Email: [email protected] ∗∗ Universitat Polit` e cnica de Catalunya; Barcelona, Catalunya, Spain. Universitat Polit` e cnica de Catalunya; Barcelona, Catalunya, Spain. Email: [email protected] ∗∗ Universitat Polit`eEmail: cnica [email protected] Catalunya; Barcelona, Catalunya, Spain. ∗∗∗ Email: [email protected] Universidad Nacional de Colombia; Bogot´ a , Colombia. Email: ∗∗∗ Email: [email protected] ∗∗∗ Universidad Nacional de Colombia; Colombia; Bogot´ Bogot´ a,, Colombia. Colombia. Email: Email: de a [email protected] ∗∗∗ Universidad Nacional Universidad Nacional de Colombia; Bogot´ a, Colombia. Email: [email protected] [email protected] [email protected] Abstract: In this work several architectures to implement repetitive controllers are compared. Abstract: In analytical this work work analysis several architectures architectures to implement repetitive controllers are compared. Abstract: In this several repetitive controllers compared. A complete is performed to forimplement these different architectures andare a simulation Abstract: In this work analysis several architectures to implement repetitive controllers are compared. A complete A complete analytical analysis is isisperformed performed for these these different different architectures architectures and and aa simulation simulation example withanalytical a power converter included. for A complete analytical analysis is performed for these different architectures and a simulation example example with with a a power power converter converter is is included. included. example with(International a power converter is included. © 2017, IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Repetitive Control, Disturbance observer, Youla parametrization, inverter control Keywords: Keywords: Repetitive Repetitive Control, Control, Disturbance Disturbance observer, observer, Youla Youla parametrization, parametrization, inverter inverter control control Keywords: Repetitive Control, Disturbance observer, Youla parametrization, inverter control 1. INTRODUCTION example is presented and finally section 8 provides 1. example presented and section 1. INTRODUCTION INTRODUCTION example isabout presented and finally finally section 88 provides provides discussionis the different approaches. 1. INTRODUCTION example isabout presented and finally section 8 provides discussion the different approaches. discussion about the different approaches. discussion about the different approaches. Repetitive Control (RC) (Wang et al., 2009; Longman, Repetitive (RC) (Wang et 2009; Repetitive Control (RC) Ahn (Wang et al., al., 2009; Longman, 2010; ChenControl et al., 2008; et al., 2007) is anLongman, Internal Repetitive Control (RC) (Wang et al., 2009; Longman, 2010; et 2008; et 2007) is 2010; Chen et al., al., (IMP) 2008; Ahn Ahn et al., al.,and 2007) is an an Internal Internal ModelChen Principle (Francis Wonham, 1976; 2010; Chen et al., (IMP) 2008; Ahn et al.,and 2007) is an Internal Model Principle (Francis Model Principle (IMP) (Francis and Wonham, Wonham, 1976; Costa-Castell´ o et al., 2012) based control technique.1976; RC Model Principle al., (IMP) (Francis and Wonham, 1976; Costa-Castell´ based RC Costa-Castell´ et al., 2012) 2012) based control control technique. RC it is speciallyoo et suited for systems subjecttechnique. to periodical Costa-Castell´ o et al., 2012)systems based control technique. RC it is suited to it is specially specially suited for for systems subject to periodical periodical references and disturbances (Ramossubject et al., 2013). During it is specially suited for systems subject to periodical references and disturbances (Ramos et 2013). references andtechnique disturbances (Ramos et al., al., 2013). During During last years this has been applied in different fields references andtechnique disturbances (Ramos et al., different 2013). During last years has applied fields last years this this technique has been been applied inordifferent fields like power electronics (Ramos et al., 2013)in mechatronic last years this technique has been applied in different fields like power electronics (Ramos et 2013) like power electronics (Ramos et al., al.,others. 