Brekken, T.

Carapellese, F.

Faedo, N.

Ferri, F.

English

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

17 October 2023POLITECNICO DI TORINORepository ISTITUZIONALEDesign and experimental assessment of an anti-windup for LTI state-constrained control of wave energy converters /Faedo, N.; Ferri, F.; Carapellese, F.; Brekken, T.. - (In corso di stampa). (Intervento presentato al convegno IFAC WorldCongress 2023 (IFAC 2023) tenutosi a Yokohama).OriginalDesign and experimental assessment of an anti-windup for LTI state-constrained control of wave energyconvertersPublisher:PublishedDOI:Terms of use:Publisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2979937 since: 2023-07-05T13:21:43ZIFACDesign and experimental assessment of ananti-windup for LTI state-constrainedcontrol of wave energy converters ⋆Nicolás Faedo ∗,1 Francesco Ferri ∗∗ Fabio Carapellese ∗Ted K. A. Brekken ∗∗∗∗ Marine Offshore Renewable Energy Lab., Dept. of Mechanical andAerospace Engineering, Politecnico di Torino.∗∗ Department of the Built Environment, Aalborg University.∗∗∗ Electrical Engineering and Computer Science, Oregon StateUniversity.Abstract: Motivated by the necessity of suitable state constraint mechanisms within lineartime-invariant (LTI) energy-maximising control of wave energy converters (WECs), this dis-cussion paper presents an anti-windup (AW) scheme for state constraint satisfaction, wherethe associated unconstrained controller is designed via impedance-matching theory for WECsystems. As in the standard (input) AW scenario, the adopted technique provides a mechanismfor ‘informing’ the (unconstrained) controller when constraints are active, so that appropriatemodifications to future control actions can be taken accordingly. The overall adopted AWtechnique is tested experimentally, on a prototype of the Wavestar WEC system, availableat Aalborg University (Denmark).Keywords: Wave energy converters, Impedance-matching, Anti-windup, State constraints1. MOTIVATIONIn the light of the inherent requirement of effective stateconstraint handling mechanisms for wave energy conver-sion (WEC) systems (see e.g. (Faedo et al., 2020)), wepresent, in this discussion paper, an anti-windup (AW)scheme for state constraint satisfaction, where the asso-ciated (unconstrained) energy-maximising control is de-signed via linear impedance-matching theory. An AWsetup is discussed, based on the structure proposed in (Ro-jas and Goodwin, 2002), which can be applied to a generalclass of non-optimisation-based WEC strategies. The AWscheme incorporates a tailored design for the associatedlimiting circuit, able to map the defined state constraintsinto an associated (time-varying) input constraint set. Thecombination between LTI energy-maximising controllerand proposed AW technique, is tested experimentally, us-ing a prototype of the Wavestar WEC system (Hansen andKramer, 2011), available within the tank-testing facilitiesof Aalborg University, Denmark. We explicitly demon-strate that the proposed AW scheme is able to consistentlyrespect the defined state constraints, having a mild impacton the overall energy absorption performance when com-pared to its unconstrained counterpart.1.1 NotationR+ and C0 are used to indicate the set of non-negative realnumbers, and the set of complex numbers with zero real-⋆ Nicolás Faedo has received funding from the European Union’sHorizon 2020 research and innovation programme under the MarieSklodowska-Curie grant agreement No 101024372.1 Corresponding author - e-mail: nicolas.faedo@polito.it.part, respectively, while C<0 denotes the set of complexnumbers with negative real-part. The Laplace transformof a function f , provided it exists, is denoted as F (s),s ∈ C. The Hermitian operator is denoted by F (ȷω)⋆, withω ∈ R. The saturation function is defined as sat∆(x) =sign(x)min(|x|,∆), ∆ ∈ R+, and x ∈ R. Finally, givena continuous-time signal g(t), its discrete-time zero-orderhold equivalent is denoted as g[k].2. EXPERIMENTAL WEC SYSTEMThe experimental system considered within this discussionpaper, illustrated in Figure 1, is a small-scale (1:20) pro-totype of the Wavestar WEC device (Hansen and Kramer,2011), tested within the basin facilities available at Aal-borg University, Denmark, as part of a larger experimentalcampaign in WEC modelling and control, executed inSeptember 2022. The system, which has been previouslyselected as a benchmark case for WEC control assessmentwithin the first edition of the Wave Energy Control Com-petition (WEC3OMP) (Ringwood et al., 2017), is essen-tially composed of a floater, connected through an arm toa pivoting point