energies Article
Application of Predictive Feedforward Compensator to Microalgae Production in a Raceway Reactor: A Simulation Study Andrzej Pawłowski 1, * ID , José Luis Guzmán 2 and Sebastián Dormido 1 ID 1 2 3
*
ID
, Manuel Berenguel 2 , Francisco G. Acíen 3
ID
Departamento de Informática y Automática, ETSII, UNED, 28040 Madrid, Spain;
[email protected] Departamento de Informática, University of Almería, CIESOL-ceiA3, 04120 Almería, Spain;
[email protected] (J.L.G.);
[email protected] (M.B.) Departamento de Ingeniería, University of Almería, CIESOL-ceiA3, 04120 Almería, Spain;
[email protected] Correspondence:
[email protected]; Tel.:+34-9-1398-7147
Received: 24 November 2017; Accepted: 2 January 2018; Published: 4 January 2018
Abstract: In this work, the evaluation of a predictive feedforward compensator is provided in order to highlight its most important advantages and drawbacks. The analyzed technique has been applied to microalgae production process in a raceway photobioreactor. The evaluation of the analyzed disturbance rejection schemes were performed through simulation, considering a nonlinear process model, whereas all controllers were designed using linear model approximations resulting in a realistic evaluation scenario. The predictive feedforward disturbance compensator was coupled with two feedback control techniques, PID (Proportional-Integral-Derivative) and MPC (Model Predictive Control) that are widely used in industrial practice. Moreover, the classical feedforward approach has been used for the purpose of comparison. The performance of the tested technique is evaluated with different indexes that include control performance measurements as well as biomass production performance. The application of the analyzed compensator to microalgae production process allows us to improve the average photosynthesis rate about 6% simultaneously reducing the energy usage about 4%. Keywords: disturbance compensation; feedforward control; process control; PID; MPC
1. Introduction Disturbance compensation is a very important aspect that needs to be considered in control system design for a particular process [1]. In such a case, the control technique should be able to maintain the controlled variable close to the reference despite the external disturbances that influence the controlled process. For this reason, most industrial processes need a personalized control scheme to achieve the performance requirements [2]. From a control system point of view, the process disturbances can be grouped as unmeasurable and measurable quantities depending on their origin. The measurable disturbances can be handled by the feedforward structure that can compensate the disturbance before its effect appears on the controlled variable. However, in many industrial applications, the control system consists of a feedback controller, since it is able to provide a set point tracking, reduces the influence of plant-model mismatch as well as compensates for process disturbances [1,3]. Due to this simple structure, the control system focuses only on one of these issues and provides weak performance for the other problems [4]. In the classical approach, the feedforward compensator is obtained as the quotient between the disturbance and the process dynamics. Nevertheless, this structure is rarely realizable, e.g., due to dead time inversion issues. This problem has been analyzed by several researchers during the past decades and drives to design more advanced tuning rules for feedforward Energies 2018, 11, 123; doi:10.3390/en11010123
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schema [5]. Other approach to measurable disturbance compensation consists in the application of the MPC (Model Predictive Control) technique. Due to simultaneous solution for feedback as well as feedforward control, this control technique is frequently used in industrial processes [6,7]. Regarding the feedforward action, the most relevant advantage of the MPC algorithm is related to the prediction mechanism that can handle future disturbance information [8,9]. Nevertheless, considering the internal relationship, the MPC feedforward and feedback co-design needs a compromise, where the design is oriented considering the most relevant aspects [10,11]. Taking into account all these properties, a new predictive feedforward compensator has been recently proposed in [12], merging advantages of classical MPC-based solutions. In this case, the compensator can be easily coupled with any feedback controller and its design is independent of other elements in the control loop. Moreover, it is also able to consider future disturbance estimations to improve a disturbance attenuation. The disturbance compensation aspects are also very important in bio-processes, where the control technique has to reduce the influence of external disturbances [13,14], and this is the case of microalgae production in raceway reactors. In such a system, the main task of the controller is to compensate for the influence of external disturbances that affect the photosynthesis rate. The microalgae culture is influenced by many factors, where the most important variables are related to: solar light viability, temperature of the medium, dissolved oxygen and medium pH [15]. In this context, a proper disturbance attenuation could improve the control system accuracy and, as a consequence, the biomass production performance. The remaining variables of the microalgae cultivation process such as dissolved oxygen and pH need to be regulated using a proper control system [16]. Both variables, dissolved oxygen and pH, are characterized by highly changing dynamics (influenced by solar radiation and photosynthesis rate) and their values should be maintained within optimal values for each strain. The photosynthesis rate is influenced mostly by solar irradiance, temperature and pH among others. It is well known that injection of CO2 has a strong influence on the pH level since it affects the growth medium. Moreover, the usage of CO2 represents important operational expense for microalgae culture and its supply excess should be avoided [17]. Taking advantage of this relationship, the control approach uses the pH value to define the amount of CO2 and time instance for its injection [18,19]. Considering all previously mentioned aspects, the disturbance compensation scheme can play a decisive role in boosting control performance and improving the rentability in biomass production process. In this study, we provide a practical evaluation of a predictive disturbance compensation technique introduced in [12] in order to highlight the most important advantages and drawbacks of such a scheme. As was shown in previous works [12,20], the main advantage of such a technique is the ability to handle the complex dynamics required for perfect disturbance compensation. In the analysis performed previously, a complete knowledge of the process model was assumed and the future disturbance signal was known a priori [20]. From the theoretical point of view, these assumptions allow us demonstrate that predictive feedforward compensator is able to compensate completely the disturbance before its effect appears on the controlled variable. Nevertheless, this assumption can be rarely achieved in an industrial control system, where the usage of simplified models is a common practice. Moreover, the future values of disturbance signals need to be estimated, which also introduces additional uncertainty resulting in performance degradation. Additionally, in this work, the original approach based on a GPC (Generalized Predictive Control) algorithm is extended, with a constraints’ handling mechanism in the feedforward controller being a new future and very important aspect from practical implementation point of view. Moreover, state of the future disturbances required for predictive mechanism is estimated using a Double Exponential Smoothing (DES) technique that was proposed in [8] for MPC control techniques where future information on disturbance signal is required. In the performed study, we use PID and MPC techniques as a feedback controllers that are coupled with the predictive feedforward compensator to show its influence on the control performance in two cases. The evaluation is performed on the biomass production process using a
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raceway photobioreactor model developed in [21]. The test bed is built using a first principle raceway reactor model as a process plant and all controllers are developed using linear model approximations obtained close to its operating point. This approach considers a plant-model mismatch allowing us create a realistic control scenario. Taking into account all these features, it is possible to obtain results similar to those obtained in the real plant. However, the implementation in simulation environments has the advantage of the repeatability of the external parameters that influence the control system. In such a way, the changes in production performance depend only on the applied control techniques and giving measurable changes in biomass production. All the analyzed control systems are evaluated using control performance as well as biomass production indexes. Additionally, the test bed system is evaluated in terms of energy usage efficacy. 2. Feedforward Compensators This section is devoted to introducing the disturbance compensation approaches used in the performed analysis. First, the classical approach is introduced and combined with a PID controller. Next, the GPC-based implicit feedback/feedforward structure is described. Afterwards, predictive feedforward compensator originally introduced in [12] is summarized. Moreover, the previous approach is extended with constraints handling mechanism and the future disturbance estimation technique is also provided. It needs to be highlighted that, in this paper, we analyze the disturbance rejection problem in a process that is affected by dead time. This issue makes the classical feedforward compensator less effective and a more complex compensator is required. This is due to dead time inversion issues (see [20] for more details). Moreover, for simplicity, the process, Pu , and the disturbance dynamics, Pv , are represented by a first order system with dead time using the following representation: Pu (s) =
Ku e−sdu , 1 + sTu
Pv (s) =
Kv e−sdv . 1 + sTv
2.1. Classical Feedforward Structure for Disturbance Compensation The block diagram of a classical feedforward disturbance compensation problem is shown in Figure 1 and consists of a feedback controller C, process Pu , a feedforward controller C f f and process Pv , which relates a measurable load disturbance v with the process output y. The remaining signals are: set point w, control signal u (composed of u FF and u FB provided by feedforward and feedback controllers). In this scheme, the load disturbance is fed through a feedforward compensator whose output is added to the feedback control action. From the process control point of view, the goal of this structure consists in a feedforward compensator design in a way that the influence of the measurable disturbance on the controlled process is minimized.
v (t )
w(t )
+
- Cff
u (t)
C
u (t)
Pv
FF
+
FB
u(t)
Pu
y (t ) +
-1 Figure 1. Feedback plus feedforward control scheme.
