Hâ Control of Anaerobic Digester for Winery Industry Wastewater

Article pubs.acs.org/IECR

H∞ Control of Anaerobic Digester for Winery Industry Wastewater Treatment R. Flores-Estrella,† G. Quiroz,‡ H. O. Mendez-Acosta,§ and R. Femat*,† †

División de Matemáticas Aplicadas, IPICyT, Camino a la Presa de San José Apartado Postal 2055, Colonia Lomas 4a Sección, San Luis Potosí, S.L.P., México ‡ Facultad de Ingeniería Mecánica (FIME), Universidad Autónoma de Nuevo León, UANL, FIME. Av.Universidad S/N Ciudad Universitaria, C.P. 66451, San Nicolás de los Garza Nuevo León, México § Departamento de Ingeniería Química, Centro Universitario de Ciencias Exactas e Ingenierías (CUCEI), U.D.G., Boulevard Marcelino García Barragán s/n, Guadalajara, Jalisco, México ABSTRACT: A robust H∞ controller has been developed to regulate the chemical oxygen demand in an anaerobic digester from the winery industry. A sensitivity analysis was performed, and the parameter set having the most significant effect on the process behavior was identified. The parameters inducing the most sensitivity in the solutions were selected as uncertain; in addition, they were related to kinetic terms and he hydrodynamic regime. Then, a control problem was formulated as robust regulation, and a controller was designed using H∞ theory to ensure robust stability. The actions of the H∞ controller are illustrated through numerical simulations. The controller was found to execute robust regulation facing parametric uncertainties and load disturbances.

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INTRODUCTION Anaerobic digestion (AD) is a process that converts organic matter into a gaseous mixture, composed of methane and carbon dioxide, through the action of a series of complex biological and enzymatic reactions. Although AD has distinct applications, it has mainly been used for wastewater treatment. Operating and controlling an AD process is not a simple task for the following reasons:1,2 (i) wastewater varies continuously in quantity and composition, (ii) biomass activity changes under the influence of internal and external factors, (iii) adequate sensors for online measurements are often lacking, and (iv) there is uncertainty in kinetic parameters. Several control methods have been proposed in recent years for AD processes. Steyer et al.3 and Méndez-Acosta et al.4 pointed out that classical control methods have not been able to face the inherent difficulties presented in AD. Moreover, classical control methods have been shown to yield unsatisfactory performance when AD is subjected to disturbances or significant set-point changes. Mendez-Acosta et al.5 proposed a linear reference-feedforward/output-feedback control that is robust in the face of parameter uncertainties and piecewise time disturbances. Another feature is that, to diminish the peaking phenomenon induced by a high-gain observer, an antireset windup scheme can be taken into account to handle saturation constraints by actuator restrictions. However, such a controller was not designed to handle disturbances with frequency components in the same interval as the AD responses, which can occur under specific operating conditions. Alcaraz et al.6 proposed an interval-based scheme to lead the AD trajectories into a desired operating interval. However, the performance of the control scheme depends on the definition of the uncertainty interval, which is heuristically defined. Adaptive schemes have also been proposed.7,8 For example, Monroy et al.7 showed robustness against load changes using an © 2013 American Chemical Society

adaptive control scheme by taking into account the nonlinearities and nonstationary features of AD. However, complete knowledge of the system parameter structure is required. Moreno et al.9 developed optimal control strategies for a biological sequencing batch reactor. The objective was the maximization of the product output against varying inlet substrate concentrations. Nevertheless, the efficiency per cycle was found to depend strongly on the initial conditions, which are often uncertain under continuous operating conditions. Guwy et al.10 showed that a neural-network-based controller was capable of maintaining stable bicarbonate alkalinity levels without overshoot during process overload. However, to apply fuzzy-based and neural network strategies, either a great deal of information or expertise in the process is required. To date, few results can be found in the open literature regarding the proposal of robust H∞ control schemes for biological processes. In fact, to the best of our knowledge, H∞ robust control has been explored only for a second-order anaerobic digestion process.11 It should be noted that, although AD processes are not effective enough for organic treatment for specific operating conditions (for instance, in the presence of suspended particles or phosphorus compounds), the AD process is essential for treatment of modern wastewater. Actually, in an overall perspective, a wastewater treatment plant consists of primary treatment, in which suspended particles are removed from the wastewater by mechanical operations such as screening and sedimentation, and secondary treatment, in which, in general, dissolved carbon- and nitrogen-containing wastewater components are removed by microbial activity. In Received: Revised: Accepted: Published: 2625