fixed at a reference frame. In equilibriumposition, the arm sits at approximately 30° with respect tothe horizontal reference frame. The system is free to movein a single degree-of-freedom, and extracts energy frompitch motion (about the reference point - see Figure 1)via the corresponding power take-off system (PTO, linearmotor/generator) sitting on the upper structural joint ofthe device arm. Though we avoid a full description of eachof the WEC components for economy of space, we referthe reader to (Ringwood et al., 2017) for further detail.Fig. 1. Experimental prototype of the Wavestar system(left) and corresponding schematic (right).3. ENERGY-MAXIMISING CONTROL DESIGNThroughout this study, and adopting the arguments posedwithin the WEC3OMP (Ringwood et al., 2017), we assumethat the prototype WEC system, described in Section 2,can be modelled in terms of a representative linear mapG : C → C, s 7→ C(sIn − A)−1B ∈ RH2, defined in termsof a minimal state-space realisation asG(s) ≡{ẋ(t) = Ax(t) +B(dθ(t) + uθ(t)),yθ(t) = Cx(t),(1)with (A,B,CT) ∈ Rn×n × Rn × Rn, and where G(s) isof relative degree 1. Equation (1) characterises the input-output (I/O) relationYθ(s) = G(s) (Dθ(s) + Uθ(s)) , (2)where yθ is the device (pitch) velocity about the referencepoint (see Figure 1), dθ is the (uncontrollable) waveexcitation torque due to the action of the wave field, anduθ is the control torque, supplied via the PTO system.Following the requirement of energy-maximisation inWEC systems, and adopting a frequency-domain ap-proach, the optimal control solution uθ for maximumpower transfer can be derived via the so-called impedance-matching principle (Faedo et al., 2022), where the WECsystem is essentially described in terms of an impedance,analogously to standard circuit theory. Briefly summaris-ing, let I : C0 → R, ȷω 7→ I(ȷω) be the impedance map ofthe prototype WEC system, defined asI(ȷω) = 1/G(ȷω). (3)With the definition in (3), the (frequency-domain) optimalcontrol solution (load), can be simply written asUθ(ȷω) = −I(ȷω)⋆Yθ(ȷω), (4)which is, in essence, an output feedback structure.Remark 1. Though one can be tempted to use the ana-lytic continuation of I⋆ to C, in order to implement thecontrol structure in (4), the resulting transfer function isinherently non-causal, due to the nature of the analyticcontinuation of the Hermitian operator (Scruggs, 2010).Remark 1 refers to a well-known issue in design and syn-thesis of wave energy control systems. One can, although,approximate condition (4) by employing tailored causaland stable control structures, i.e. implementable. In par-ticular, within this study, we consider a first-order bipropercontroller, defined asK(s) = α1s/(s+ α2), (5)where the set of parameters A = {α1, α2} are uniquelycomputed as the solution of the interpolation equationK(ȷωI) = I(ȷωI)⋆, (6)with ωI ∈ R+ a suitably selected interpolation frequency.Remark 2. For the case of the prototype WEC systemconsidered, the set A ⊂ R+ (see also Section 5), and hence(5) is both stable and minimum-phase.4. AW FOR STATE CONSTRAINT HANDLINGWithin this section, we introduce the adopted AW schemefor state constraint satisfaction, tailored for the WECprototype presented in Section 2, and the control structuredefined in Section 3. Throughout this section, we considerthe discrete-time equivalents {Gd(q),Kd(q)} (with q theforward-shift operator) corresponding with the transferfunctions G(s) and K(s) in equations (2) and (5), respec-tively, computed via a standard zero-order hold procedurewith a sufficiently small sampling time Ts ∈ R+. Further-more, we write Gd in terms of a state-space realisationGd(q) ≡{x[k + 1] = Adx[k] +Bd (dθ[k] + uθ[k]) ,yθ[k] = Cdx[k],(7)where the triple of matrices (Ad, Bd, Cd) stem directlyfrom (1). We further assume that (7) is subject to a set ofstate constraints defined in terms of the velocity yθ, i.e.yθ[k] = Cdx[k] ∈ Y ≜ [−∆,∆] ,∆ ∈ R+. (8)Fig. 2. General AW scheme for input constraints.Figure 2 illustrates a general scheme for AW in the (stan-dard) case of input constraints (see e.g. (Galeani et al.,2009)). In particular, following an analogous procedure tothat in (Rojas and Goodwin, 2002), and given that thecontroller Kd is biproper, stable, and minimum-phase (seeRemark 2), we can decompose its transfer function asKd(q)−1 = h∞ +H(q), (9)where h∞ is the high frequency gain of Kd(q)−1, and H(q)is both stable and strictly proper.Focusing on the development of an AW scheme for theWEC state constraint case, we note that the set Y defines acorresponding input constraint set U. The latter, although,is inherently time-varying, as discussed in the