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Using block algebra, it can be seen that the analytical expression for disturbance compensation is given by: Pv − C f f Pu y = , (1) v 1 + CPu and, from this expression, it can be observed that the disturbance can be completely attenuated when C f f = Pv /Pu . Nevertheless, the resulting compensator is rarely realizable, since it could contain unstable or non-causal dynamics, and thus other developments should be investigated. In a first approach, low-order feedforward and feedback controllers are considered. In this case, we consider a PI or PID controller as the feedback controller, such that: 1 C (s) = K 1 + + sTd , (2) sTi where K refers to proportional gain, Ti is the integral time and Td is the derivative time (Td = is set to 0 if a PI controller is used). Note that, in a real implementation, features like filters, anti-windup and limitations must be added [1]. In addition, we also assume a low-order feedforward compensator (a lead-lag filter with a dead time), such that: C f f (s) = K f f
1 + sB f f −sL ff, e 1 + sT f f
(3)
and K f f , −1/B f f , −1/T f f and L f f refers to the static gain the zero, the pole and the dead time, respectively. Note that more complex structures are seldom used [22] in practice. Taking into account the aforementioned process transfer functions, it is possible to achieve perfect disturbance compensation by making K f f = Kv /Ku , B f f = Tu , T f f = Tv , L f f = −ρ and ρ = du − dv , where du is process time delay between input and output, and dv is time delay disturbance and output. 2.2. Generalized Predictive Controller This section briefly describes the GPC algorithm with intrinsic feedforward capabilities, that is when the GPC takes explicitly into account the dynamics of the measurable disturbances [23]. As is well known, GPC consists of applying a control sequence that minimizes a multistage cost function of the form: N
J=
∑
j= N1
δ( j)[yˆ (t + j|t) − w(t + j)]2 +
Nu
∑ λ( j)[∆u(t + j − 1)]2 ,
(4)
j =1
where yˆ (k + j|t) is an optimal system output prediction sequence performed with data known up to discrete time t, ∆u(t + j − 1) is a future control increment sequence obtained from cost function minimization with ∆ = (1 − z−1 ), N1 and N are the minimum and maximum prediction horizons, respectively, Nu is the control horizon and δ( j) and λ( j) are weighting sequences that penalize the future tracking errors and control efforts, respectively, along the horizons. The horizons and weighting sequences are design parameters used as tuning knobs. The reference trajectory w(k + j) can be the set point or a smooth approximation from the current value of the system output y(t) towards the known reference by means of a determined filter [23]. In Equation (4), the j-step ahead prediction of system output with data up to time t, yˆ (k + j|t), is computed using the following CARIMA model [23]: A ( z −1 ) y ( t ) = z − d u B ( z −1 ) u ( t − 1 ) + z − d v D ( z −1 ) v ( t ) +
e(t) , ∆
(5)
where v(t) is the measured disturbance at time t, e(t) is a zero mean white noise, A, B and D are adequate polynomials in the backward shift operator z−1 and du and dv are the dead time in samples of the process and disturbance dynamics, respectively. Notice that the effect of the measurable disturbances is represented in Equation (5) by the term z−dv D (z−1 )v(t). Consider a set of N-ahead
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predictions in Equation (5) where different horizons are used for output prediction and disturbance estimation, N, and future control action, Nu , respectively [23]. The prediction equation can be expressed as: y b = Gu + Hv + f; f = Gp ∆u(t − 1) + Hp ∆v(t) + Sy(t), (6) Version December Version December 22, Version 2017December 22, submitted 2017 Version submitted 22, toDecember Energies Version 2017 to submitted December Energies 22, 2017 to submitted Energies 22, 2017 submitted to Energies to Energies 6 of 20 6 of 20 6 of 20 where Gp , Hp and S are matrices of polynomials that are used to calculate the contribution to future Version December 22, 2017 submitted to Energies 6 of 20 process outputs of the past and present values of the process output, disturbance and control signal where upwhere up = where where where upy(t) = Δu(t − Δu(t 1), vupp−= =1), Δv(t) Δu(t vp =u − and 1), =ypvΔu(t and = − Δv(t) = yp1), for Δu(t =vand y(t) simplicity. = −yfor 1), Δv(t) vpy(t) Hence, and = Δv(t) for yp simplicity. the = Hence, and y(t) solution ypfor the =Hence, simplicity. solution of y(t) thefor the GPC of simplicity. solution Hence, the GPC the ofHence, the solution GPC th pΔv(t) p= p p =simplicity. values (see [23] for more details). Then, Equation (6) becomes: controllercontroller when no when controller constrains no constrains when are controller considered no constrains are controller when considered is obtained no areconstrains when considered is obtained minimizing no constrains areisminimizing considered obtained J with are considered minimizing respect is J obtained withtorespect u,isJminimizing which obtained with to u, respect leads which minimizing Jto:with toleads u, respect which to: J with leads to respe u, to: wh where up = Δu(t − 1), vp = Δv(t) and yp = y(t) for simplicity. Hence, the solution of the GPC Version December Version December 22, Version 2017 submitted 22, December 2017 Version submitted to 22, Energies December 2017 Version tosubmitted Energies 22, December 2017 to Energies submitted 22, 2017tosubmitted Energies to Energies 6 of 20 6 of 20 6 of 20 y b= Gu + G + H v−p(G += ,to− (7) −1 + Hv −1 u −1 Sy p −1 u,which pG pw controller when no constrains are considered is=obtained J− with respect leads to: w −(G G w Sy − G (G Sy G uppG G Sy w Hv − G G (G G Hv Sy uHp − −vpG G ) Sy H G Hv u− − G GG G HpHv vpp(8) )− −G GHv H (8) (8) u = K−1u(G K u =minimizing K u = K u K p− pp ppu p− pw p p pv pp)− pu p v− p )G H pv
−1 where uuppu ∆u t(G 11), )Q ,G. vppG = ∆v (Finally, )and and = yreceding )Q=the for simplicity. Hence, solution ofthe the w − Sy − G G uyQ G − G H vΔv(t) )strategy, (8) ==Δu(t K where where where u u= where ufor = Δu(t = − Δv(t) 1), = vwhere =G Δv(t) yG. 1), = v− and Δu(t y(t) =G y(Hv Δv(t) − = 1), simplicity. y(t) Δu(t and vhorizon for = yto simplicity. = 1), Hence, y(t) vpand = for the yΔv(t) Hence, simplicity. solution =the y(t) and the for of ysolution Hence, the = simplicity. y(t) GPC of for the solution simplicity. Hence, GPC the Hence soluti GPC p= p Δu(t p pto p pFinally, p−= p= pt+ ppaccording ptto pG. p p− phorizon pvalue where K= where Qλ(uK where K Q where K− where K = + G Finally, G G. Finally, + according + receding G G. according the horizon receding according to strategy, only the the receding only first to strategy, the the horizon receding first only of value u strategy, is the horizon of first uonly value strategy, isofthe the of first uonl is λ + λaccording λpthe λ Finally, −1 −1 −1 −1 −1 u, GPC controller when no constraints are considered is obtained minimizing J with respect to controller controller when no constrains when controller no constrains are when considered controller no are constrains considered is when obtained controller are no is constrains considered minimizing obtained when no are minimizing constrains is J obtained considered with respect are J minimizing with is considered to obtained u, respect which J minimizing with is to leads obtained u, respect which to: minimizing J leads to with u, which to: respect J leads with to u, to Δu(t) isif ,first Δu(t) given by: isoffirst given , matrix Δu(t) by:is , Δu(t) by: Δu(t)by: is given by: re computed, computed, Δu(t). Hence, computed, Δu(t).ifHence, k Δu(t). iscomputed, theiffirst kHence, is row the computed, Δu(t). first of if kmatrix row isHence, the Δu(t). ofK first matrix if,row kHence, isK the of matrix k is row K the row K ofgiven matrix K is ,given where K which = Qλ +leads G G. to: Finally, according to the receding horizon strategy, only the first value of u is −11 −1 −1 −1 of u Sy −1 Sy ) − ,kG Δu(t) by: computed, Δu(t). Hence, if k isΔu(t) row Kw (G w − G (G Sy − G G (G G w −− G G G (G Hv uu − − w G G −kG G Hv H (G uSy w G − G H GHv G vp− Sy )− u G − − G G H(8) G )uvpp− −kG GG (8) HHv )pkG G(9) (8 u = K =−1 K uu = K = K p= pp− p= p pu p p ppHv pu p pv Δu(t) kG =matrix Δu(t) kG Δu(t) kG = kG kG w Sy − − w G kG −G Sy G kG w − Hv kG − − w Sy kG − H −G kG kG Sy H vp− kG G uvHv (9) kG (9) Hv H −pv− H uthe = first K= (uG= wK −kG G Sy GkG Gppis u−given − G Hv G H vv− )p− ,kG (8) p− pu pΔu(t) ppp− p− p pHv pppkG pv pG pkG pp− pkG ppH ppp p− p p− pkG pp