August 20, 2012 December 17, 2012 January 26, 2013 January 27, 2013 dx.doi.org/10.1021/ie302233t | Ind. Eng. Chem. Res. 2013, 52, 2625−2632

Industrial & Engineering Chemistry Research

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some plants, a tertiary treatment step is added to achieve better purification results.12 This means that bioprocesses are often used as the secondary or tertiary step. Hence, control of chemical oxygen demand (COD) is fundamental in the overall rationale of watewater treatment. In this article, an H∞ controller is proposed to ensure robust stability in the regulation of the COD concentration for continuous AD processes. This article is organized as follows: A brief description of the mathematical model related to the AD process used in this work is presented in the next section. The control problem is formulated in the section Robust H∞ COD Regulation. The section Robust Synthesis describes the controller synthesis using H∞ theory. Numerical evaluations are presented to illustrate the implementation of the controller in the AD process. Finally, some concluding remarks are made.

scalars x3,in and x4,in (g/L) represent the inlet concentrations. The dilution rate D (h−1) is defined as the ratio of the inlet flow rate Qin (L h−1) to the reactor volume V (L). The fraction of biomass in the liquid phase is given by the constant parameter α belonging to the interval [0, 1], where α = 0 corresponds to an ideal fixed-bed reactor whereas α = 1 corresponds to ideal continuous-flow stirred-tank reactor. The AD system of eqs 1 includes the growth rates of acidogenic and methanogenic bacteria, denoted by μ1(x3) and μ2(x4), respectively. Such growth rates are described by the Monod and Haldane kinetics such that14

MODEL DESCRIPTION Bernard and co-workers13 proposed a sixth-order model that has been successfully used in the design and validation of several control schemes.5 The model describes the dynamics of acidogenic and methanogenic biomass, organic, and volatile fatty acids substrates, alkalinity, and total inorganic carbon (xi, i = 1, ..., 6). According to Mendez-Acosta et al.,5 the sixth-order model has equilibrium (x* ∈ R6+) such that the AD operation with active biomass can be found for different parameter values. That is, there exist operating conditions implying x1, x2 ≠ 0, xj,in > xj > 0, j = 3, 4, 5, 6, for all t ≥ 0 and any initial state in a physically realizable domain, which means that AD wastewater operates under “normal operating conditions” (NOC). The following propositions summarize relevant results. Proposition 1.5 Consider the anaerobic digestion model proposed by Bernard et al.13 Assuming that the inlet composition, xj,in, is piecewise constant, it can be shown that there exists a unique equilibrium point x* ∈ R6+ for any constant pair (α, D*) under NOC. In addition, such an equilibrium point is contained in the closed set Ω ∈ R6+ = {x ∈ R6+ = {x ∈ R6+|xmin ≤ x ≤ xmax ; xmin > 0 and xmax < ∞} ⊂ R6+, which contains all NOC. In this expression, xmax is the concentration vector obtained when D̅ is used, whereas xmin is obtained for D̲ . Proposition 2.5 Let x* ∈ Ω be the equilibrium point of the anaerobic digestion model proposed by Bernard et al.13 for any constant pair (α, D*) such that D* ∈ [D̲ , D̅ ]. Then, under NOC, such an equilibrium point is locally stable. A reduced-order model for AD wastewater treatment has been used and implemented as well.2 According to this approach, in this contribution, the reduced Bernard et al.’s model is taken up again for control purpose. The fourth-order model is described by the dynamical system of equations