2013) or or mechatronic mechatronic systems (Park et al., 2005) among like power electronics (Ramos et al., 2013) or mechatronic systems systems (Park (Park et et al., al., 2005) 2005) among among others. others. systems et al., others. The idea(Park behind RC2005) is toamong include inside the controlThe idea behind RC is to include inside the The ideagenerator behind RC to include inside the iscontrolcontrolloop the of a isperiodical signal, which a high The ideagenerator behind RC isperiodical to include inside the iscontrolloop the of a signal, which loop the generator of a periodical signal, which is aa high high order dynamic systems. Applying conventional stabilizing loop the generator of a periodical signal, whichstabilizing is a high order dynamic systems. Applying order dynamic systems. Applying conventional techniques to this type of systemsconventional would implystabilizing obtaining order dynamic systems. of Applying conventional stabilizing techniques this would imply obtaining techniques to this type type of systems systems would imply obtaining very high to order controllers which would entail huge techniques to this type of systems would imply obtaining very high order controllers which would entail huge very high order controllers which would huge computational resources to implement these entail controllers. very high order controllers which would entail huge computational to implement controllers. computational resources to architectures implement these these controllers. Due to this, RC resources uses specific and anti-windup computational resources to implement these controllers. Due to RC specific architectures and anti-windup Due to this, this,(Ramos RC uses usesand specific architectures and Wang, anti-windup techniques Costa-Castell´ o, 2013; 2016) Due to this,(Ramos RC usesand specific architectures and Wang, anti-windup techniques Costa-Castell´ oo,, 2013; techniques (Ramos andprofit Costa-Castell´ 2013; Wang, 2016) 2016) which allow to take from the steady-state nice techniques (Ramos and Costa-Castell´ o, 2013; Wang, 2016) which allow the nice which allow to take profit from the steady-state steady-state nice properties of to RCtake and profit obtainfrom implementation structures which allow to take profit from the steady-state nice properties RC obtain structures properties of reducing RC and and computational obtain implementation implementation structures which allowof requirements to a properties of RC and obtain implementation structures which allow which allow reducing reducing computational computational requirements requirements to to aa minimum. which allow reducing computational requirements to a minimum. minimum. minimum. Since the seminal work by Inoue et al. (1982), which introSince by al. which Since the seminal work by Inoue Inoue etRC, al. (1982), (1982), which introintroducedthe theseminal plug-in work architecture foret many architectures Since the seminal work by Inoue etRC, al. (1982),architectures which introduced the architecture for duced the plug-in plug-in architecture forinput-output RC, many many architectures have been proposed, both in the (Chen and duced the plug-in architecture for RC, many architectures have been both in (Chen have been proposed, proposed, both in the the input-output input-output (Chen and Tomizuka, 2014, 2015) and state-space (Wu et and al., have been proposed, both in the input-output (Chen and Tomizuka, 2014, state-space (Wu Tomizuka, 2014, 2015) 2015) and the state-space (Wu et et al., al., 2014) formalisms. In thisand workthe most relevant input-output Tomizuka, 2014, 2015) and the state-space (Wu et al., 2014) this work input-output 2014) formalisms. Inare thisreviewed work most most relevant input-output based formalisms. architecturesIn andrelevant compared. The com2014) formalisms. Inare thisreviewed work most relevant input-output based architectures and compared. The comcombased and The parisonarchitectures is illustratedare in reviewed the case of a compared. power inverter. based architectures are reviewed and compared. The comparison is illustrated in the case of a power inverter. parison is illustrated in the case of a power inverter. parison is illustrated in theascase of a power This paper is organized follows: sectioninverter. 2 describes This is as section 22 describes This paper is organized organized as follows: follows: section describes most paper relevant concepts about internal models used in This paper is organized as follows: section 2 describes most about internal used most relevant concepts about internal models models used in in RC, inrelevant section concepts 3 the series architecture is described, most relevant concepts about internal models used in RC, in 33 the series architecture is RC, in 4section section the structure series architecture is described, described, in5 section the plug-in is introduced, in section in RC, in 4section 3 the structure series architecture is described, in section is is introduced, in section section section 4 the the plug-in plug-in structure is introduced, in a disturbance observer approach used, in section 6 the55 section 4 the plug-in structure is is introduced, in section 5 aacontroller disturbance observer in 66 the disturbance observer approach approach used, in 7section section the parametrization is used,isinused, section a numerical a disturbance observer approach is used, in section 6 the controller controller parametrization parametrization is is used, used, in in section section 77 aa numerical numerical controller parametrization is used, in section 7 a numerical
2. 2. 2. 2.
