Practical MPC with robust dead-time compensation applied to a solar desalination plant

Transcription

1 Practical MPC with robust dead-time compensation applied to a solar desalination plant Tito L. M. Santos Lidia Roca Jose Luiz Guzman Julio E. Normey-Rico Manolo Berenguel Departamento de Automação e Sistemas, Universidade Federal de Santa Catarina, Caixa Postal 476, CEP 884-9, Florianópolis, Santa Catarina, Brasil ( tito,julio@das.ufsc.br). Convenio Universidad de Almería - Plataforma Solar de Almería, Ctra. Senés s/n, 4 Tabernas, Almería, Spain ( lidia.roca@psa.es). Dep. de Lenguajes y Computación, Universidad de Almería, Ctra. Sacramento s/n, 41, Almería, Spain ( {joguzman,beren}@ual.es). Abstract: A practical model predictive control (MPC) for solar collector plants is proposed in this paper. By exploiting the non-linear structure of the solar collector model, the optimization problem is posed as a quadratic program similarly to a linear MPC. As applied in some recent publications, a robust dead-time compensation scheme is used to improve robustness. A different state-space interpretation of the robust compensation scheme is also presented. Simulation results including comparisons with other control schemes are shown in order to improve discussions. Keywords: Dead-Time Compensator, Robustness, Solar Plants, Model Predictive Control 1. INTRODUCTION The importance of renewable energy resources is increasing significantly. Solar collector fields are attractive alternatives in order to use energy in a sustainable way. AQUASOL plant at the Plataforma Solar de Almería (SPAIN), for instance, makes use of solar radiation in a combined solar and fossil multi-effect distillation plant for the continuous production of fresh water (Alarcón-Padilla et al., 5). In this kind of process, solar radiation cannot be manipulatedandaspectssuchasdailysolarcycle,cloud level and atmospheric conditions have to be considered as part of the control problem (Roca et al., 9). Moreover, the typical non-linear behavior justifies the interest for advanced control techniques (Camacho et al., 7a). Due to the non-linear behavior, model predictive control (MPC) is a natural candidate in order to improve performance in solar plants. Actually, there exist an extensive literature which consider MPC strategies in solar plants context (Camacho et al., 7b). Most of these works do not consider delay compensation explicitly because, as it is well known, delays are naturally compensated in MPC strategies. However, it was already shown that robustness can be significantly improved by using a suitable dead- This work has been partially funded by the following projects: PHB9-8 financed by the Spanish Ministry of Education; CNPq-BRASIL; CAPES-DGU /1; Spanish CICYT and EU- ERDF funds under contract DPI C5-4; AQUASOL Project (Contract Nr. EVK1-CT1-1) financed by CIEMAT and the European Commission. time compensation scheme (Normey-Rico and Camacho, 7; Gálvez-Carrillo et al., 7). In some recent papers (Roca et al., 9; Torrico et al., 1), the usefulness of this robust dead-time compensation scheme was verified in practice. In this work, a practical MPC algorithm is proposed to be used in the robust dead-time compensation context. A variable change is used in order to describe implicitly the non-linear behavior by considering a modified linear model. If compared with some recent publications: i) the resultant linear model is stable, so that it is not necessary to consider additional conditions on the robustness filter tuning (Roca et al., 9); ii) the optimization problem is solved once at each sampling instant and the computational effort is equivalent to a linear MPC problem (Gálvez-Carrillo et al., 7; Torrico et al., 1); and iii) it is not necessary to re-identify a linear model for every change in the operation point (Ayala et al., 1). The advantage of this approach, if compared with other recent published MPC strategies (Roca et al., 9; Torrico et al., 1), is that the robustness improvement can be analyzed by using an additive uncertainty interpretation. This analysis is useful since it can be applied as an auxiliary tool in order to define robust tuning parameters. The paper is organized as follows. The model of the solar collector field is presented in Section, its simplified statespace description is proposed in Section 3, Section 4 is devoted to present the MPC algorithm, some comparative Copyright by the International Federation of Automatic Control (IFAC) 499