following.Consider system (7) subject to the state constraint in (8).Since Gd is strictly proper with relative degree 1, a one-step ahead prediction of yθ can be computed asyθ[k + 1] = CdAdx[k] + CdBd (dθ[k] + uθ[k]) , (10)where R ∋ CdBd ̸= 0. Solving for uθ in equation (10), wecan define the following transition map Γ : R×Rn×R → RΓ(yθ, x, dθ) = (CdBd)−1(yθ − CdAdx)− dθ. (11)Note that the function Γ, as defined in (11), can beeffectively used to map the state constraint set Y to anequivalent time-varying input constraint set U(x, dθ) =Γ(Y, x, dθ). In particular, for a given value of yθ, theinduced input constraint set can be explicitly written asU ≜[∆l(x, dθ),∆u(x, dθ)]⊂ R, (12)with∆l(x, dθ) = (CdBd)−1(−∆− CdAdx)− dθ,∆u(x, dθ) = (CdBd)−1( ∆− CdAdx)− dθ,(13)where, clearly, it follows that yθ ∈ Y ⇔ uθ ∈ U, ∀k ∈ N.With the time-varying set derived in equation (13), an AWscheme, effectively able to handle the state constraint Y in(8), can be defined in terms of the limiting logic presentedin Figure 3. In particular, the adopted AW is, in essence, atwo-step procedure, which can be summarised as follows:• Let x[k] and dθ[k] be the current state of system (7)and wave excitation input, respectively, and uθ[k] thecontrol law required by the controller Kd. A one-stepprediction ŷθ[k + 1] can be computed via (10).• Constraint violations in accordance with the set Y aresubsequently detected in the prediction ŷ by applyingthe saturation map sat∆(·). Based on the saturatedvalue of ŷ[k + 1], an allowed control action uθ[k] canbe back-calculated using the map Γ(·, x, dθ) in (11).Remark 3. Clearly, if no constraint violation is detected,uθ = uθ, i.e. the unconstrained control solution, associ-ated with the energy-maximising controller Kd, is directlyapplied to the WEC. If, on the contrary, the constraint onyθ is active, uθ takes yθ to the limit of the set Y.Fig. 3. Adopted AW limiting logic.Remark 4. Effective implementation of the discussed AWstructure requires instantaneous knowledge of the state-vector x and the wave excitation torque dθ, in order toeffectively back-calculate admissible control actions via(11). Such estimates can be computed in following stan-dard unknown-input estimation techniques (Peña-Sanchezet al., 2019).5. EXPERIMENTAL RESULTSWe present, in this section, an experimental assessmentof the AW scheme detailed in Section 4, comprising theenergy-maximising control design as in Section 3, andapplied to the prototype described in Section 2. Forperformance assessment, we consider an irregular sea-statecharacterised in terms of a JONSWAP spectrum, with 2a significant wave height Hs = 0.063 [m], typical peakperiod Tp = 1.412 [s], and enhancement factor γ = 3.3.The wave duration is set to 300 [s], which corresponds withmore than 150 typical peak periods. The sampling time,considered both for data-acquisition, and correspondingdiscretisation of system/controller, is set to Ts = 0.005 [s].Within this study, the WEC model (1), characterising thebehaviour of the experimental prototype, is obtained usingblack-box system identification procedures. To summarise,a set of down-chirp signals with different amplitudes (inthe interval [1, 4] [Nm]) is injected into the system asinput (torque) signals, generating a corresponding set ofoutput (velocity) responses. Each I/O pair is used tocompute a frequency-domain empirical transfer functionestimate (Ljung, 1998), and their corresponding averageis used as target data-set for subspace-based identificationtechniques.With respect to the energy-maximising controller K(s),as defined in equation (5), the interpolation frequency ωIis chosen in terms of the energetic period associated withthe sea-state considered, i.e. ωI = 2π/(0.9Tp) producing acorresponding set of parameters α1 = 14.41 and α2 = 5.11.Concerning the specifics associated with the state AWscheme itself, the constraint set Y, characterising the statelimitation in (8), is set to ∆ = 0.4 [rad/s]. Note that thisvalue is conservative for this device (and sea-state), andhas been chosen to clearly illustrate the capabilities ofthe proposed state AW in experimental scenarios, underpotentially severe constraint limits.The results of applying the proposed AW scheme can beappreciated in Figure 4, where both the unconstrained(dotted), and constrained (solid), output response of theWEC are illustrated, together with each associated controlinput. Note that both responses are obtained in separateexperiments using the same wave realisation, being theformer only driven by the unconstrained controller K(s),while the latter effectively incorporates the proposed AW,with all its