INTERNAL INTERNAL INTERNAL INTERNAL
some some some some
MODEL MODEL MODEL MODEL
I(z) I(z) I(z) I(z)
E(z) + E(z) + E(z) + E(z) +
+ + + +
σ · W (z) σ ·· W W (z) (z) σ σ · W (z)
H(z) H(z) H(z) H(z)
Y (z) Y (z) (z) Y Y (z)
Figure 1. Internal model used in Repetitive Control. Figure Figure 1. 1. Internal Internal model model used used in in Repetitive Repetitive Control. Control. Figure 1. Internal model used in Repetitive Control. The internal model is the core element of a RC system The internal model is core aa RC system The internal model by is the the core element element ofcan RC systema and it is composed an element whichof generate The internal model by is the core element ofcan a RC systema and it is composed an element which generate and it is signal. composed by 2anshows element canofgenerate a periodic Figure the which structure a generic and it is composed by an element which can generate a periodic signal.used Figure shows the the structure structure of of a a generic generic periodic signal. Figure 22 shows internal model in RC: periodic signal. Figure 2 shows the structure of a generic internal model model used used in in RC: RC:σH(z)W (z) internal internal model used I(z)in=RC:σH(z)W σH(z)W (z) (z) , σH(z)W I(z) = = 1− σH(z)W (z)(z) ,, I(z) I(z) = 11 − − σH(z)W σH(z)W (z) (z) , where σ = {−1, 1}, W (z) the time(z) delay function, and 1 −is σH(z)W where σ = {−1, 1}, W (z) is the time delay where is σ= 1}, W (z) isAs thean time delay function, function, and H(z) a {−1, low-pass filter. example, for σ =and 1, where is σ= {−1, 1}, W (z) isAs thean time delay function, and −N H(z) a low-pass filter. example, σ H(z) low-pass filter. = As 1ana example, forgenerator, σ = = 1, 1, W (z) is= az −N and H(z) N-periodicfor H(z) is a1z −N low-pass filter. = As an example, forgenerator, σ =− 1, N W (z) and H(z) N-periodic W (z)= = = N z −N and H(z) = 11foraa σN-periodic generator, 2 I(z) , is obtained and = −1, W (z) = z N W (z) =z 11z−1 and H(z) = 1 a N-periodic generator, − −N 2 I(z) = ,, 1,is obtained and = (z) zzand 2 I(z) H(z) = zzN is the obtained and for for σ σgenerator = −1, −1, W W(Gri˜ (z) = = −1 N1−1 −N and = odd-harmonic n o ´ I(z) = zN −1 , is obtained and for σ = −1,−1W (z) = z 2 and H(z) = 1, the odd-harmonic generator (Gri˜ n o ´ and and H(z) = o1,, 2005) the odd-harmonic generator o and Costa-Castell´ is obtained I(z) = N−1 (Gri˜ . n´ and H(z) = 1, the odd-harmonic generator o and 2 +1(Gri˜ −1 Costa-Castell´ .. n´ Costa-Castell´oo,, 2005) 2005) is is obtained obtained I(z) I(z) = = zz N N −1 +1 . Costa-Castell´o, 2005) is obtained I(z) = z N22 +1 For those internal models obtained with H(z) = 1, z 2 +1 For internal those internal internal models obtained infinite with H(z) H(z) = the 1, For those obtained with = 1, the model, models I(z), introduces gain at For internal those internal modelsintroduces obtained with H(z) = 1, the model, gain at the internal model, I(z), infinite gain gain at the the signal frequency andI(z), all itsintroduces harmonics.infinite This high at the internal model, I(z), introduces infinite gain gain at the signal frequency all its signal frequency and and all be its aharmonics. harmonics. This high gain at at high frequencies might problem inThis the high presence of signal frequency and all its harmonics. This high gain at high frequencies frequencies mightthis be gain problem in the the presence of high might be aa problem in presence uncertainty. To reduce a low-pass filter, H(z), of is high frequencies mightthis be a problem in the presence of uncertainty. aa low-pass H(z), is uncertainty. To reduce this gain gain low-pass filter, H(z), is usually used.To A reduce null-phase low pass filter isfilter, usually used uncertainty. To reduce this gain a low-pass filter, H(z), is usually used. A null-phase low pass filter is usually used usually used. A null-phase filter et is al., usually used (Gri˜ n´o and Costa-Castell´ o, low 2005;pass Escobar 2014). usually used. A null-phase low pass filter is usually used (Gri˜ n´o´o and and Costa-Castell´ Costa-Castell´oo,, 2005; 2005; Escobar Escobar et et al., al., 2014). 2014). (Gri˜ n (Gri˜ n´o and Costa-Castell´o, 2005; Escobar et al., 2014).