Δu(t) =λ+kG −Finally, Sy − G uthe − kG Hv − kG H (9) where K == Q GG G. according to the receding horizon strategy, only the first value of uofisthe where Kunconstrained where Q K =w where Q K = Q where K Q K = Q + G. 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In consequence, different in unconstrained unconstrained control law unconstrained control (equation law 9) unconstrained (equation control is used. law 9) Inunconstrained (equation isconsequence, control used. 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Δu (w(t), and signals Thus, considered y(t), Δv(t)). Δu (9) and Thus, signals Δv(t)). y(t), (9) (w(t), Δu Thus, and Δv (9 f[z wwith: G Sy= − (G G uwith: G G Sy Hv − G G G v(G (8) G Gand −y(t), G u m(z) = with: K =(G u K = m(z) kG =z (z) kG m(z) = with: kG m(z) = kG m(z) = kG p− where ppzconsidered (z) pG (z) p (z) p [z (z) where = , zp (z) (z) ,u, .where ,.G z.H =p.ppp.u[z .p(z) ,p,J)− zzleads . (z) .with where .,.z.Hv .],respect = , n(z) zp[z ,− .(z) .G .= .],.K ,H −kG .u, = zn(z) ,(8) , .S.w −kG ],.,and z.− n(z) . .G l(z) , zto: S ,= .Sy and .=−kG .p. − ],.l(z) .n(z) , z S= =p u],−kG n(z) =S = pw p− pp p pz[z p )= pl(z) zwhere z− z z− zv zu oller when no constrains controller areand considered when no is constrains obtained are minimizing Jz= with is obtained respect minimizing to which to: to which leads −1 y ( t ) , ∆u ∆v ( t ) ). Thus, Equation (9) can be expressed as: can expressed (G w − G Sy − G G u − G Hv − G H v ) (8) u = Kcan−kG be expressed be as: can be as: expressed can as: be expressed can be as: expressed as: p p p p p Gp .−kG Moreover, αpG and Moreover, α are, andG respectively, α+2 .αare, Moreover, respectively, Gαpf.polynomials, Moreover, αp respectively, andpolynomials, αfαincluding are, respectively, polynomials, αfincluding the are, coefficients respectively, polynomials, the including coefficients ofpolynomials, the theincluding past coefficients of the past including the of coefficient the the past −kG p . Moreover, p. α pf−kG p and p and G−kG dfup+1 dare, U +N = with:Km(z) pz (z) where , z duaccording , horizon . . .. ..to , zstrategy, ], n(z) = Sand where = Qλ= K= according Qpλz (z) where K horizon = −kG Q +kG Gwhere G. Finally, +G G. to[z the Finally, receding the receding only the first +strategy, value G G. of l(z) Finally, u only is =theaccording first valuetoofthe u is receding horiz −1 −1 λ − G Sy (G − Sy Hv − G H p vp ) (G w − G G u w G − G Hv − G − G H G v ) u − G (8) (8) u = K u = K p p p p p p p p H ) and H future ) and (−kG future H ) H) and (−kG values future H H) ) of (−kG values and the future H disturbance. H) of ) the and values (−kG disturbance. future of Depending H) the (−kG values disturbance. Depending H) of on values the the disturbance. relation Depending on of the the between relation disturbance. on Depending the between the relation Depending on the between the relation on the th (−kG (−kG (−kG (−kG (−kG p p ( t ) + n ( z )−1 −1 ∆u (the tf)first = m (z)of w ypfirst (,= tΔu(t) )+ + l (zΔu(t) )pincluding ∆u (+ tcomputed, )l(z)Δu(t) + (Δu(t) α−1 (, zcoefficients )of + α(α (fz(z) ))∆v t+p)if(α ,n(z)y(t) GΔu(t). Moreover, αpifp kand α are, respectively, polynomials, the of the ppn(z)y(t) K=Q + G G. according tothe receding horizon strategy, only the first value is + (z))Δv(t) + (+ α (z))Δv(t) (z)the + first α+p+(z))Δv(t) (10) +(10) αp(α (z))Δv(t) (10 αp ( Δu(t) = m(z)w(t) Δu(t) = + Δu(t) n(z)y(t) + m(z)w(t) n(z)y(t) l(z)Δu(t) + = n(z)y(t) + m(z)w(t) (α + = +u+ l(z)Δu(t) m(z)w(t) l(z)Δu(t) l(z)Δu(t) (α(10) + p .Finally, is given by: Δu(t) is given by: Δu(t computed, Hence, computed, is Δu(t). row Hence, matrix if m(z)w(t) k is the K row of matrix K Δu(t). Hence, kpast row of matrix K λ−kG ff (z) f is f (z) f (z),+ and and the input-output and the input-output process time, input-output the dead d and ,time, such theinput-output dead dfactors process , such time, are factors input-output dcalculated dead , such are time, factors calculated dead d , such are time, calculated factors du , such a disturbance-output disturbance-output dead timedead timethe ddead disturbance-output time dprocess time dead ddead time dprocess v process vdead v and u −1dvdisturbance-output disturbance-output Hpif )Finally, and future (−kG H) of, Δu(t) the disturbance. Depending the relation between the is given by: uted, Δu(t). k is the first row of matrix K eK= Qλ(−kG where Kaccording = Q +Hence, G G. to the G.values receding Finally, according horizon strategy, todthe receding only the horizon firston value strategy, of udvisu+2 only theufirst value of u is d +Nu d +N λ +G 1 d d2u +1 du +2 ddU dUu+N +1 dd +Ndu +1 du +2 uu+1 uu+2 u dU +Ndu +1 uU+2 U U d + d + + N with: kG m(z) = kG m(z) = with: = kG (z) where p (z) where = p− pu [z (z) where ,Sy z= [z pways. p−1 ,zkG (z) ..= .,.kG z.configuration .where = ..p,this [z .− .zp .(z) .(z) ],,.in n(z) ,nand zρΔu(t) ,= .work, [z p .0.],z−kG (z) .− n(z) .in .kG ,ρd,zzvthis = S = and .1, ], .in l(z) .n(z) .),this z(ρS .> = and zp,f .the .and ],.ρα.> n(z) = .(nd S,G zand = n( = )> α 1, )(9) 1, the α)(nd > inΔu(t) in=kG different ways. ways. the configuration For in different For in analyzed ways. different configuration For in this work, in For ρp > configuration analyzed this (nd and + analyzed (nd 0−kG work, + and dkG work, + 0− dl(z) > 0p+ d−kG = Δu(t) = w Sy − kG G w kG −with: kG Hv − vwork, Hv − kG (9) H v[z − Sy uand − kG with mdifferent (m(z) zdead ) with: = pzp− (in zwith: )kG ,different where pways. ((z) zkG )configuration =m(z) [(z) ,kG zanalyzed ,the .− .analyzed ,zuH z,pthe ]0z,.where (= z)> = S> l.(nd z,= )> = zFor z zthe z.(z) vthe v.kG p,f v], p,f pz= pz pm(z) pzkG pw p,and p,−kG pl(z) zthe the process input-output dead d.G factors disturbance-output time , Δu(t) ispgiven by: , Δu(t) ispgiven by:are calculated uted, Δu(t). Hence, computed, if k isthe first Δu(t). rowdHence, ofand matrix if kK is−1 the first row of matrix Ktime, u , such v Δu(t) = kG w − kG Sy − kG G u − kG Hv − kG H v (9) −−kG GG . Moreover, α and α are, respectively, polynomials, including the coefficients of the past . Moreover, G . Moreover, α and G α . α Moreover, are, and respectively, α G are, α . Moreover, and respectively, polynomials, α G are, . α Moreover, respectively, polynomials, and including α are, α polynomials, and respectively, including the α coefficients are, the respectively, including polynomials, coefficients of the the past polynomials, coefficients including of the past the including of coefficie the pas −kG −kG −kG −kG polynomials polynomials are as polynomials follows: are as follows: polynomials are as follows: polynomials are as follows: are as follows: pp p fppf p pp p p f pp fp pp p f f > 1, the measurable α in different ways. For configuration analyzed this 0 andunconstrained (nd + pdv ) of p,f of the disturbances This unconstrained control Thisthe unconstrained law control lawin knowledge thework, GPCρconsiders of>future This measurable knowledge disturbances future law GPC considers knowledg of the GPC of control considers Δu(t) (= kG w Δu(t) kG Sy kG GH w −and kGfuture Sy Hv − kG G H u v − kG Hv − (9) kG H vvalues (9) H ) and future Hp p−)(−kG and (−kG future H) values H) of H values (−kG )(−kG and disturbance. of future H) H the values ) disturbance. (−kG and Depending future of the H) (−kG disturbance. values Depending on the of H) the relation Depending disturbance. the between ofrelation thebetween on disturbance. Depending the the between relation the Depending onbetween the relatio tho (−kG −(−kG H )p(−kG and future (= −kG H) values of the disturbance. Depending on the relation pu p the p p p p on p− pp)(−kG p p −(d −(dv +nd−1)−(dv +nd−1) −1 −1 −1−(dv +nd−1) −1 −1 v +nd−1) −(d v +nd−1) polynomials are as follows: unconstrained control law ofand the GPC considers of future measurable disturbances α α α α α (z) = α + = α α z + (z) + α . = . z . α + α + + . (z) α . . + z = z α α + (z) . + . . α + = z α z α , + . α . z . , + z α + . . . , + z αare z , must be, so and references. Notice that references. an optimization Notice problem that an optimization must be solved problem using and must a references. quadratic be solved cost Notice using function, that a quadratic an optimization cost problem pknowledge pp(z) p p p p p p p p p p p p p p p p pfunction, 1 2 1 2 1 2 v) 1 1 (nd+d (nd+d ) 2 time, (nd+d ) ,2 such(nd+d ) (nd+d ) v v v v the disturbance–output dead time d and the process input–output dead d factors the and time the input-output process input-output the dead process time, input-output dead dthe process time, factors ddead input-output ,usuch process are time, factors calculated duinput-output ,dead such are calculated time, factors ddead , such time, calculated