components (i.e. energy-maximising controller,state/input observer, and associated constraint satisfac-tion logic). It can be straightforwardly seen that the AWtechnique is effectively able to enforce the imposed stateconstraint, being always within the limit considered, whilethe unconstrained controller consistently violates the max-imum velocity value. Furthermore, note that, as expectedfrom the arguments posed in Remark 3, the AW strategyprovides an input correction only when the system isclose to attain the constraint limit, by requiring an ‘extra’torque to assist constraint satisfaction, yet being virtuallythe same as the unconstrained energy-maximising solutionwhen the state constraint is not active.Remark 5. Though small differences can be noted betweenunconstrained and constrained control solutions when the2 This sea-state corresponds with sea-state N°5 of the WEC3OMP.Fig. 4. Experimental WEC motion and associated control input for both unconstrained, and state-constrained AW case.Fig. 5. Absorbed mechanical energy.state constraint is inactive, these can be explained by thefact that both controllers are tested in separate experi-ments, where the same wave has to be realised by theassociated wavemaker, which, although highly repetitive,presents slight differences in the effective wave generation.Finally, Figure 5 presents an appraisal in terms of energyabsorption, showing that the AW has a minimum influenceon the overall loop performance, by virtue of the specificlimitation logic, which modifies the unconstrained optimalsolution only when strictly necessary.6. CONCLUSIONSWe present, in this discussion paper, an AW scheme forstate constraint handling in LTI energy-maximising con-trol of WEC systems. The technique provides a mechanismfor ‘informing’ the (unconstrained) controller when stateconstraints are active, so that appropriate modifications tofuture control actions can be taken accordingly. The com-bination between LTI energy-maximising controller andproposed AW technique, is tested experimentally, using aprototype of the Wavestar system. We demonstrate thatthe proposed AW scheme is able to consistently respectthe defined state constraints, having a mild impact in theoverall energy absorption performance. Future work willconsider other LTI control structures for K(s), includingthose of a broadbanded nature, reported in e.g. (Garćıa-Violini et al., 2020).REFERENCESFaedo, N., Carapellese, F., Pasta, E., and Mattiazzo, G.(2022). On the principle of impedance-matching forunderactuated wave energy harvesting systems. AppliedOcean Research, 118, 102958.Faedo, N., Garćıa-Violini, D., Peña-Sanchez, Y., and Ring-wood, J.V. (2020). Optimisation-vs. non-optimisation-based energy-maximising control for wave energy con-verters: A case study. In European Control Conference(ECC), 843–848.Galeani, S., Tarbouriech, S., Turner, M., and Zaccarian,L. (2009). A tutorial on modern anti-windup design. In2009 European Control Conference (ECC), 306–323.Garćıa-Violini, D., Faedo, N., Jaramillo-Lopez, F., andRingwood, J.V. (2020). Simple controllers for waveenergy devices compared. Journal of Marine Scienceand Engineering, 8(10), 793.Hansen, R.H. and Kramer, M.M. (2011). Modelling andcontrol of the Wavestar prototype. In Proceedings ofthe 9th European Wave and Tidal Energy Conference,EWTEC 2011. University of Southampton.Ljung, L. (1998). System identification. In Signal analysisand prediction, 163–173. Springer.Peña-Sanchez, Y., Windt, C., Davidson, J., and Ringwood,J.V. (2019). A critical comparison of excitation forceestimators for wave-energy devices. IEEE Transactionson Control Systems Technology, 28(6), 2263–2275.Ringwood, J., Ferri, F., Ruehl, K.M., Yu, Y.H., Coe, R.G.,Bacelli, G., Weber, J., and Kramer, M. (2017). Acompetition for WEC control systems. In Proceedings ofthe 12th European Wave and Tidal Energy Conference27th Aug-1st Sept 2017, 831, 1–9. European Wave andTidal Energy Conference 2017.Rojas, O.J. and Goodwin, G.C. (2002). A simple anti-windup strategy for state constrained linear control.IFAC Proceedings Volumes, 35(1), 109–114.Scruggs, J. (2010). On the causal power generation limitfor a vibratory energy harvester in broadband stochasticresponse. Journal of Intelligent Material Systems andStructures, 21(13), 1249–1262.

Design and experimental assessment of an anti-windup for LTI state-constrained control of wave energy converters

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Design and experimental assessment of an anti-windup for LTI state-constrained control of wave energy converters

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