where the nominal parameter values π0 ∈ Π = {α, μ1max, μ2max, KS1, KS2, KI2, k1, k2, k3} are real constants whoses value are uncertain. The results in the same sense as established in propositions 1 and 2 have to be derived before using system of eqs 1 to design a feedback control. This is because two states, related to biogas production and alkalinity, are not being considered in the reduced-order model in eqs 1. If this is proved, the problem of controlling AD allows one to ensure that the sixth-order model and system of eqs 1 have some correspondence at equilibrium. This is relevant to a discussion of whether the two models are indistinguishable in designing robust control. That is, if propositions 1 and 2 are applicable to system of eqs 1, then the equilibrium of eqs 1 is a projection of the equilibrium of the sixth-order model onto a four-dimensional subspace, and such a projection captures the dynamical features of the Bernard et al.13 model. Results in this direction are summarized in the following section. Proposition 3. Consider the model in eqs 1 with kinetics given by eqs 2. Assume that the inlet composition xj,in for j = 3 and 4 is piecewise constant. Then, there exist constants α ∈ [0, 1] and D* ∈ [D̲ , D̅ ] ⊂ R+ involving NOC, such that model in eqs 1 has a unique equilibrium point x* ∈ Ωr = {x ∈ R4+: x3 < x3,in, x4 < x4,in}, which is locally stable. Actually, the components x* ∈ Ωr ⊂ R4+ of the equilibriumpoint coordinates for system of eqs 1 obey the expressions

μ1(x3) = μ1max μ2 (x4) = μ2max

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x1* =

x4 x4 + KS2 + (x4 /KI2)2

(2)

(x3,in − x3*)D k1μ (x3*) 1

x 2* =

(x4,in − x4*−)D + k 2μ1(x3*)x1* k 3μ (x4*−) 2

αDKS1 x3* = (μ1max − αD)

x1̇ = [μ1(x3) − αD]x1 x 2̇ = [μ2 (x4) − αD]x 2

(3)

and the solution of the equation

x3̇ = (x3,in − x3)D − k1μ1(x3)x1 x4̇ = (x4,in − x4)D + k 2μ1(x3)x1 − k 3μ2 (x4)x 2

x3 x3 + KS1

αDx4*2 + x4*(αDKI2 2 − μ2max KI2 2) + αDKS2KI2 2 = 0 (1)

(4)

where the scalars x4+ * and x4− * denote solutions of the last second-order equation. Note that, if the functions μ1 and μ2 stand for the Monod and Haldane kinetics, respectively, the equilibrium x* ∈ Ωr ⊂ R4+ for system of eqs 1 lies in a projection of the subspace Ω ⊂ R6+, that is, Ωr ⊂ Ω.5,13 Note that the properties of kinetic functions μ1 and μ2 define the

where the state variables are as follows: x1 represents the acidogenic bacteria concentration (g/L); x2 is the methanogenic bacteria concentration (g/L); and x3 and x4 denote the chemical oxygen demand (COD, g/L) and volatile fatty acids concentration (VFA, mmol/L), respectively. The positive real 2626

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existence and dynamical features of the equilibrium x*. Another issue to be noticed is the interpretation of equilibrium. As mentioned before, for the case of vinasse wastewater treatment, μ1 and μ2 denote Monod and Haldane kinetics, respectively, in eqs 2, but for other wastewaters with distinct organic matter sources, μ1 and μ2 can include different substrate inhibition effects. Equations 3 and 4 are general expressions for the equilibrium of any AD processes governed by the fourth-order model in eqs 1. Next, an H∞ control problem is formulated for the vinasse wastewater treatment considering the AD nonlinear model defined in eqs 1.