2 simulation results are shown in Section 5 and the final remarks are presented in Section 6.. SIMPLIFIED SOLAR FIELD MODEL A simplified model of the solar desalination plant will be presented in this section. As explained in Roca et al. (8), solar field outlet temperature, T of ( o C), may be modeled by considering the dynamic of a hypothetical equivalent absorber tube with the same behavior as the whole solar field. This equivalent absorber tube is characterized by an equivalent mass flow, ṁ eq, that depends on the inlet solar field mass flow, ṁ F, number of operative solarfield loops, n l, number of collector groupsin each one ofthe loops,n c,numberofparallel-connectionsineachone of the groups, n cp, and the number of absorber tubes in each collector, n a, as expressed in: ṁ F ṁ eq = (1) n l n c n cp n a The solar-field outlet of this equivalent absorber tube varies depending on irradiance, I (W/m ), ambient temperature, T a ( o C), solar-field inlet temperature, T if ( o C), and on the equivalent mass flow, ṁ eq (kg/s) (Roca et al., 9; Torrico et al., 1). As described in Torrico et al. (1), the loop outlet temperature is the variable employed for control purposes. Then, assuming the collector group shown in shady gray in Fig. 1, it can be considered two sources of delay: a delay d TiF is caused due to the inlet temperature sensor position and a temperature-flow dead-time d c, which was verified experimentally, can be caused by more than one reason as will be detailed in the following. As the inlet temperature sensor, T sif ( o C), is located at the beginning of the loop, there is a transport delay, d TiF (s), in the incoming water, which can be estimated at each sampling time as a flow-dependent delay as described in Normey-Rico et al. (1998): L = dtif T if (t) T sif (t d TiF ) () ν(t)dt T i=n 1 s f(k i) = L (3) A cs i= where T s (s) is the sampling time, ν is the velocity rate (m/s), index n is equal to the delay (in sampling times), f(k) (m 3 /s) is the flow rate at sampling time k, LA cs is the product of the length, L (m), and the cross-section area of the loop pipe, A cs (m ), and the integral of ν is approximated by a discrete time sum that accounts for different flow rates. Then, n is found as it is the iteration number at which the flow sum reaches LA cs /T s 1. It is important to mention that this delay is not included directly in the control algorithm. The disturbance, T if, is estimated every sample time using past measures of the inlet temperature, T sif, and evaluating the delay d TiF with the past flow rates. This temperature estimation is then an approximation of the solar-collector inlet water temperature and thus, it is used in the controller predictor model. 1 A detailed explanation of this delay evaluation is included in Roca et al. (9); Torrico et al. (1) Table 1. Process parameters Symbol Name Value A a absorber cross-section area e-4; m C p specific thermal capacity 419 J kg 1 oc 1 d c input-output dead-time 3-5 s H thermal losses coefficient 4 J s 1 K 1 L eq equivalent absorber tube length 5.67 m β I irradiance model parameter.14 m ρ water density 975 kg m 3 Furthermore, tests have demonstrated the existence of a nominal temperature-flow delay of d c =4s with 5% variability (d c [3,5]s). This observed delay comes from different causes: a water transport delay in the group collector connections, an output-flow delay caused by the heat transfer dynamics of the differential flow elements inside the absorber tubes, and hydraulic upsets in the collector groups and absorber tubes. To illustrate this variabledelay,fig.showstwodifferent experimentalstep changes of the water flow at different operating points. Thus, the simplified model may be described by ρ C p A a T of(t) = β I I(t) H ( T(t) T a (t)) t L eq C p ṁ eq (t d c ) ToF(t) T if (t) L eq (4) where the absorber temperature is modeled as the mean temperature, T ( o C), between inlet and auxiliary water temperatures: T(t) = T of(t)+t if (t) (5) and the other parameters are shown in Table 1. This kind of approximated lumped-description model has demonstrated to produce promising results in comparison with real data dealing not only with variations in the water flow and inlet temperature, but also with strong changes in irradiation (Roca et al., 8; Torrico et al., 1). 