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I(z)
+
R(z) +
σ · W (z)
+
−
G(z)
Gc (z)
H(z)
Y (z)
3. SERIES APPROACH The first approach to RC consists in placing the internal model in series connection with the plant. Additionally a stabilizing controller needs to be introduced to guarantee closed-loop stability (Figure 2). In this case the controller becomes: C(z) = I(z)Gc (z) the complementary sensitivity and sensitivity function become: I(z)Gc (z)G(z) T (z) = 1 + I(z)Gc (z)G(z) 1 . S(z) = 1 + I(z)Gc (z)G(z) Usually, for minimum-phase plants kr Gc (z) = G(z) and :
(1)
kr σW (z)H(z) T (z) = 1 + (kr − 1)σW (z)H(z) 1 − σW (z)H(z) . S(z) = 1 + (kr − 1)σW (z)H(z) As expected, S(z) has in the numerator the denominator of I(z). For non-minimum phase plants it is necessary to use phase cancellation techniques (Ye et al., 2009; Tomizuka, 1987) when designing Gc (z). In case of systems with multiplicative uncertainty: G(z) = anchez-Pe˜ na and Sznaier, 1998), Gn (1 + Wum (z)∆(z)) (S´ the robust stability condition is: m kr σW (z)H(z) Wu (z) . 1 + (kr − 1)σW (z)H(z) ∞
4. PLUG-IN APPROACH
C(z) I(z)
σW (z)
R(z) +
H(z)
Gx (z)
+
−
this internal controller is to guarantee closed-loop stability and robustness to the control system without the internal model. Later, the internal model I(z), and the stabilizing controller, Gx (z), are plugged in to the previous closedloop system. In this case the closed-loop transfer function can be constructed in terms of the closed-loop transfer function without the internal model:
Figure 2. RC : Series approach.
+
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+
+ Gc (z)
G(z)
Y (z)
Figure 3. RC : Plug-in approach. The most popular approach in RC is the plug-in architecture (Figure 3). This architecture introduces the internal model to a previously existing control system defined by an internal controller Gc (z) and the plant G(z). The goal of
1 Gc (z)G(z) , To (z) = 1 + Gc (z)G(z) 1 + Gc (z)G(z) and a modifying term: So (z) =
1 − σW (z)H(z) . 1 − σW (z)H(z) (1 − Gx (z)To (z)) So the closed-loop transfer functions are: SM od (z) =
S(z) = So (z)SM od (z) (1 − σW (z)H(z) (1 − Gx (z))) To (z) T (z) = . 1 − σW (z)H(z) (1 − Gx (z)To (z))
For minimum-phase plants 1 the most popular form of for the stabilizing controller is: kr . (2) Gx (z) = To (z) With this selection the closed-loop function becomes: 1 − σW (z)H(z) 1 + (kr − 1) σW (z)H(z) (To (z) − σW (z)H(z) (To (z) − kr )) . T (z) = 1 + (kr − 1) σW (z)H(z) S(z) = So (z)
The following two conditions guarantee closed-loop stability (Ramos et al., 2013): (1) To (z) must be stable (Gc (z) can be designed to fullfil it) (2) W (z)H(z) (1 − kr ) ∞ < 1 (kr can be selected appropriately ) Even thought these conditions are only sufficient it has been proved that they are close to the necessary ones in practice (Songschon and Longman, 2003). It is important to emphasize that in (2) the inversion of To (z) is required while in (1) it is required the inversion of G(z), as To (z) is a closed-loop system its uncertainty should be less than that of G(z). Additionally, it is important to visualize that the sensitivity function in the series approach and the plug-in one are the same except the So (z) term. In case of multiplicative uncertainty the robust stability condition becomes: m (To (z) − σW (z)H(z) (To (z) − kr )) Wu (z) < 1. 1 + (kr − 1) σW (z)H(z) ∞ This condition is quite similar to the one obtained in the series approach but it contains To (z). This term can be shaped using Gc (z), so it is simpler to fulfill this constrain than the one obtained in the series approach. 1 For nonminimum phase plants a phase cancellation approach is usually used
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5. DISTURBANCE REJECTION APPROACH
degrees of freedom they might increase the controller complexity.