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ROBUST H∞ COD REGULATION The main AD objective in vinasse treatment is to decompose the dissolved organic matter present in wastewater from vinery Figure 3. Frequency domain of Urel(ω) and max {Urel}. ω∈[ω,ω]

Figure 1. Block diagram for the control synthesis. We,Wu, and Wn are weighting functions for the control design. WpΔ represents an unstructured multiplicative uncertainty (||Δ|| ≤ 1) whose maximum frequency response is captured by the weighting function Wp. The nominal plant Pnom is derived from linearization of system of eqs 1 at the equilibrium point described by eqs 3 and 4.

Figure 4. Frequency responses for K8th (solid line) and K4th (dashed line) controllers.

Figure 2. Frequency domain of nominal plant Pnom(ω) (solid line) and the set of perturbed models P(ω) (dashed lines).

fermentation (red wine, tequila, etc.). The outlet concentration of organic compounds must comply with environmental and safety regulations. The pollutants are usually measured in terms of soluble COD concentrations. Therefore, one of the key issues in wastewater treatment by means of AD is COD regulation through feedback control. The existence, uniqueness, and stability of solutions (including equilibrium points) are important properties that must be studied prior to the design of a feedback control scheme. Proposition 3 states fundamental mathematical issues toward robust control synthesis such that the dilution rate, D = F/V, is specified and the output is regulated in the face of inlet concentration disturbances and model uncertainties. In this work, the COD regulation problem was addressed by taking into account the following aspects: (i) the presence of fluctuations, called disturbances, such as the fact

Figure 5. Frequency response for nominal plant Pnom (solid line), controller K4th (dashed line), and closed-loop sensibility function S (dash-dotted line).

that the concentration of organic compounds in the inlet stream is unmeasured and unknown and (ii) the complexity of the digester and the uncertainty of its kinetic model in relation to AD operating conditions. A desirable specification for the control design is to attenuate the effect of noisy measurements, which are unavoidable in a wastewater plant. We explore H∞ theory as a suitable framework for solving the control problem of regulating COD as an alternative to previously proposed control schemes. As demonstrated later in this article, H∞ 2627



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Figure 8. Parameter percentage variations around the nominal values used in numerical implementation for set-point regulation in AD nonlinear model using H∞ controller K4th: arbitrary percentage variation (±8%) in parameters (A) α, (B) μ1max, and (C) μ2max. For case studies iii-a and iii-b, the arbitrary percentage variations are given by the continuous lines and open circles, respectively.

Figure 6. Simulation of the regulation problem in AD using H∞ controller K4th: (A) outlet COD concentration in AD (solid line) and reference COD concentration (dashed line), (B) load disturbance induced in x3,in (dashed line), and (C) dilution rate D (solid line).

Figure 9. Simulation of the regulation problem with parameter variation in the AD nonlinear model using H∞ controller K4th (reference COD = 2 g/L): (A) outlet COD concentration and (B) dilution rate D. For case studies iii-a and iii-b, the results are given by the continuous lines and open circles, respectively. The inlet disturbance signal in x3,in is presented in Figure 7B as well. Figure 7. Simulation of the set-point tracking problem in AD using H∞ controller K4th: (A) outlet COD concentration in AD (solid line) and reference COD concentration (dashed line), (B) inlet disturbance signal in x3,in (dashed line), (C) dilution rate D (solid line), and (D) peak dilution rate D (solid line) for the [674.5, 676.5] time period in panel C.

theory allow for the parameterization of a stabilizing controller such that internal stability of the closed-loop system is ensured; the achievement of output regulation; and the attenuation of the effects of unknown disturbances, noisy measurements, and model uncertainties. Problem Statement. Roughly speaking, the design procedure consists of the following steps: (i) linearization of the nonlinear process model at an operating point, (ii) derivation of a state-space representation of the uncertain model, (iii) description of performance specifications for the control using the frequency response of weighting function at the response frequency, and (iv) control design through H∞ synthesis. As usual in process control, Figure 1 shows the block diagram of the feedback control system. Because the AD system of eqs 1 has a locally stable (unique) equilibrium point x* (with coordinates given by eqs 3 and 4), we search for a suboptimal control input u = D − D*, under NOC, at x* ∈ Ωr ⊂ R4+. Such a control is synthesized under nominal parameter values π0 ∈ Π