3. PROPOSED PREDICTION MODEL Now, a simplified model will be proposed in order to consider a linear representation by means of a variable change. Initially, Eq. (4) will be rewritten by using Eq. (5) so that it turns into T F (t) =.5H T of (t) T of(t) T if (t) ṁ eq (t d c ) ρc p A a L eq ρa a L eq +.5H (T a (t) T if (t))+ β I I(t). (6) ρc p A a L eq ρc p A a In this case, T of (t) is the process variable, ṁ eq (t d c ) is themanipulatedvariabledefinedinthepastandi(t),t a (t) and T if (t) are measurable disturbances. In this paper, it is assumed that measurable disturbances are constant in the future despite the fact that it would be possible to consider particular prediction models as in Camacho et al. (1997, Chapter 5) for instance. This idea was not applied in this work for the sake of controller simplicity.. = ṁ eq (t)) has no As the current control action (u(t) effect between T of (t) and T of (t + d c ), an estimation for T of (t+d c ) can be obtained from a filtered Smith predictor which is useful to improve robustness (Normey-Rico and 491

3 T T of (t) Hot water Cold water L T if (t) T sif (t-d TiF ) T sif (t) T Fig. 1. One loop layout of AQUASOL solar field Camacho, 7). Hence, by considering Eq. (6) d c instants ahead, it is obtained T of (t+d c ) =.5H T of (t+d c ) ρc p A a L eq T of(t+d c ) T if (t+d c ) ṁ eq (t) ρa a L eq +.5H (T a (t+d c ) T if (t+d c )) ρc p A a L eq + β I ρc p A a I(t+d c ). In fact, T if (t+d c ), T a (t+d c ) and I(t+d c ) are not known at t but, due to the assumption that disturbances are constant in the future, they can be approximated by T if (t), T a (t) and I(t) respectively. Moreover, the estimation of T of (t+d c ) will be detailed in next subsection. In this case, by defining the virtual control action given by v(t). = (T of (t+d c ) T if (t))ṁ eq (t); a vector of measurable disturbances expressed by p(t). = [I(t) T a (t) T if (t)] T ; and the current state x(t). = T of (t+d c ), the followinglinearstate-spaceprediction model is derived ẋ(t) = Φx(t)+Γv(t)+Πp(t) where Φ =.5H/(ρC p A a L eq ), Γ = 1/(ρA a L eq ) and Π = [β I.5H/L eq ]/(ρc p A a ). With no thermal inversion in the solar field, T of (t+d c ) T if (t), so the virtual control law is valid in the state space defined by the operation range (Roca et al., 9). Forasufficientlysmallsamplingperiod,T s,itisreasonable to consider that disturbances are almost constant between subsequent sampling instants. In this case, zero-orderhold discretization can be also used to approximate measurable disturbance effect in such a way that x(k +1) = Ax(k)+Bv(k)+Mp(k) with A = e ΦTs, B = Ts e Φθ dθγ and M = Ts e Φθ dθπ. Hence, it is necessary to present the robust estimation strategy for x(k) = T of (k T s +d T s ) with d = d c /T s. 3.1 Robust dead-time compensation A filtered Smith predictor which is a robust dead-time compensation scheme (Normey-Rico and Camacho, 7) will be introduced in order to estimate the current state value. Consider that ˆx(k) is the robust prediction for x(k), andx(k) isanominal open-loopprediction,v(k) = (x(k) T if (t))u(k) isthe realvalue forvirtualcontrolwhich isnot known because x(k) is not measurable at k. Then, it can be defined ˆv(k) = (ˆx(k) T if (t))u(k) (7) as the estimated virtual control, which is used for control purposes. Uncertainties such as plant-model mismatch, unmeasured disturbances, and measurement noise can be lumped together as an additive state disturbance (w(k)) in order to describe the real state evolution by using the nominal model representation. In this case, real and nominal state evolution are described respectively by: x(k +1) =Ax(k)+Bv(k)+Mp(k)+w(k), (8) x(k +1) =Ax(k)+Bˆv(k)+Mp(k). (9) As the model is open-loop stable, the filtered Smith predictor is directly obtained from ˆx(k) = F r (z 1 )(x(k d) x(k d))+x(k) (1) where F r (z 1 ) is a robustness filter as will be further explained (Normey-Rico and Camacho, 7). As a consequence, the evolution of the predicted (estimated) system can be represented by ˆx(k +1) = Aˆx(k)+Bˆv(k)+Mp(k)+ŵ(k) (11) where ŵ(k) is the additive disturbance for the prediction. The importance of this