factors du , disturbance-output disturbance-output dead disturbance-output time dead dv and time disturbance-output disturbance-output dv and dead time ddead time dv and v dprocess vdead v and u , such uthe uare eferences. Notice that an optimization problem must solved using a solution quadratic cost function, unconstrained control This law unconstrained of thedifferent GPC considers control law knowledge ofbe the GPC ofexplicit future considers measurable knowledge disturbances of future future measurable disturbances when constraints are taken when into constraints account are since taken there into is account no since there when is [24]. noin constraints explicit For solution are analysis taken [24]. the into For account future since analysis there theis no explicit −(d −1 v +nd−1) calculated in ways. For the configuration analyzed this work, ρ > 0 and ( nd + d ) > 1, v α (z) = α + α z + . . . + α z , ) > 1, the ) α > 1, the α ) > 1, the αvp,) in and different in ways. different For in ways. the different configuration For the ways. in configuration different For analyzed the in ways. configuration different in analyzed this For work, ways. the in configuration analyzed this ρ For > work, 0 the and in configuration ρ this (nd analyzed > work, 0 + and d ρ (nd in analyzed > this + 0 d work, and in (nd ρ this > + work, d 0 and ρ (nd > + 0 d and p part p2 following p(nd+d v p,f v p,f 1 the future and the future has andpfollowing part the future has and form: part thehas future form: and following the partfuture has following part has following form: form: v ) form: constraints are taken into account since there is used. no explicit solution [24]. For future analysis the eferences. Notice that and anreferences. optimization Notice problem that an must optimization be solved using problem av different quadratic must beterms cost solved function, using arepresented quadratic cost function, unconstrained control law unconstrained (equation 9) control is law In (equation consequence, 9) is used. the In consequence, unconstrained in (9) the control are different law terms (equation in (9) 9) are is represented used. In consequence, th the α polynomials are as follows: p, f polynomials polynomials are as follows: polynomials are as follows: are polynomials as follows: polynomials are as follows: are as follows: (duzsolution −d (d (d )isu −d (d)+1 −d (dare )−d )+1 (du(d−d (d ) vu)+1 −d (d )+N −d (d (d ) −d −d )+1 (d (d −d )+N (du(9) −dv )+N (d v ) signals u −d v(w(t), v u vu vanalysis u −d vthe v Thus, uoperator vfuture u u vv)+N uu−d vv)+1 u −dv )+N and thelaw future has following form: strained (equation is used. In consequence, the different terms in (9) represented constraints are taken when into constraints account since there taken into no= explicit account since there [24]. no For explicit future solution [24]. For analysis the ascontrol polynomials inpart the operator as9)polynomials zare that multiply the operator considered that multiply considered y(t), as polynomials Δu signals and Δv(t)). (w(t), in the y(t), Δu (9) and z that Δv(t)). multiply Thus, considered signals αinf is α α α α (z) αff(z) z = α + z (z) α = z α + z (z) z + = . + α . . α + z (z) α z + = . . z . α + + z α α + z . . . + α z z + . . . + α + z . . . + α z f1f f1 f1 ff1 f1ff1 fN f1 ffN1 ffN fN fN 1 1 −(d −(dv +nd−1) −(dv +nd−1) −(dv +nd−1) −1 −1 −1 −1 −1 −1 Δu and −( dvv+nd−1) +nd(9) −1)−(dv +nd−1) ynomials inbethe operator z that multiply considered signals (w(t), y(t), Δv(t)). Thus, nstrained lawunconstrained (equation 9) iscontrol used. Inlaw consequence, (equation 9) the is used. different In consequence, terms in (9) are the represented different terms in (9) are represented α α α α α (z) = α (z) + α = z α + (z) . α . . = + z α + + . . α . (z) + z z α = + α . . + . z (z) + , α α = z α + + , . z . α . + z α + . , . . z + α ,z α ( z ) α α + . α , cancontrol expressed as: can be expressed as: can be expressed as: p p p p p p p p p p p p p p p p p p p p 1 2 1 2 1 2 1 2 1 2 2 (nd+d ) (nd+d ) (nd+d ) (nd+d ) (nd+d ) 1 (du −dv )+1 (d+ud−d (nd v v )+N v v v v u −dv ) αwhere (z) =118 αfrefers z (ddegree +ofαdegree + . to .D . the + αdegree zto f refers fthe fN 1 1 z to 118 nd to the where tond118 the where polynomial 118 the nd of the where refers degree polynomial nd of from refers the polynomial equation D from the of(w(t), the degree equation (5). D polynomial from Moreover, of (5). equation polynomial Moreover, Dpolynomials from (5).equation Moreover, polynomials DThus, from referring (5). equation polynomials Moreover, referring (5). Moreover, polynom referring elynomials expressedinas:the operator as where polynomials z 118 that multiply in nd the considered operator zrefers signals that multiply (w(t), y(t), considered Δu and signals Δv(t)). Thus, y(t), (9)the Δu and Δv(t)). (9) and the future and the part future has and following part the has future following form: and part the has form: future following and part the has form: future following part has form: following form: +disturbances α (z))Δv(t) + αp= (z))Δv(t) (10) Δu(t) = Δu(t) n(z)y(t) + m(z)w(t) l(z)Δu(t) + n(z)y(t) + (ααfcan l(z)Δu(t) + (αα Δu(t) m(z)w(t) += + l(z)Δu(t) (α past 119 αtopm(z)w(t) and past 119 future α to+ past α values α119p =and to α past future values disturbances 119α to αof and disturbances values future and of+ αbe future grouped values be αof into values disturbances can be= of into grouped α disturbances α can =αfbe into α . p grouped Additionally, +αcan α .beαAdditionally, into + α(10) into Additionally, αp + α α=f+. α A and the future part has following form: pcan f (z) p+ ffuture fof fpast p(z) fgrouped f n(z)y(t) pgrouped f .= e expressed as: nd 119 canto beto expressed as: 118 where refers the degree ofpthe polynomial D fromp equation (5). fMoreover, polynomials referring (z) + α (z))Δv(t) (10) Δu(t) =120m(z)w(t) + n(z)y(t) + l(z)Δu(t) + (α f p theprediction 120 the prediction mechanism 120 the prediction mechanism of 120 the the mechanism prediction of120 the the intrinsic compensator prediction of mechanism intrinsic mechanism of be the compensator also can intrinsic used of be also to compensator can intrinsic include used be also compensator future include can disturbance be to future also include can used disturbance be future also to uinclude used disturbance future incl (dintrinsic −d (du −d (dvu)the −ddcompensator )+1 (dcan (d −d (d (duuuthe −d ))+1 )+N (d (duuto −d −d ) used (d )+N )+N to v +2 u v +2 v )+1 u vv)+N v )+1 (du ααduz(z) +1 +N +1 d.dvvUvα +N du+ +1 u uv+2 119with: to past values of disturbances be into α α.ff.p+ Additionally, du,u− )) p )+ ((z) )+ α(z) α z.(z) αdvvfUdα z+ .d. u−d .. )..= + z+ .u−d + α + α zz(z) . z.−d .(d + αzf1+ z−d . .uv.−d +v )+1 α z (d.,= .z−d .d+ α,f.N.z.(d. u.−d vuα u,− f m(z) f= m(z)αp=and kGfuture m(z) kG pzwith: (z)αwhere p= =pzα[z(z) z=ddcan .ff(z) .grouped .z(.d= .= z+ [z ],+ ,uwith: n(z) ,= .α −kG .= .d,α z− Sv and ], n(z) where ==α+ (z) S = and [zfl(z) ., fkG f f (where f+ f11α fz(z) fz fp fN f−kG 1,zα 1z 1z 1 1 1 zα z+ N.z N Npz N + α(z) ,fl(z) ((z) zl(z)Δu(t) )m(z)w(t) = + ααf+ f= f 11zestimations, f1N (z) α (z))Δv(t) + (10) α (z))Δv(t) (10) Δu(t) = + n(z)y(t) Δu(t) + + + (α n(z)y(t) + l(z)Δu(t) + (α 1 f p f p 121 m(z)w(t) estimations, 121 estimations, v, 121 through estimations, v, the through 121 polynomial v, the through polynomial 121 α estimations, . the v, In polynomial through such α . a In v, scheme, the such through α polynomial a . scheme, the In the such implicit polynomial α a the scheme, . feedfroward implicit In such α the . a feedfroward In scheme, implicit compensation such a feedfroward the scheme, compensation implicit the feedfroward compensation implicit feed c f f f f f d +1 d +2 d +N u u U the prediction mechanism intrinsic can be also used to future m(z)120=−kG kG pzp(z) where pz−kG (z) =of , zrespectively, , compensator .