Figure 10. Frequency responses for nominal plant Pnom (solid line), H∞ controller (dashed line), and RFOF control designed by MéndezAcosta et al.5 (dash-dotted line). Note that, if a disturbance has frequency components lower than 10−1, then (i) the nominal plant responses and (ii) the control response are expected to be larger in RFOF control than in the K4th controller. As a consequence, the tradeoff between robust COD regulation and control-action magnitude is expected to be superseded by H∞ control.

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Figure 11. Simulation of the regulation problem with a component in frequency (10−1 rad/h) in AD nonlinear model using H∞ controller K4th and the linear feedforward/feedback control of Méndez-Acosta et al.:5 (A) error signal of the H∞ controller, (B) error signal of linear feedforward/ feedback control, (C) dilution rate of the H∞ controller, and (D) dilution rate of linear feedforward/feedback control.

0.031 h−1, KS1 = 7.1 g/L, KS2 = 9.28 mmol/L, KI2 = 16 mmol/L, and the nominal inlet concentrations xin = [0, 0, 16, 68.78]. From eqs 7, the nominal state-space realization becomes

to derive a nominal plant Pnom(s). Then, Pnom is excited through kinetic and inlet concentrations parameters to stimulate its frequency response and to identify a family of plants PΔ = [I + Wp(s) Δ(s)]Pnom(s) toward control design in the face of disturbances and uncertainties. Note that the inputs u, n, and x3,in denote the dilution rate D − D*, noisy measurements, and inlet COD, respectively. The measured output y represents the COD concentration, x3 − x*3 , measured at the outlet stream, and z1 and z2 are auxiliary output signals used to specify control requirements on control (u) and error (e) signals, respectively. The generalized plant G is derived from the input/output relations such that z = Gd, where the output vector is z = [z1 z2|e]T and the input vector is d = [n|u]T valued at s = ωj for ω ∈ [ω̲ ,ω̅ ]. Then, G is given by the equation

A sensitivity analysis was performed on the nonlinear model in eqs 1 to find a first-order estimation of the parameter changes that can provoke significant variations in solutions (see Appendix I).16 The parameters with significant effects were found to be α, μ1,max, and μ2,max, which agrees with the physical interpretations in AD.13 Arbitrary values within ±10% of the nominal parameter values of α, μ1,max, and μ2,max are representative for designing robust COD regulation through H∞ control. The entries of the matrices in eq 8 were changed within such an interval to derive a family of uncertain plants. That is, the uncertainty in AD was shown through changes in parameter values and compared with respect to the frequency response of the nominal plant Pnom on ω ∈ [ω̲ , ω̅ ] = [1 × 10−4, 1 × 10−2]. This can be represented as a relative uncertainty, Urel, with the mathematical description

where Wu, We, and Wn are defined above. The designed controller is denoted as K. Formally, the described problem can be addressed by a classical H∞ controller.15 Thus, if it exists, K can be parametrized such that the closed-loop transfer function is minimized

Urel(ω) =

(6)

Nominal and Uncertain Plants. Proposition 3 states the existence of a unique equilibrium point x* ≠ 0 for any real parameter vector. This fact allows one to find a nominal plant at the point x*. Now, by considering u = D − D* and the COD concentration as the measured variable, i.e, (y = x3 − x*3 ), the following nominal representation of model in eqs 1 is derived

max {Urel} = max

ω∈[ω,ω]

ω

P(ω) − Pnom(ω) Pnom(ω)

(10)