robust compensation strategy lies in the fact that ˆx(k) is a robust estimation of x(k), so thatf r (z 1 ) canbe usedtoattenuate thepredictionerror. Robustness filter importance can be analytically evaluated through the relationship between w(k d) and ŵ(k). As shown in appendix, ŵ(k) given by ŵ(k) = F r (z 1 )[w(k d)+b(x(k d) ˆx(k d))u(k)]. This result is important to demonstrate the effect of F r (z 1 ) over w(k d). In other words, this filter can improve robustness attenuating the effect of the unexpected mismatch over the cost function. Another advantage of this interpretation is that w(k d) can be calculated at k +1 because it is given by by w(k d) = x(k+1 d) [Ax(k d)+bv(k d)+mp(k d)] where x(k + 1 d) = T of (k + 1). As a consequence, by collecting a real data set, it is possible to make a fast Fourier transformation off-line using a sequence of w(j). This information may be useful to define F r (z 1 ) directly from the frequency response of the additive disturbance which was not considered in previous approaches. This relationship is closely related to those obtained for linear systems (Santos et al., 1). However, the term B(x(k d) ˆx(k d))u(k) appears in this case because it 4911

4 Flow T of Temperature [ºC] d c =5 s Flow [l/s] Temperature [ºC] d c =33 s Flow [l/s] Relative time [s] Fig.. Experimental tests to evaluate the delay between outlet water temperature and inlet water flow. is necessary to use ˆv(k) instead of v(k) in the prediction scheme (v(k) is unknown at k). Actually, this additional uncertainty is also attenuated by the robustness filter. It is important to remark that if the closed-loop system is stable and F(1) = 1, in the presence of constant disturbances (w(k) w ss ), x(k) x ss and (x(k d) ˆx(k d)) (see Normey-Rico and Camacho (7) for details). 4. PROPOSED MPC STRATEGY A practical MPC strategy will be proposed in this section in order to control the solar desalination plant using a linear MPC strategy. In this case, it is necessary to define prediction model, cost function and desired constraints. As it is desired to reject constant disturbances, the positional prediction model, presented in Eq. (11), will be converted into an incremental one. Hence, subtracting (11) at k + 1 from the same equation at k, it is obtained an augmented representation is obtained where χ(k) = [ˆx(k) ˆx(k 1)] T is the augmented state and its dynamic is described by ˆχ(k +1) = A χˆχ(k)+b χ ˆv(k)+M χ p(k) (1) ˆT of (t+d) = C χˆχ(k) (13) where A χ = [ ] [ ] [ ] T [ ] 1+A A B 1 M, B 1 χ =, C χ =, M χ =. As already explained, measured disturbances are considered to be constant in the future p(k+j) = for j 1. Now, consider a general MPC problem. The decision variable is an auxiliary control increment represented by ˆv(k) = [ ˆv(k k) ˆv(k +1 k)... ˆv(k +N u k)] and the optimization problem is given by min ˆv(k) s.t. N [ˆT of (k +d+j k) T ref ] + j=1 N u 1 j= λ[ ˆv(k +j)] T min ˆT of (k +d+j k) T max, 1 j N, u min u(k +j k) u max, j N u 1. The problem is that it is desired to impose control constraints over u(k) instead of ˆv(k). Due to the nonlinear nature of the problem, the relation between u(k+j k) and ˆv(k+j k) depends on T of (k+j k) which also depends on future control actions that were not already defined at k (u(k k), u(k+1 k),..., u(k+j 1 k)). As a consequence, this control problem has nonlinear constraints even if the variable change is considered. Similarly to the idea of constraint mapping applied in Roca et al. (9), it is possible to use an approximated solution based on the control increments computed at last MPC iteration (k 1). The idea is to use, at k, ˆx(k) and ˆv(k k 1),..., ˆv(k + N u 1 k 1) (which were computed at k 1) to obtain an approximated prediction for the future states ˆx(k +j k). Let x(k +j k 1) be an approximation for ˆx(k+j k) obtained from the prediction model (1) with ˆv(k k 1),..., ˆv(k + N u 1 k 1) and ˆx(k) known. In this case, it is possible to define a transformation matrix T (k) = diag (ˆx(k) T if (k)) 1 ( x(k +1 k 1) T if (k)) 1. ( x(k +N u 1 k 1) T if (k)) 1 that can be used to map control constraints. Consider that the constraints on u(k) = [ u(k k) u(k +1 k)... u(k +N u k)] are originally expressed by R u u(k) s u (k), then u(k) T (k) ˆv(k) (14) if ˆv(k + j k 1) ˆv(k + j k) in such a way that it is