α. p. .and . . ,polynomials, zαf are, ], respectively, n(z) = −kG S include and = G . Moreover, αp and G[z α .f Moreover, are, including polynomials, the G coefficients Moreover, including ofdisturbance the αpthe past andcoefficients αf are, respectively, of the pastpolynomials, −kG pthe p .l(z) 118 122where 118 nd where nd 118 where degree to nd the of118 refers the degree where polynomial to ofthe the nd 118 polynomial where D from of to nd the the equation refers D polynomial degree from tocompletely (5). of the equation the Moreover, degree D polynomial from (5). ofequation Moreover, polynomials D from (5). polynomials Moreover, equation referring D from (5). polynomials referring equation Moreover, referring polyno Moreo is included 122 refers is included in to control 122 is+1 inincluded law, control however 122 in law, is control included however the 122 law, isdegree in included the control however disturbance cannot in law, the control however disturbance becannot law, the be however cannot completely disturbance attenuated the be polynomial disturbance completely attenuated cannot as was be shown attenuated cannot as completely was in be shown completely asattenuated was in(5). shown attenu as in w the drefers du +2 daUdisturbance +N drefers d(5). +N udegree u +2implicit the Uthe refers to Dthe Equation Moreover, polynomials referring v, future through thekG polynomial αoff,.the such scheme, feedfroward compensation Gp .121Moreover, α)where and αnd are, respectively, coefficients of m(z) = estimations, kG H with: m(z) = pzp(z) where =H [z (z), zpolynomials, where .the p .In .z (z) .polynomial . . ,including zH) = [z ],u +1 n(z) , zfrom = .disturbance. .(−kG . .on . .S,the zand ],past n(z) =future = −kG Srelation andvalues l(z) =of the the (−kG H) future (−kG disturbance. values Depending ofdthe the,−kG H relation Depending and between (−kG on the the H) between disturbance. D (−kG (−kG pand fp z (z) zvalues p )pand p )l(z) 119 123to [9,13]. past αpThis to[9,13]. and past future 119 α and to future values αp119 of to values disturbances future past 119 of αparameter αpis and to values past future be αof grouped α and disturbances values future be into grouped of αstructure α can disturbances values = be into αin of + αthe α disturbances = can . α.into Additionally, be + α grouped α = .that can Additionally, αpfor be into + grouped αfα.GPC. = Additionally +GPC ααfor = . 123 isfuture due 123 This [9,13]. isλpast due parameter This 123 to λαand is [9,13]. parameter that 123 to isThis λ used [9,13]. that in due iscan the This used tointernal λ that is in due the iscan structure to internal λ in parameter that the of internal the used that GPC. of is the Itused internal in of the Itthe structure was internal shown It of structure was the that shown for ofαinto Itthe was p to f due fdisturbances pparameter f used fis pgrouped fwas pshown f GPC. pthat fsh 119 past α and αα values of disturbances can be into αare = αdstructure + αGPC. Additionally, is future included in control law, however disturbance cannot be completely attenuated as was shown pu ,time Hp122 and (−kG H) values the disturbance. Depending on the relation between the .) disturbance-output Moreover, αto and α G are, .time Moreover, respectively, αffpthe polynomials, and αftime are, the polynomials, coefficients including offactors the past the coefficients of the past G G fdin the process input-output and thedead process time, input-output dgrouped dead time, calculated such and the are process calculated input-output dea dead dofv and dead drespectively, disturbance-output dead p−kG f pdisturbance-output p vincluding u , such vfactors 120 124the prediction 120 the prediction mechanism 120 the mechanism prediction of the 120intrinsic the mechanism of the prediction compensator 120 intrinsic the of mechanism prediction the compensator intrinsic can be mechanism of also compensator the can used intrinsic be also to of include the can used compensator intrinsic be to future also include compensator used disturbance can future to be include also disturbance can used future be to also include disturbanc used futu to λ > 124 0, feedforward λ > 0, 124 feedforward λ compensator > 0, feedforward 124 compensator λ will > 0, 124 never feedforward compensator λ will > perform 0, never feedforward compensator will perform as a never classical as compensator perform a will classical disturbance never as a will disturbance perform classical compensation never as disturbance perform a compensation classical structure. as a compensation disturbance classical structure. disturbance compensati structure. c to λ parameter intrinsic the mechanism of the compensator can be also used to include disturbance [9,13]. This isprediction that isFor used the internal GPC. Itdways. that for+ and the process input-output dead du ,structure such are calculated bance-output dead time ddue Hp123)inand future (−kG H) H values )different and of future the (−kG disturbance. H) values Depending of work, the on disturbance. relation Depending between on the relation between G (−kG )ρthe >>shown 1, αfuture 1, the αp,fin this work, different ways. inpthe configuration ways. analyzed thein configuration in time, this analyzed ρthe >factors 0ofin and inthe different this (nd work, + For 0 the and the (nd configuration dv ) > the analyzed v For vwas p,f 121 125 estimations, estimations, v, through 121 compensator estimations, v, the through polynomial 121The the v, estimations, through polynomial α ..only 121 In the such estimations, v, polynomial αfthrough .aobtained In scheme, such v, α athrough the .scheme, In implicit such the aobtained polynomial feedfroward α scheme, implicit . 0, In the feedfroward αcompensation afimplicit .scheme, In as such feedfroward compensation a the scheme, implicit compensation the feedfroward implicit The121 125 ideal The compensator ideal 125 The ideal can 125 becompensator obtained can ideal be 125 The compensator can ideal when be only compensator λ when = canthe 0,be only as λ obtained = was when can 0, shown as bethe λonly was = shown when [13]. assuch only was λin Nevertheless, = [13]. when shown 0, Nevertheless, λin was =the [13]. 0, shown asNevertheless, was in the shown [13]. Neve inthe [1 ffobtained fpolynomial fin estimations, v, through the polynomial α In such a scheme, the implicit feedforward compensation 124 λ > 0, feedforward compensator will never perform as a classical disturbance compensation structure. ) > 1, the α ferent ways. For the configuration analyzed in this work, ρ > 0 and (nd + d and the process input-output the process time, duinput-output , such factors are time, calculated dp,f such factors are calculated rbance-output dead time disturbance-output dead time dv anddead polynomials are asdvfollows: polynomials are as follows: polynomials are follows: v dead u ,as 122 126 included 122 inincluded control 122scheme in is law, included control however 122 law, in is the control however included disturbance 122 law, the is incharacterized however included disturbance control cannot law, the in be control disturbance cannot completely however law, becharacterized the completely cannot however attenuated disturbance the attenuated completely asco-design disturbance cannot was shown as be attenuated was completely cannot in shown be asand completely in was attenuated shown as ain resulting 126 is resulting control 126 control resulting is scheme 126 characterized control resulting scheme characterized 126 control resulting by isimplicit scheme by control co-design implicit is scheme characterized by co-design for implicit isfeedback for co-design by feedback implicit andbefeedforward for by and implicit feedback part co-design for and and feedback feedforward part for and feedback feedforw part and and isisincluded in control law; however, the disturbance cannot be completely attenuated as feedforward was shown The ideal compensator can beFor obtained onlywork, when was shown [13]. omials as follows: the αNevertheless, > 1, the αp,f −1 ferent125are ways. For the in configuration different ways. analyzed theinconfiguration this ρ λanalyzed >=0 0, andasin (nd this + work, dv ) >in ρ1,> 0 and p,f (nd + dv ) the −(d −(d −1 −1 +nd−1) vin v +nd−1) in[9,13]. [9,13]. This is to the λ parameter that is used in the internal structure of the GPC. It was 123 127 123 This [9,13]. isp (z) duedue This 123 tobandwidth λin [9,13]. is parameter due to This λ 123 parameter that is [9,13]. due is used to This 123 λ that parameter in [9,13]. the is used due internal to This that λ the parameter is structure is internal due used to in λ of that structure the parameter the internal is GPC. used of that It in the structure was the GPC. is internal used shown of It in was the that the structure GPC. shown internal for It that of was structure the for shown GPC. of that It the was fo the high 127 the bandwidth high 127 the the high feedback 127 bandwidth in the the feedback high loop, 127 in bandwidth what the loop, feedback high results what bandwidth in the loop, results very feedback what in aggressive in the very results loop, feedback aggressive changes what in very loop, results aggressive changes in what control in very results in signal. changes control aggressive in very Due signal. in control aggressive changes Due signal. in changes contro Due α α α = α + α z + (z) . . . = + α + α z z + . . . + , α z , (z) = α + α z + . . . + α p1 p2 by implicit p p1(nd+dv )p2 for feedback and p(nd+d p2 p(nd+dv )G v) 126 are resulting control schemeare is as characterized co-design feedforward partp and p1 nomials as follows: polynomials follows: −(d −1 +nd−1) v shown that, for λ > compensator 0, feedforward will never perform as aactions, classical disturbance αλp (z) =124 αimplicit + α0, zfeedforward + .