Weighting Functions for Control Design. In what follows, details on the weighting functions are described to discuss, in turn, the solution proposed for the robust control problem formulated in the preceding section. In addition, a physical interpretation is made of each weighting function in Figure 1. The upper bound of the relative uncertainty Urel, given by max {Urel}, can be approximated by a weighting function

x ̇ = Ax + Bux(t0) = x0 y = Cx

(9)

where P(ω) represents the corresponding plant for each specific value of parameters α, μ1,max, and μ2,max. It should be noted that Urel can be approximated as WpΔ, where ||Δ|| ≤ 1. Figure 2 shows the frequency response of both nominal plant Pnom(ω) and family of plants Φ = (I + Urel)Pnom containing the set of plants P(ω). Figure 3 shows the frequency response from eq 10 and the upper bound of the relative uncertainty Urel computed from the expression

|| Tzd || = max ω σ[P1,1 + P1,2K (I − P2,2K )−1P2,1] ⎡ z1 ⎤ ⎢ z ⎥ ⎡ P1,1 P1,2 ⎤⎡ n ⎤ ⎥⎢ − ⎥ ⎢ 2⎥ = ⎢ ⎢− ⎥ ⎢⎣ P2,1 P2,2 ⎥⎦⎢⎣ u ⎥⎦ ⎣e ⎦

P(ω) − Pnom(ω) Pnom(ω)

(7)

where A = (∂f i/∂xk)|x* and B = (∂f i/∂u)|D*, for i, k = 1, 2, 3, 4, and C = [0, 0, 1, 0]. We take the following as nominal values of parameters π0 ∈ Π:5,14 D* = 0.02 h−1, α = 0.5, k1 = 42.14, k2 = 116.5 mmol/g, k3 = 268 mmol/g, μ1,max = 0.05 h−1 , μ2,max =

ω∈[ω,ω]

Wp(ω), which can be identified as a rational function.15 Wp is

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derived by finding a stable, minimum-phase, nth-order transfer function. An iterative least-squares fitting method was used for the search of a system matrix to capture the frequency response of eq 10. Thus, we have Wp(ω) = Wp,N(ω)/Wp,D(ω), where Wp,N(ω) = 1.193 × 10−5s3 + 4.695 × 10−4s2 + 4.298 × 10−3s + 5.971 × 10−5 and Wp,D(s) = s3 + 0.237s2 + 0.015s + 1.593 × 10−4. Because the control input should be implemented by a servo mechanism, which allows the manipulation of the inlet flow rate in AD, a frequency constraint is taken into account through the weighting function Wu(ω) with ω ∈ [ω̲ , ω̅ ]. Thus, a performance weighting restriction is included as ||WuK(1 + PK)−1|| ≤ 1 ⇒ ||1/Wu|| ≤ ||K(1 + PK)−1, from which Wu = [s + (ωb/Mu)]/(εus + ωb) with εu = 0.05, ωb = 100, and Mu = 0.1. The weighting function Wu involves the following interpretation: Wu is bounded by Mu = 0.1 for low frequencies, Mu is related to the maximum dilution rate D − D* = 0.1 h−1, and εu = 0.05 is a bound at high frequency that is related to the highfrequency gain. Now, ||We(1 + PK)−1|| ≤ 1 ⇒ ||1/We|| ≤ (1 + PK), from which We = [(s/Me) + ωb]/(s + ωbεe) with Me = 10 and εe = 0.005. Note that the sensitivity function (1 + PK)−1 is a criterion for robust performance to the disturbance attenuation problem. Finally, the weighting function for noise is derived to have Wn = (s + ωnεn)/[(s/Mn) + ωn] with ωn = 103, εn = 0.01, and Mn = 0.1. The weighting function Wn is physically interpreted as follows: A disturbance in measurements can be caused by both resolution and response of sensors. That is, the available technology for COD measurements has typical responses from 1.5 to 15 min (equivalently, from 0.069 to 0.0069 rad/s) with a resolution of COD (g/L) from 10 to 0.2. Without loss of generality, εn is arbitrarily chosen to be 0.01 rad/s, whose value can be adjusted for specific equipment. Robust Synthesis. The suboptimal control problem was numerically solved by means of the robust Control Toolbox of Matlab using the standard Riccati solution (linear matrix inequalities method).17 The approximated stabilizing controller K(s) was derived after an iterative numerical process (hinfsyn command), ensuring nominal internal stability with γ = 0.1517. We assume that the system model is described by the set of multiplicative perturbations Λ = {(I + WpΔ)Pnom: Δ ∈ RH∞}. Then, the condition for robust stability is accomplished with || WpPnomK(1 + PnomK)−1||∞ = 0.3452 ≤ 1 (see theorem 8.5 in ref 17). The stabilizing full-order controller is eighth-order and is given by K8th(s) = K8th,N(s)/K8th,D(s), such that K8th,N(s) = 13.33s7 + 2.8 × 104s6 + 2.677 × 105s5 + 1.03 × 106s4 + 1.38 × 103s3 + 7328s2 + 115.6s + 0.544 and K8th,D(s) = s8 + 1266s7 + 2.722 × 105s6 + 6.366 × 105s5 + 4.645 × 104s4 + 8.745 × 105s3 + 5.677 × 104s2 + 975.5s + 4.907. The Hankel’s values from the full-order controller are given by σ8th = [0.1946, 0.1530, 0.0211, 0.0041, 0.0032, 2.459 × 10−7, 1.636 × 10−8, 4.409 × 10−10]. To seek a low-order controller, a balanced model truncation by the square-root method was applied to the full-order controller and, as a result, the following controller was obtained: K4th(s) = KN(s)/KD(s), such that KN(s) = 20.57s3 + 1232s2 + 1233s + 138.6 and KD(s) = s4 + 147.6s3 + 2947s2 + 3977s + 1180. The Hankel’s values from the reduced-order controller are σ4th = [0.1946, 0.1530, 0.0211, 0.0041]. Figure 4 shows quite similar frequency responses for both controllers K8th and K4th. Figure 5 shows the frequency response for the nominal plant Pnom, controller K4th, and closed-loop sensibility function S. The proposed control scheme was evaluated under the following cases: (i) set-point regulation with nominal parameter values π0