possible to map the constraints on u(k) to ˆv(k). The most importantpointis that,forthe firstcontrolelement, which is actually applied at k, there is no approximation error. This happens because u(k) = (ˆx(k) T if (k)) 1ˆv(k) where T if (k) and ˆx(k) were already measured at k. In other words, the constraints on u(k) are effectively kept because the mapping between u(k) and ˆv(k) is used inversely to obtain u(k) at the same iteration. It is interesting to point out that the transformation matrix T (k) can also be used to easily rewrite the cost function in terms of u(k) instead of ˆv(k). In practice, if the MPC strategy is implemented in such a way that (ˆx(k) T if (k)) is kept in a narrow range, this change does not have a significant effect. Note that, if it is used N u = 1 as in Torrico et al. (1), the solution of the proposed strategy is equal to the ideal non-linear algorithm because the optimization problems are equivalents. This is valid only for N u = 1 because there is no approximation error for the first control action. 491

5 Algorithm 1. Practical MPC algorithm 1: function u(k)=mpc(t of (k),t if (k),t a (k),i(k)) : Set x(k d) = T of (k) 3: Compute ˆx(k) from Eq. (1) 4: Compute x(k +j k 1) using model (1) 5: Rewrite control constraints using Eq. (14) 6: Obtain the optimal solution ˆv (k k) 7: Apply u(k) = ( ˆv (k k)+ ˆv(k 1))(ˆx(k) T if (k)) 8: end A pseudo-code of the proposed MPC strategy is presented in Algorithm 1. The differences between this algorithm and an usual linear MPC are that it is necessary to do the dead-time compensation explicitly and it is necessary to rewrite the control constraints. However, these steps are very simple and require low computational resources. 5. SIMULATION STUDY The simulation was performed considering the non-linear model presented in Eq. (4). In order to illustrate the robust compensation effect, the temperature-flow deadtime was set to 5s for simulations purposes but it was considered d c = 4s as the nominal dead-time. The measured disturbances, shown in Fig. 3, were extracted from a real data set. In the real plant, the main purpose of the control system is to maintain the difference between outlet and inlet temperatures within the range of 5 o C and o C to optimize collector efficiency and avoid stress in the collector materials. Thus, the reference for the outlet temperature was set to T if + T where T is a desired temperature increment. In order to emphasize the reference change effect, T is switched between and 15 o C at each hour during the simulation. In this simulation, four loops are operatives (n l = 4, n c = 7, n cp = 3, n a = 7). Thus, inlet water flow should be greater than 1. l/s and smaller than 4.4 l/s meanwhile output temperature constraints are T min = 4 and T max = 9. Similarly to Ayala et al. (1), the control weighting was normalized as λ = λ n k p where k p = B/(1 A) is the static gain of the nominal model because, in this case, the selection of λ n does not depends on the process gain. As T is not constant in this simulation, the transformation matrix is used in order that λ is a weighting for the real I(W/m ) Ta ( o C) Tif ( o C) Fig. 3. Real disturbances obtained from AQASOL plant Table. Tuning parameters Strategy N N u λ F r(z 1 ) FL-GPC NEPSAC 5 1 PMPC 5 8 k p.195z z (1.95z 1 ).8 1.9z z 1 controlactioninstead ofthe virtualone.as aconsequence, the real control effort is uniformly weighted during the whole simulation. The practical MPC (PMPC) is compared with the feedback linearization GPC (FL-GPC) (Roca et al., 9) and the nonlinear extended prediction self-adaptive control (NEPSAC) (Torrico et al., 1). These recent published controllers were tested in simulations and in real experiments providing promising results. The controller tuning parameters are shown in Table. In the FL-GPC, the robustness filter zero is used to guarantee the internal stability of the predictor because the resultant linear model is integrative. The parameters λ = and N u = 1 were used in Torrico et al. (1) in order to reduce computational burden. As shown in Fig. 4, it is possible to obtain as good result as those from FL-GPC and NEPSAC. The PMPC advantages come from its implementation simplicity and additivedisturbanceinterpretation.moreover,ifcompared with (Torrico et al., 1), the optimization problem is solved only once at each sampling time. Although the frequency response information of w(k) was not used in this simulation study due to lack of space, it can be used as an additional tool in order to improve robustness filter tuning. 