C(z) z −m G(z)−1
+
+
6. YOULA PARAMETRIZATION
z −m
C(z)
Q(z)
R(z) +
E(z)
Gc (z)
+
+
+
D(z) + G(z)
R(z) +
Y (z)
−
−
+
F (z)
G(z)
Y (z)
+ G(z)
Figure 4. RC : Disturbance rejection approach. During last years a great effort in the disturbance rejection mechanisms has been made (Chen et al., 2016). In this context a RC has been proposed based on this methodology (Chen and Tomizuka, 2014) and its characteristics are shown in this section. The controller scheme for an m-relative degree minimum phase plant, G(z), is shown in Figure 4. This system uses a baseline controller, Gc (z), plus a disturbance observer composed by the plant model and a filter, Q(z). The sensitivity function for this closed-loop system is: 1 − z −m Q(z) S(z) = . 1 + G(z)Gc (z) In order to transform this in a RC it is necessary to choose Q(z) appropriately, so that the denominator of the internal model, I(z), appears in the numerator of S(z). Consequently, Q(z) must be isolated from the equation: Ns (z) 1 − z −m Q(z) = So (z) (1 − σW (z)H(z)) 1 + G(z)Gc (z) Ds (z) where Ns (z) and Ds (z) are polynomials that must be fixed (this selection requires So (z) to be stable, which can be achieved by selecting and appropriate Gc (z)). So: (1 − σW (z)H(z)) Ns (z) − Ds (z) . Q(z) = z m Ds (z) A simple option is choosing Ns (z) = 1 and Ds (z) = 1 − α · σ · W (z)H(z) with |α| < 1 which generates: (−(1 + α) · σW (z)H(z)) Ns (z) . Q(z) = z m 1 − α · σ · W (z)H(z) Although this is not the conventional shape of Q(z) in disturbance observer based control it allows to reject periodical signals and behave as RC. Finally the controller becomes: Gc (z) + Q(z)z m G(z)−1 C(z) = 1 − z −m Q(z) This controller has a degree of freedom which corresponds to the value of α. This value plays a similar role to kr in the plug-in approach. In case of multiplicative uncertainty the robust stability condition becomes: Wum (z)T (z) ∞ < 1, with 1 − σW (z)H(z) T (z) = 1 − So (z) . 1 − α · σW (z)H(z) Clearly, other selections for Ns (z) and Ds (z) are possible. Although these other options might provide additional
Figure 5. RC : Youla parametrization. It is well-known that all stabilizing controllers for a given stable plant, G(z), can be written in terms of the Youla parametrization (S´anchez-Pe˜ na and Sznaier, 1998) shown in Figure 5. In this case, the sensitivity and complementary sensitivity functions are: S(z) = 1 − F (z)G(z), T (z) = F (z)G(z) where F (z) is a stable system. And the controller is (z) C(z) = 1−FF(z)G(z) . In case of minimum-phase plants it is possible to select F (z) = F (z)G−1 (z), so the closed-loop functions become: S(z) = 1 − F (z), T (z) = F (z). In order to impose that the controller behaves as a RC it is necessary to impose the appropriate shape for F (z). It is necessary that: Ns (z) S(z) = 1 − F (z) = (1 − σW (z)H(z)) Ds (z) with Ns (z) and Ds (z) arbitrary elements. So Ds (z) − (1 − σW (z)H(z)) Ns (z) . F (z) = Ds (z) A simple solution is Ns (z) = 1 and Ds (z) = 1 − α · σ · W (z)H(z) with |α| < 1, which generates: (1 − α) · σ · W (z)H(z) . F (z) = 1 − α · σ · W (z)H(z) This option generates the following closed-loop transfer functions: (1 − α) · σ · W (z)H(z) 1 − α · σ · W (z)H(z) 1 − σ · W (z)H(z) . S(s) = 1 − α · σ · W (z)H(z) Which can be considered a generalized version of the series approach (section 3). T (z) =
Finally, robust stability condition for multiplicative uncertainty is : Wum (z)T (z) ∞ < 1. Clearly, this approach can generate more complex closedloop transfer function, in particular the closed-loop poles
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d
d¯ if
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vf _
_
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vo _
d
PWM
d
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vref
Figure 7. Sensitivity function for RC based on plug-in and disturbances observer approaches.