∈ Π; (ii) set-point tracking with nominal parameter values π0 ∈ Π; and (iii) set-point regulation with two set parameter variation Π ⊃ πδ = {α(1 ± δ), μ1max(1 ± δ), μ2max(1 ± δ)}, where α, μ1max, and μ2max are nominal values and 0 > δ > 0.1. The numerical evaluations of the controller execution for cases i and ii are shown in Figures 6 and 7, respectively. The parameter changes induced in the third case are shown in Figure 8. Figure 9 shows the results under parameter changes depicted in Figure 8. Disturbances around the nominal inlet COD concentration (x3,in = 16 g/L) were attenuated in both cases with nominal parameter values π0 ∈ Π. Figure 8 shows the percentage variations of two set parameters around nominal values in α, μ1max, and μ2max in the AD model. Note that the arbitrary percentage parameter variation intervals were ±8%. Figure 9 shows that, despite disturbances around the nominal inlet COD concentration (x3,in = 16 g/L) (the disturbance is described in Figure 6B) and arbitrary parameter variations around nominal values (±8%), controller K4th is capable of achieving good performance in the face of set-point regulation. Because AD is an essential step in overall wastewater treatment, COD control is very important. Consequently, linear approaches have been developed to handle robust regulation of COD. Because linear controllers are desired for implementation at the industrial scale, a scheme has been derived from the linear approach to nonlinear (geometrical) control.5 The proposal by Méndez-Acosta et al.5 has the structure of reference-feedforward/output-feedback (RFOF) control, whose robustness allows for the handling of modeling errors, load disturbances, and even saturation constraints in the control input. Actually, as an effect of RFOF control, COD is robustly regulated in a very precise manner. However, because RFOF control is designed as an approach to the exact inversion of an AD plant, the control actions can have a high magnitude under disturbances with certain frequency components. This implies that the tradeoff of regulating robustly and having lowmagnitude control actions cannot satisfactorily be addressed. This fact is depicted in Figure 10. Note that, if disturbance enters AD with frequency components lower than 10−1 rad/h, the nominal plant has a frequency response. Then, comparing the control responses, K4th control has a lower response than RFOF control. This is derived from the weighting functions included during H∞ synthesis. Hence, even if RFOF control induces better COD regulation, a lower magnitude can be induced in control actions through H∞ synthesis. As a direct consequence, the tradeoff between robust COD regulation and control action requirements is superseded with H∞ synthesis. It should be noted that, for frequencies larger than 10−1 rad/h, the response of the nominal plant decays. This fact led us to complementary criteria on the selection between RFOF- and H∞-based controllers. Figure 11 shows the time evolution of both RFOF and K4th controllers. A disturbance of 10−1 rad/h was entered into the AD process with the nonlinear model in eqs 1. Although the RFOF control permits less error signal, the control action is degraded in the presence of components of a certain frequency. Contrarily, H∞ control allows one to address the tradeoff between robust regulation and control constraint by weighting functions.