6. CONCLUSION A practical MPC algorithm was proposed to control solar collector plants. The algorithm is based in a control variable change which allows to consider implicitly the non-linear process behavior by using a modified linear model. The cost function and the control constraints can be rewritten in terms of the real manipulated variable by using an approximated transformation matrix. Moreover, robust delay compensation effect was analyzed in a additive state-space disturbance context. A simulation example, using a real disturbance data set, was used to compare the proposed approach with some related works. To study different tuning procedures for the robustness filter and to apply the proposed algorithm in the AQUASOL plant are possible issues for future work. REFERENCES Alarcón-Padilla, D., J.Blanco, S.Malatao, I.Maldonado, and P.Fernández (5). Design and setup of a hybrid solar seawater desalination system: the AQUASOL project. In Proceedings of the ISES 5 Solar World Congress. Ayala, C., Roca, L., Guzman, J.L., Normey-Rico, J.E., Berenguel, M., and Yebra, L.J. (1). Local model predictive controller in a solar desalination plant collector field. Submitted to Renewable Energy. 4913

6 Outlet temperature ( o C) Set Point PMPC 5 FL GPC NEPSAC Inlet flow (l/s) PMPC 1.5 FL GPC NEPSAC Fig. 4. Simulation using disturbance entries of a sunny day Camacho, E.F., Berenguel, M., and Rubio, F. (1997). Advanced control of solar plants. Springer Verlag. Camacho, E., Rubio, F., Berenguel, M., and Valenzuela, L. (7a). A survey on control schemes for distributed solar collector fields. part i: Modeling and basic control approaches. Solar Energy, 81(1), Camacho,E., Rubio,F., Berenguel,M., and Valenzuela,L. (7b). A survey on control schemes for distributed solarcollectorfields. partii: Advanced controlapproaches. Solar Energy, 81(1), Gálvez-Carrillo, M., Keyser, R.D., and Ionescu, C. (7). Application of a Smith predictor based nonlinear predictive controller to a solar power plant. In Proceedings of 7th IFAC Symposium on Nonlinear Control Systems. Pretoria, South Africa. Normey-Rico, J.E. and Camacho, E.F. (7). Control of dead time processes. Springer Verlag. Normey-Rico, J., Bordons, C., Berenguel, M., and Camacho, E. (1998). A robust adaptive dead-time compensator with application to a solar collector field. In 1st IFAC International Workshop on Linear Time Delay. Roca, L., Berenguel, M., Yebra, L.J., and Alarcón-Padilla, D. (8). Solar field control for desalination plants. Solar Energy, 8, Roca, L., Guzmán, J.L., Normey-Rico, J.E., Berenguel, M., and Yebra, L.J. (9). Robust constrained predictive feedback linearization controller in a solar desalination plant collector field. Control Engineering Practice, 17, Santos, T.L.M., Normey-Rico,J.E., and Limón, D. (1). Explicit input-delay compensation for robustness improvement in mpc. In Proceedings of the 9th Workshop on Time Delay Systems. Prague, Czech Republic. Torrico,B.C., Roca, L., Normey-Rico,J.E., Guzmán, J.L., and Yebra, L.J. (1). Robust nonlinear predictive control applied to a solar collector field in a solar desalination plant. IEEE Transactions on Control Systems Technology. In press. Appendix A. ROBUSTNESS FILTER EFFECT Here,itisdesiredtorelateŵ(k) withthe disturbancesthat appears at k d. Thus, it is obtained, from Eq. (11), that ŵ(k) = ˆx(k +1) [Aˆx(k)+Bˆv(k)+Mp(k)]. (A.1) By replacing Eq. (1) in (11) for both instants k and k+1, leads to ŵ(k) =F r (z 1 )(x(k d+1) x(k d+1))+x(k +1) A{[F r (z 1 )(x(k d) x(k d))+x(k)] Bˆv(k) Mp(k)}. (A.) Then, simplifying Eq. (A.) by means of the equality presented in Eq. (9) gives ŵ(k) =F r (z 1 )(x(k d+1) x(k d+1)) A[F r (z 1 )(x(k d) x(k d))]. (A.3) Now, Eqs. (8) and (9) can be replaced at k d+1 in order to express w(k d) as ŵ(k) =F r (z 1 ){Ax(k d)+bv(k d)+mp(k d) +w(k d) [Ax(k d)+bˆv(k d)+mp(k d)]} A[F r (z 1 )(x(k d) x(k d))]. (A.4) Hence, the final expression can be obtained eliminating the terms that are both added and subtracted in Eq. (A.4) resulting in ŵ(k) = F r (z 1 )[w(k d)+b(x(k d) ˆx(k d))u(k)]. 4914