>.between . 0, + αcompensator z 128 ,implicit 124 128 > feedforward >the 124 λthe feedforward will λrelation never > 0,compensator compensator 124 feedforward will perform λnever > 0, as perform feedforward will acompensator classical never as perform disturbance classical compensator will never as disturbance asuch compensation classical perform will never compensation as aperform structure. classical compensation disturbance structure. aacannot classical compens structure to 0, the 128 relation implicit to the relation implicit 128 to between feedback the implicit feedback to and between the relation feedforward and feedback between feedforward relation actions, andafeedback between feedforward such actions, structure and feedback feedforward structure cannot anddisturbance such feedforward be actions, cannot structure used as as be such actions, used structure asbe such a disturban used cannot structu as a pλ p2128 p124 1to (nd+d v) 127 thethe high bandwidth the feedback loop, results in+nd−1) very aggressive changes in control signal. Due form: and future part hasin and following the future form: haswhat following form: and the−(d future part has following −(d −1 part +nd−1) v −1 vonly compensation structure. The ideal compensator can be obtained when λ = 0, as was shown =129alone αThe αpalone z compensator + .α.compensator .alone (z) + = αcompensator zwhat α zlimits +alone .,its . ideal .limits + α z= This ,limits 125 129α The ideal 125 ideal The can ideal be129αobtained 125 can be ideal only obtained can when compensator The be only λ obtained = when 0, can asapplicability. λonly was be obtained 0, when shown as can was λbe in only =shown [13]. obtained 0, when asNevertheless, was in only λ[13]. shown = 0,addressed Nevertheless, asinThis the λ was [13]. =addressed shown 0,can Nevertheless, as the was inparticular [13]. shown Ne th stand stand 129 stand disturbance disturbance stand alone 129 stand compensator disturbance what applicability. disturbance compensator what its limits compensator its what particular applicability. This what particular its issue applicability. limits This can issue particular be itswhen applicability. can be issue particular This be issue addressed can p (z) pcompensator p pcompensator pThe p2125 pcompensator 1 +disturbance 2125 1 v+ (nd+d ) (nd+dv ) 128 part to the relation and feedforward actions, such structure cannot beco-design used as a for feedback e future hasimplicit following form: between feedback in [13]. Nevertheless, the resulting control scheme is characterized by implicit (d )scheme (d −d (d −dis )feedforward (d (dufeedforward −d )+N (d )+N (d )for (dand −d )+1 u −d vfeedforward u characterized v )+1 u vpredictive u−d vthat v)+1 u −d v u −dv u vand 126 130resulting control resulting scheme 126 control resulting is characterized control 126 is resulting scheme by 126 implicit control resulting characterized by scheme co-design implicit control is co-design for by characterized scheme implicit feedback is for characterized co-design and feedback by implicit feedforward for and by co-design feedback feedforward implicit part and and co-design feedback feedforward part for feedback part feedfo with126 130 predictive with predictive 130 feedforward with predictive 130 compensator with feedforward predictive compensator 130 that with was compensator that recently was recently introduced compensator was introduced recently compensator in [12] that was and introduced in [12] which recently that and was is in which introduced summarized [12] recently and is summarized introduced which in [12] is and summarized in which [12] and αf (z)compensator = α f1 z +αα + issue . . . +can αf be zαaddressed + αf z + . . .is+ f f(z) f1 z + . . . + αThis f1Nzz particular f (z) = αf1 z 1z = 129 standhas alone disturbance what limits itsαapplicability. he future part following and the form: future part hasand following form: and feedforward part the high bandwidth invthe feedback loop, whichNresults in very aggressive 1 (d −d ) (d −d )+1 (d −d )+N u v u v u the high 127 bandwidth high feedback inbandwidth 127 the the feedback high 127 bandwidth loop, results the feedback high what in in bandwidth very results loop, theaggressive feedback what in very inresults the aggressive loop, changes feedback inwhat very inchanges results loop, aggressive control what inin signal. very control results changes aggressive Due signal. ininvery control changes Due aggressive signal. in cont chan Du 131the following 131bandwidth in following section. 131 inthe section. following 131 in section. following 131 in following section. section. α127 =inhigh α +compensator α + .was .loop, . +recently αwhat zthe f (z) f127 fin fin 1z 1z N 130 withnd predictive feedforward that introduced in [12]equation and which isthe summarized 118 where refers to 118 the where degree nd of refers the polynomial to the degree D from of the equation polynomial (5). 118Moreover, D where from nd polynomials refers (5). to Moreover, referring degree of polynomials the polynomial referring D from equation (d −d ) (d −d )+1 (d −d ) (d −d (d −d )+N )+1 (d −d )+N u v u v u v u u v v u v to=theα128 to therelation implicit 128 the relation feedback theα+ and implicit 128 feedforward to relation and implicit feedback feedforward between and feedforward actions, structure such and feedback actions, feedforward cannot structure and such be cannot used feedforward structure actions, asbea used such cannot actions, as structure abe used suchcanno as stru α128f (z) + αf1αzto (z) =implicit αf128 +zbetween . to . relation .+ αzbetween z the +actions, . . relation .+ αsuch zbetween fimplicit fbetween ffeedback ffeedback 1z 1 N f1 N 131 section. nd119refers tofollowing theαpdegree of119thetoαpolynomial from equation (5).ofMoreover, polynomials to in past and future values αp129and ofD disturbances future α129f compensator values can grouped disturbances into 119 α can to = past belimits αThis αreferring + and α .applicability. future into Additionally, ααf=what values αThis + of α canbe beaddressed groupe fpast pgrouped p particular fwhat p be f .disturbances 129 stand alone 129 stand disturbance alone disturbance stand compensator alone disturbance stand whatbe alone limits 129 compensator what stand disturbance its applicability. limits alone its what disturbance compensator applicability. its compensator This limits issue particular its can applicability. limits issue particular addressed can itsAdditionally, applicability. beThis issue addressed particular can This issue particu ca α120 and αdegree values ofthe disturbances can be grouped into αbeD =also αp used + αthe . include Additionally, et nd to the118 of the refers polynomial to the degree D from ofequation theofpolynomial (5). Moreover, from polynomials equation (5).referring Moreover, polynomials referring the future prediction mechanism 120 nd prediction of the intrinsic mechanism compensator the intrinsic can compensator 120 prediction can be future also mechanism used disturbance to include of the intrinsic future disturbance compensator can be al p refers f where fto 130 with predictive 130 with predictive feedforward 130 withfeedforward predictive compensator 130 with feedforward compensator predictive that 130 was with recently compensator that feedforward predictive wasintroduced recently feedforward that compensator was introduced in recently [12]compensator and thatinintroduced which was [12]recently and isthat summarized which inwas [12] introduced recently isand summarized which in introduced [12] is summarized and in which [12] ediction mechanism of the intrinsic compensator can be also used to include future disturbance st α and future 119 α to values past α of disturbances and future α can values be grouped of disturbances into α = can α be + grouped α . Additionally, into α = α + α . Additionally, 121p estimations, fv, through 121 estimations, v,insection. through such polynomial a scheme, the . In implicit estimations, scheme, v, thethrough compensation implicit the feedfroward compensation αf . In such a scheme, th p the polynomial f αf . Inthe p121such f afeedfroward p f polynomial f 131 in following 131 in following section. 131 following 131 section. in following 131 inα section. following section.