Figure 6. Circuit diagram. could be arbitrarily placed by choosing and appropriate value for Ds (z). Alternatively, these degrees of freedom could be used to optimize any criteria such as robustness. This increase in the controller complexity would increase the computational resources required to implement the controller. 7. NUMERICAL EXAMPLE The numerical example is based on the system depicted in Figure 6. It consists of a Voltage Source Inverter (VSI) which transforms a DC supply, Vdc , in an AC power source, vo . The duty cycle, d, of a Pulse Width Modulation (PWM) signal is the control action used to switch the power transistors and the produced signal is filtered by an LC network to obtain the output AC voltage vo (t). The control objective is providing a sinusoidal waveform voltage signal with specific amplitude and frequency (vref (t)). At the same time, it is needed to reject disturbances caused by the load. Nonlinear loads, as the one produced by full-bridge diode rectifiers, are of special interest since they produce disturbances with high harmonic content. A way of measuring the system performance is through the Total Harmonic Distortion (THD), thus a low THD in voltage waveform (vo (t)) is desirable.
Additionally, the control action is Vf (s) = (2d − 1)Vdc , where d ∈ [0, 1] is the duty cycle of the PWM signal and Vdc is the DC input source. The PWM signals has a frequency of 10KHz, and the duty cycle is updated each period, so a sampling time of Ts = 0.0001s is used.
The system transfer function is based on the LC filter circuit equations to obtain: (3) Vo (s) = Gp (s)Vf (s) + Gd (s)Io (s), with RC , Gp (s) = 2LCRC s2 + (2CRL RC + 2L)s + (2RL + RC ) (4) and 2LRC + 2RL + RC , Gd (s) = − 2 2LCRC s + (2CRL RC + 2L)s + (2RL + RC ) (5) where Gp (s) is the plant transfer function, Gd (s)Io (s) is the disturbance signal caused by the load, L = 300 µH and C = 80 µF are the inductive and capacitive part of the filter, RL = 0.1 Ω and RC = 8200 Ω are the parasitic resistance of inductance and capacitance respectively.
Although the inverter is designed to feed generic loads, it is difficult to design a control system without some assumptions about the concrete load being used. The load has been modeled by means of an inductance and a resistor in series placed in parallel with the capacitor. For this reason, an impedance Z(s) to relate the current introduced to the load Io (s) and the voltage supplied to it Vo (s). This impedance is as can be seen in equations (6) and (7): (6) Io (s) = Z(s)Vo (s) 1 (7) Z(s) = L · s + RL Taking this into account the nominal model is constructed as: Gp (s) Gn (s) = . 1 + Gd (s)Z(s) This model will be the one used to design the controller.
Figure 8. Sensitivity function for RC based on series and Youla parametrization approaches.
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closed-loop poles. In fact, a relation between kr and α can be established, α = 1 − kr , so that the closed-loop poles from different architectures are placed in the same location. For consistency with previous cases, α = 0.3 is choosen. An element present in all these architectures is the lowpass filter. This element’s most relevant goal is introducing robustness in the high frequency range. In this work the following filter has been used: 0.25z 2 + 0.5z + 0.25 , (8) H(z) = z it has null-phase, a gain close to 1 in low and medium frequency range and a important attenuation in the high frequency range. Figure 9. Bode diagram of 1/T (z) for all four approaches.
Figure 10. Bode diagram of 1/T (z) for all four approaches.
Finally, the plug-in approach (section 4) and disturbance rejection one (section 5) contain an internal controller in charge of providing robustness. To prevent increasing the complexity and allowing a better comparison a proportional controller like the following one has been selected: Gc (z) = 0.1. (9) Taking this into consideration and noting that the discretetime plant obtained when applying the z-transform to Gn (s) is minimum-phase plant, all the controllers introduced in this work are completely defined. Figure 7 shows the frequency response of the sensitivity function for the plug-in approach and the disturbance observer one. As it can be seen, both responses are similar, have small values for the working frequency and all the harmonics. The effect of the filter can also be seen, the small values at harmonics increase as frequency increases. Additionally, as the sensitivity function does not take values over 6dB it can be stated that both of them are quite robust. Figure 8 shows the sensitivity function for the series and Youla architectures. As it can be seen, both schemes provide the same result and it is quite similar to the one observed in Figure 7.
The goal in this type of system is feeding the load with a sinusoidal voltage of frequency f = 50 Hz and amplitude √ 220 2 V. Consequently, the period signal of to be tracked 1 is Tp = 50 = 0.02s. Consequently, the discrete time period, N , can be computed as N = Tp /Ts = 200. This allows to obtain a good reconstruction of the continuous time signal.