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CONCLUDING REMARKS A robust H∞ control design of AD is proposed for controlling COD in vinasse wastewater. Uncertainty from parameter variation within the AD nonlinear model representation was characterized using relative uncertainty. The robust controller 2630



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was evaluated for robust regulation acting on the nonlinear AD plant. In addition to robust regulation, the robust controller was evaluated for set-point tracking (servo control) as well. A sensitivity analysis was performed to identify the system parameters affecting AD solutions. Identified parameters were assumed to be the source of parametric uncertainty. Then, weighting functions were designed to capture the effect of the parametric uncertainty, through a relative multiplicative uncertainty. The parameters are related to kinetics and hydrodynamic regime. That is, robust stability was assessed using weight functions such that the effects of both uncertain kinetics and hydrodynamic regime can be compensated, achieving regulation of effluent COD. For a clearer exposition of the results, a discussion is motivated in comparative context. We show that, although the previous RFOF control permits less error signal, its control action is degraded in the presence of components of certain frequency. Contrarily, the H∞ control allows one to address the tradeoff between robust regulation and control constraint through weighting functions. Numerical simulations show that the robust H∞ controller is capable of addressing robust COD regulation and the tradeoff between COD regulation and control actions. Because the robust controller is linear, experimental implementation is possible; hence, results in this direction will be reported elsewhere.

Figure 13. Solution of sensitive equation Ṡf for x3 and x4. The figure shows that state x3 is sensitive to variations in parameters α and μ1,max, whereas state x4 is sensitive to variations in parameters α, μ1,max, and μ2,max. As in the case of x1 and x2, the effects of the remaining parameters on x3 and x4 are negligible.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.A.F.-E. thanks CONACyT for financial support under Scholarship Grant 160117 and FONCICYT for financial support under BITA consortium (S-3146). G.Q. thanks PAICYT-UANL for financial support under Grant IT546-10.

APPENDIX I. SENSIBILITY ANALYSIS For a set of parameters: Π = {α, μ1max, μ2max, KS1, KS2, KI2, k1, k2, k3}, the sensitivity function for solutions is Sḟ = ASfSf + BSf where Sf = [∂x/∂π]π0, ASf = [∂f(x)/∂x]x*, BSf = [∂f(x)/∂π]π0 ∈ R9×9, and the nominal parameter π0 ∈ Π. Figures 12 and 13 show the solutions of the sensitivity equation for states x1, x2, x3, and x4 with respect to parameter

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Figure 12. Solution of sensitive equation Ṡf for x1 and x2. The figure shows that state x1 is sensitive to variations in parameters α and μ1,max, whereas state x2 is sensitive to variations in parameters α, μ1,max, and μ2,max. From these results, the sensitive parameter set is defined as {α, μ1,max, μ2,max}. The effects of the remaining parameters on solutions are negligible.

set Π. The parameters having the most significant effects on dynamic behavior of a nonlinear model are α, μ1max, and μ2max.

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