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changes in control signals. Due to the implicit relation between feedback and feedforward actions, such structure cannot be used as a stand alone disturbance compensator, which limits its applicability. This particular issue can be addressed with a predictive feedforward compensator that was recently introduced in [12] and which is summarized in the following section.
Predictive Feedforward Compensator Version December Version22, December Version 2017 submitted December 22, 2017 tosubmitted Energies 22, 2017 submitted to Energies to Energies 6 of 20 6o From the previously presented approaches, it can be deduced that the analyzed solutions provide some advantages and also drawbacks. In [12], it was found that advantages of previously introduced where up =where where up − Δu(t u−p 1), = Δu(t vp = Δv(t) =1), Δu(t vand −y1), Δv(t) and = Δv(t) for yp simplicity. = and y(t)y for=Hence, simplicity. y(t) for thesimplicity. solution Hence, the ofHence, the solution GPC theofsolutio the G p = p =vpy(t) schemes can be merged using a predictive feedforward compensator that isp derived from the GPC controller no constrains controller when noare constrains when considered constrains areisconsidered obtained are considered minimizing isconfiguration. obtainedisJminimizing obtained withItrespect minimizing Jshown with to u, respect which J with leads to respect u, which to: toleads u, w algorithm withcontroller implicit when feedforward action using ano specific algorithm was that an improved disturbance compensator can in the feedback structure obtained −1 by setting −1 the error (Gbe w (G − GG (G GSy upp− −G GSy G Hv − − G G GG HpHv vpp )− − G GHv Hp v−p )G H(8) u = K−1 u =− KG uSy= K− pw pw pu p− pu p vp ) equal to zero. This assumption can be confirmed by the GPC control law provided in Equation (8). Taking into account the set set y(according tthe )Finally, for j =according dreceding + 1,tostrategy, . the . . , horizon treceding + donly + strategy, N, and where Kthat =Q where Kpoint = where QFinally, K Qw G G. G=to according G. +j)G=to G. receding tot + the horizon the horizon firstthe only value strategy, theoffirst uonly isvalue the firs of λ + λis+ λ (Finally, −1 −1 −1 results in: system is in a computed, steady state, then u = ∆u ( t − 1 ) = 0. Equation (8) representing control law p computed, , matrix Δu(t) is , Δu(t) by: Δu(t)by: is given by: Δu(t). computed, Hence, Δu(t). if k isHence, the Δu(t). first if k row Hence, is the of matrix if first k is row the K offirst row K ofgiven matrix K is ,given −1 = kG Δu(t) = w kG −− kG kG w Syp− kG SyG −upkG − − kG kG Gpu H Hv vp− kG kG Hv Hp−vpkG(9) H p vp uΔu(t) = (Qλ=+kG GΔu(t) G)− (− GSyw Hv GG H v− )−.kG (11) p− pu ppkG pHv p pp−
This Thisaunconstrained control Thisfeedforward unconstrained law control of the compensator GPC law control of considers thelaw GPC ofknowledge considers the GPC considers knowledge future measurable knowledge ofonly future of measurable disturbances future measurable disturban Following thisunconstrained equation, typical structure can of be obtained when andpreviously) references. and and, references. Notice and that references. an Notice optimization thatNotice anλ optimization problem that an optimization must problem be solved must problem using be solved must a quadratic be using solved cost a quadratic using function, a quadratic cost funct λ = 0 (as shown as a consequence, Q = 0 results in: when constraints whenare constraints taken wheninto constraints areaccount taken into are since taken account there into issince account no explicit theresince issolution nothere explicit is [24]. nosolution explicit For future [24]. solution analysis For [24]. future theFor analysis futur 1 −1 u (equation = −G−law Hv Glaw H9) visp ,used.9) (12) pconsequence, unconstrained unconstrained control law unconstrained control 9) control (equation is−used. In (equation Inisconsequence, used. the different In consequence, the terms different in (9) theare terms different represented in (9) terms are represen in (9) a as polynomials as polynomials in the operator as polynomials in the z that operator multiply in thezoperator that considered multiply z thatsignals considered multiply (w(t), considered signals y(t), Δu (w(t), signals andy(t), Δv(t)). (w(t), Δu and Thus, y(t),Δv(t)). Δu (9) andThus, Δv( where H and Hp refer to future and past disturbances, and G is the process dynamics. Then, it results be expressed canstructure be as:expressed can where be expressed as: the disturbance as: in a commoncan feedforward dynamics are divided by the process
dynamics with a reversed sign. In =the case when future disturbance are in +l(z)Δu(t) α (z))Δv(t) (z) + α (10) Δu(t) m(z)w(t) Δu(t) = +m(z)w(t) n(z)y(t) Δu(t) =+ m(z)w(t) +l(z)Δu(t) n(z)y(t)+++ n(z)y(t) l(z)Δu(t) (αknown, +pthe (αffirst +term (αp (z))Δv(t) f (z) + f (z) + αp (z))Δv(t) Equation (12) is also considered, v(t + j) for j = t + d + 1, . . . , t + d + N. In such a case, it is possible du +2 du +1 +2 dUdu +1 du +2 dU +N to use the predictive mechanism for disturbance compensation a[z of with: m(z) = with: kGm(z) = where kG m(z) kG p (z) ppz= (z) where =pz[z(z) pduz+1 (z) where , zfor =p ,zwide .(z) . . . ,. range = z. ,duz[z ,+N . . ],.,systems, z.n(z) . . , z,= . . including .−kG . ],. .n(z) , zdSU +N and = ],−kG l(z) n(z)=S = and −kG l(z zwith: z(z) Moreover, those with inversion can respectively, be polynomials, easily used with different Moreover, Gp−kG .αMoreover, Gαpthe Moreover, αdeveloped respectively, αfαare, and respectively, polynomials, αf are, including polynomials, the including coefficients including the of coefficients the the pastcoefficien of the −kG Gproblems. p . −kG p and f. are, p and p scheme focuseson the disturbance feedback controllers, since it only compensation. ) and future H ) (−kG and future H H) ) and values (−kG future of H) the (−kG values disturbance. H) of values the disturbance. Depending of the disturbance. on Depending the relation Depending on the between relation on the the between relation (−kG H (−kG (−kG p p p and time the process input-output thedprocess input-output dead process time,input-output ddead time, factors dead du , such are time, calculated factors du , such arefactors calcul disturbance-output disturbance-output dead disturbance-output time dvdead dead dv and time v and the u , such Future Disturbance Estimation the > + 1, dthe in different ways. in different Forinthe ways. different configuration Forways. the configuration For analyzed the configuration in analyzed this work,in analyzed ρthis > 0work, and in this (nd ρ >work, +0 dand (nd >1,0+ and dvα)(nd v )ρ> p,f v) In previous works, [8,9,24], a study of different forecasting techniques was performed in order polynomialspolynomials are as follows: polynomials are as follows: are as follows: to extend the functionality of the MPC approach. The availability of the future signal values are −(d v +nd−1)−(dv +nd−1) αp (z) accuracy, = αp1 α+p (z) αwhich z −1ααp+results + . . .α= + α + .α . z. performance + z −1αv +nd−1) ... + , z −(d αof z, , used to improve the prediction mechanism in−1 thev )MPC p2 = p2 z p(nd+d p+ p(nd+d 1p (z) 1better (nd+dv ) v )p2 controller [9]. The previous studies show that relatively lowest prediction errors were reached using and the future and part thehas future and following the partfuture has form: following part has following form: form: the Double Exponential Smoothing technique and therefore this forecasting method was also used in the study performed in this work. (du(d −d (d (duu−d )+N −dv )+N (du −dv )+N u −dv ) u −d v ) v )+1(du −d v)+1 αf (z) = αf1 α z (d ααfα1fzf(z) +1 zαf+1 z.(d.v u). −d +vα)+1 + . .−d . v+ αf+N z. (d . .u+ α fN z f (z) = + fN 1 z = αf 1 zz The Double Exponential Smoothing (DES) technique is characterized by two equations: 118 where nd 118refers where to118nd the where refers degreeto nd ofthe refers thedegree polynomial to the of the degree D polynomial from of the equation polynomial D from (5).equation Moreover, D from(5). equation polynomials Moreover, (5). Moreover, polynomials referring polynom refer S = φx + ( 1 − φ )( S + b ) , (13) k k k − 1 k − 1 119 to past α 119 to past future 119αpto αand values future αp and ofαfdisturbances future valuesαof disturbances can beofgrouped disturbances can be intogrouped αcan= beαinto + α f .