Figure 9 shows the closed-loop time response for all described architectures when a sinusoidal reference of the working frequency is introduced in the system. As it can be seen, after a small transient all closed-loop schemes converge to the reference with null-steady state error. No relevant differences can be observed.
In the following, 4 different RC systems, corresponding to the previously introduced architectures will be designed. In each approach different parameters have been tuned.
Finally, the attention is turned to robustness eases. In this type of application, the real load is not know. A way to analyze the robustness of the control system against changes in the load characteristics is to model the variation in the load as a multiplicative uncertainty Wum (z) in the plant and perform a robust stability analysis. The robust stability condition is Wum (z)T (z) ∞ < 1, which can be analyzed frequency by frequency as : 1 |Wum (ejωTs )| < . (10) |T (ejωTs )| Figure 10, shows the frequency response of T (z)−1 . As it can be seen it is always over 0dB in all the frequency range, and specially in the high frequency range. Consequently, a 100% change in the impedance at each frequency can be handled. It can be stated that the closed-loop system is very robust. Additionally, it is possible to see that all architectures provide a similar robustness.
In the series (section 3) and the plug-in (section 4) approaches, kr is the most relevant degree of freedom. It is related with the most relevant closed-loop poles. Approximately, these are the solutions of z N = 1 − kr . These poles are homogeneously distributed over a circle of radius |1 − kr |, which directly defines the closed-loop settling time (Yeol et al., 2008; Garimella and Srinivasan, 1994). Due to this, the selection of kr is a trade-off between settling time and robustness. A reasonable value is kr = 0.7. In the disturbance rejection (section 5) and the Youla parametrization (section 6) a parameter called α is introduced. Similarly to kr , α is directly related with the
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8. CONCLUSIONS In this paper most relevant architectures used to implement RC in input-output form have been reviewed and analyzed for the case of minimum-phase plants and its applications to the case of an inverter has been illustrated. As it can be shown, most architectures achieve similar results. Obtaining good steady-state results and reasonably robustness conditions. Although it has not been shown in this work, the plug-in approach and the disturbance observer contains an additional degree of freedom which can be used to slightly improve robustness and transient behavior. Youla parametrization is the most generic by far because it can be used to reduce all the other approaches. Finding new approaches based on Youla parametrization which improve transient and robustness with a limited complexity increase is a current research area. ACKNOWLEDGEMENTS This work was partially supported by the spanish Ministerio de Educaci´ on project DPI2015-69286-C3-2-R (MINECO/FEDER) and the catalan AGAUR project 2014 SGR 267. REFERENCES Ahn, H.S., Chen, Y.Q., and Moore, K.L. (2007). Iterative Learning Control: Brief Survey and Categorization. IEEE Transactions on Systems Man & Cybernetics Part C, 37(6), 1099–1121. Chen, W.H., Yang, J., Guo, L., and Li, S. (2016). Disturbance-observer-based control and related methods : An overview. IEEE Transactions on Industrial Electronics, 63(2), 1083–1095. doi: 10.1109/TIE.2015.2478397. Chen, X. and Tomizuka, M. (2014). New repetitive control with improved steady-state performance and accelerated transient. IEEE Transactions on Control Systems Technology, 22(2), 664–675. doi: 10.1109/TCST.2013.2253102. Chen, X. and Tomizuka, M. (2015). Overview and new results in disturbance observer based adaptive vibration rejection with application to advanced manufacturing. International Journal of Adaptive Control and Signal Processing, 29(11), 1459–1474. doi:10.1002/acs.2546. Acs.2546. Chen, Y., Moore, K.L., Yu, J., and Zhang, T. (2008). Iterative learning control and repetitive control in hard disk drive industry—a tutorial. International Journal of Adaptive Control and Signal Processing, 22(4), 325–343. doi:10.1002/acs.1003. Costa-Castell´ o, R., Olm, J.M., Vargas, H., and Ramos., G.A. (2012). An educational approach to the internal model principle for periodic signals. International Journal of Innovative Computing, Information and Control, 8(8), 5591–5606. Escobar, G., Mattavelli, P., Hernandez-Gomez, M., and Martinez-Rodriguez, P.R. (2014). Filters with linearphase properties for repetitive feedback. IEEE Transactions on Industrial Electronics, 61(1), 405–413. doi: 10.1109/TIE.2013.2240634. Francis, B. and Wonham, W. (1976). The internal model principle of control theory. Automatica, 12(5), 457 – 465. doi:10.1016/0005-1098(76)90006-6.
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