= into Additionally, αp + α α=f . αAdditiona p and fpast f values pgrouped p + αf . 120 the prediction 120 themechanism prediction 120 the prediction of mechanism the intrinsic mechanism of the compensator intrinsic of thecompensator intrinsic can be also compensator used can be to also include canused befuture also to include used disturbance tofuture include disturba futur bk = γ ( S k − S k − 1 ) + ( 1 − γ ) bk − 1 , (14) 121 estimations, 121 estimations, v, through 121 estimations, the v, through polynomial v,the through α polynomial . In the such polynomial α a scheme, . In such α the . a In scheme, implicit such a the feedfroward scheme, implicit the feedfroward compensation implicit feedfroward compensa f f f where xk 122 is actual signal Sis is the unadjusted forecast, bk isthe the estimated trend,beφ completely isattenuated the shown k in is included 122 in isvalue, control included 122 law, included control however in law, the control however disturbance law, the however cannot disturbance be disturbance completely cannot be cannot attenuated completely as was attenuated as in was show as smoothing123parameter for data, and γ is the smoothing parameter for trend. The one sampling instance [9,13]. This 123 [9,13]. is due123 toThis λ[9,13]. parameter is dueThis to λthat isparameter due is used to λ parameter in that theisinternal usedthat in structure is theused internal inofthe the structure internal GPC. of Itstructure was the GPC. shown of Itthe that was GPC. for shown It was thats ahead prediction is obtained by xˆ k+1 = Sk + bk and the m− sampling periods forecast is obtained 124 λ > 0, feedforward 124 λ > 0,124feedforward compensator λ > 0, feedforward compensator will nevercompensator perform will never as awill perform classical neverasperform disturbance a classical as acompensation disturbance classical disturbance compensation structure. compensa struct usign xˆ k+m = Sk + mbk . The initial values for Sk and bk were set to Sk = x1 and bk = ( x1 + x2 + x3 )/3 125 The ideal 125 compensator The ideal 125 The compensator canideal be obtained compensator can be only obtained can whenbeλonly obtained = 0, when as only was λ =when shown 0, asλin was =[13]. 0, shown asNevertheless, was in shown [13]. Nevertheless, inthe [13]. Nev as recommended in [8,9]. Moreover, the values for φ and γ ∈ (0, 1) can be obtained via optimization 126 resulting126control resulting scheme 126 control resulting is characterized scheme controlis scheme characterized by implicit is characterized co-design by implicitfor byco-design implicit feedbackco-design for andfeedback feedforward for and feedback feedforward part and and feedfor part techniques as described in [8]. 127 the high 127 bandwidth the high 127in bandwidth the high feedback bandwidth in the loop, feedback what in theresults loop, feedback what in very loop, results aggressive what in very results changes aggressive in very in control aggressive changes signal. inchanges control Due in signal. contr 128 to the implicit 128 to the relation 128 implicit tobetween therelation implicit feedback between relation andfeedback between feedforward and feedback feedforward actions, andsuch feedforward actions, structure such actions, cannot structure be such used cannot structure as abe canno used 129 stand alone 129 disturbance stand alone 129 stand compensator disturbance alone disturbance compensator what limits compensator its what applicability. limitswhat its applicability. limits This particular its applicability. This issue particular can This be particular issue addressed can be issue addres can 130 with predictive 130 withfeedforward predictive 130 with predictive feedforward compensator feedforward compensator that was recently compensator that was introduced recently that was inintroduced recently [12] and introduced which in [12]isand summarized inwhich [12] and is summar which i 131 in following 131 in section. following 131 in following section. section.
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Control Signal Saturation Constraints Due to its operation principles, the predictive feedforward compensator is able to handle the constraints into the optimization procedure. As shown perviously, the analyzed compensator is derived from GPC structure and keeps the ability to handle the process constraints in an independent way. This characteristic is important from a practical point of view since, in a classical feedforward compensator, this issue is not considered. Nevertheless, due to the fact that feedforward compensator works as an independent block from the main feedback control loop, it is necessary to take into account the value of the control signal computed by the feedback controller. In such a way, the available change in the control action is determined and considered in the optimization procedure. Due to the physical limitations of control signal (actuator saturation limits, umin ≤ u(t) ≤ umax ), it is necessary to include this data into an optimization procedure, where control signal u(t − 1) = u FB (t − 1) + u FF (t − 1) considers the part that corresponds to the feedback controller u FB and u FF from the feedforward compensator. In a general way, these constraints can be expressed as R∆u 6 c where: " R=
T −T
#
" ;c =
lumax − lu(t − 1) −lumin + lu(t − 1)
# ,
and T is N × N lower triangular matrix of ones, l is a vector, 1 × N of ones. In such a way, saturation limits can be expressed as a function of inequalities on control signal increments: lumin ≤ T∆u + u(t − 1) ≤ lumax . 3. Microalgae Production in a Raceway Reactor The experimental raceway reactor used for modelling purposes is located at experimental station “Las Palmerillas” (Almería, Spain—property of CAJAMAR Foundation). The modelled reactor has 100 m2 surface area and is composed of two 50-m long (1 m wide) channels forming U-shape bends (see Figure 2) [25]. The reactor operates at a constant depth (0.2 m) to provide desired performance and considering power consumption issues. Resulting reactor volume is 20 m3 . The raceway photobioreactor facilities can be set up to use different carbon dioxide sources in order to provide a flexible platform to evaluate different systems and techniques. Nevertheless, for the modelling purposes, the plant was configured to use flue gases. The flue gas used for pH control was taken from industrial heating boiled (diesel-fueled with the average gas composition was: 10.6% CO2 , 18.1 ppm CO, 38.3 ppm NOx, and 0.0 ppm SO2 ). In this operation scheme, exhaust gas was refrigerated to the environment temperature and stored in a 1.5 m3 pressure tank (compressed to 2 bar) with automatic regulation of pressure. The compressed flue gas was supplied to the reactor through a 150 m pipeline of 40 mm diameter. Finally, the injection system uses the solenoid on/off valve and input flow was measured with a digital flow meter (detailed infirmation for raceway setup can be found in [17,25]). The injection instant as well as aperture time is provided by the control algorithm used for pH control. Taking into account the limitation of the actuation system that is used in the pilot-scale raceway reactor, it is necessary to convert the continuous control signal into the on/off actions of solenoid control valve. To this end, the PWM (Pulse Width Modulation) technique is used to translate the signal provided by the controller into a train of pulses with modulated width. The value of the pulse width is determined by the control system and can vary between 0 and 100%. Moreover, the modulation frequency was set to 0.1 Hz being optimal for the main controlled variable. The next section summarizes the raceway reactor modelling that is presented based on the development from [21].
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0.65 m
1m
50 m
