Anti-windup Feedforward/Feedback Control for a Class of Nonlinear Systems

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1 Anti-wdup Feedforward/Feedback Control for a Class of Nonlear Systems H. O. Méndez-Acosta a, R. Femat c,*, D. U. Campos-Delgado b, V. González-Alvarez a a Depto. Ingeniería Química, Universidad de Guadalajara, 44860, Guadalajara-Jalisco, Meico. b Fac. de Ciencias, Universidad Autónoma de San Luis Potosí, 78290, San Luis Potosí-S.L.P., Meico. c Depto. Matemáticas Aplicadas y Sistemas Computacionales, IPICyT, Apdo. Postal , San Luis Potosí-S.L.P, Meico. Abstract: In this paper an anti-wdup scheme is proposed for two robust feedforward/feedback control laws: one lear and the other nonlear. Sce the control laws present an observer-based structure, the anti-wdup scheme results a natural manner by takg to account the actuator constrats. It is shown that the anti-wdup scheme improves the performance of the control laws. The resultg strategies are implemented an anaerobic digester for wastewater treatment. Performance is illustrated by numerical simulations. Copyright 2003 IFAC Keywords: Anti-wdup; Robust Feedforward/Feedback Control; Wastewater Treatment. INTRODUCTION All practical control systems are subject to constrats on the manipulated puts which are imposed due to the limited capacity of the actuators (valves, pumps, etc.). Particularly, if the control put constrats are not takg to account the controller design, it may lead to significant deterioration of the control performance and even cause closed-loop stability. Such a performance-stability degradation is due to the so called wdup phenomenon. This phenomenon is typically ehibited by dynamic controllers with slow or unstable modes (Doyle et al., 987). Recently, numerous anti-wdup schemes for a class of nonlear systems have been proposed. It has been shown that most of them present a observer-based structure (Walgama and Sternby, 990; Kothare et al., 994). Such a structure can be implemented by constructg the difference between the computed value of the manipulated put and the actual put enterg to the process, and feedg back this signal through a static ga to the controller dynamics (A/ ström and Wittenmark, 984). On the other hand, the put-output (I/O) learizg control combed with the optimization-based strategies has been applied antiwdup strategies for nonlear systems. A lear antiwdup scheme that preserve the feedback learization of nonlear systems subject to constrats was proposed by Doyle, (999). Kurtz and Henson, (997) combed an I/O learizg controller with a lear model predictive controller to regulate constraed nonlear systems. An alternative anti-wdup scheme for nonlear systems was derived from a fite-horizon optimization (Valluri et al., 998). Also, an etension of the nonlear ternal model control to systems with constraed puts and measured disturbances has been proposed (Kendi and Doyle, 998). Kothare et al., (994) proposed a novel anti-wdup scheme to suppress the gap between the saturated actuator and the computed control put. This scheme comprises an tegral-type compensation which replaces the conventional optimization algorithm. A method to design the controller gas and a modified nonlear observer-based structure to attenuate the effect of wdup has been proposed by Kapoor and Daoutidis, (999). More recently, a modified I/O learizg feedback control was derived by defg a specific trackg error form and troducg a fleible tung parameter, which is directly obtaed by solvg an algebraic calculation (Wu, 2002). In this paper, an anti-wdup scheme is proposed for two feedforward/feedback control laws derived for a class of nonlear systems. The obtaed anti-wdup scheme has a natural observer-based structure, which * Correspondg author. rfemat@ipicyt.edu.m

2 results from feedg back the measure of the constraed put to the controller. As a consequence, the tung of the anti-wdup scheme is simple. In addition, such a scheme shows an improvement the performance of the control laws. The rest of the paper is organized as follows. Section II contas the class of nonlear systems and the problem formulation. The development of the anti-wdup scheme is presented Sect. III. In Section IV, an anaerobic digester for wastewater treatment is used to show the performance and robustness of the proposed scheme. Fally, some conclusions are poted out Section V. 2. A CLASS OF NONLINEAR SYSTEMS AND PROBLEM FORMULATION 2. The Class of Nonlear System The present work is focused towards nonlear systems with the followg properties: (P) The relative degree r is well-posed and equal to one at a pot /0 ú n (i.e., r = ). (P2) The nonlear system is mimum-phase (i.e., stable zero dynamics). (P3) Only the output y = h() is available for feedback, where h() is a smooth function. Then, nonlear systems with the above-mentioned properties can be written the followg canonical form (Isidori and Byrnes, 990): z& = (, z ν) + β(, z ν) u (.) & ν = ξ(, z ν) (.2) where L f h() = "(z,<), L g L 0 f h() = $(z,<). u 0 ú represents the control put, z 0 ú stands for the output state and < 0 ú n! denotes the ternal dynamics. In addition, subsystem (.2) is stable when y = 0 (Property P3). 2.2 Problem Statement The objective of this paper is to address the regulation problem of the nonlear systems the form () when the control put u is constraed. To this end, let us suppose that the followg assumptions hold: (A) The system () is disturbed by a disturbance w which is uncerta and unmeasured. (A2) Under disturbances w, the Lie derivatives of subsystem (.), affected by the disturbance w, has the form: L f h() = R(z,w) and L g L 0 f h() = ((z,w); where R(z,w) = az + n(z,w) with a given constant a 0. (A3) The control put u is subject to constrats. That is, u sat = sat(u) and its mathematical representation is given by: uma, u uma sat( u) = u, um < u < u (2) ma um, u um where the bounds u m and u ma are known. In the net section two controllers are designed under the above-mentioned assumptions. Each controller respectively results from the analysis of the followg study cases: (C) Only a nomal value of the perturbation w is known (e.g., w) = (w ma + w m ) / 2). In this case, the function ((z,w) is given by: ((z,w) = ()(z,w) ) + (#(z,w); where ()(z,w) ) is known and (#(z,w) is unknown and bounded. (C2) The function ((z,w) is given by: ((z,w) = () + (#(z,w); where (#(z,w) is unknown and bounded. () is a constant different from zero such that the sign (()) = sign (((z,w)) at / 0 ú n. 3. ANTI-WINDUP SCHEME Under assumptions (A)-(A3) the system () with the properties (P)-(P3) can be written as follows: z& = az + ϕ(, z w) + γ (, z w) sat() u (3.) & ν = ξ(, z ν, w) (3.2) where a 0 and n(z,w) represents the nonlear uncerta terms, which are related to the function R(z,w). It should be noted that the block diagram Fig. does not clude the actuator dynamics. Such a dynamic can affect the closed-loop performance a physical implementation. Fortunately, the anti-wdup performance can be tested by standard SIMULINK package cludg actuator dynamics (Bohn and Atherton, 995). Now, regard to the stability analysis, it had been reported that sgle loop systems with anti-reset wdup schemes are robust compared to its nomal put-output response (Glattfelder and Schaufelberger, 983). Such results were reported for lear plants; however, an etension to nonlear systems can be obtaed by considerg the cross section defed by nonlear constrats and control surfaces (Alonso and Banga, 998). 3. An Anti-wdup Scheme For A Nonlear Control In a previous work, Méndez-Acosta et al., (2003), by departg from the case (C) proposed an output feedback nonlear control with an uncertaties dynamic estimator for nonlear systems of the form (3) when the control put is not constraed. Such a controller is given as follows: z$& = az + $ ηnl + γ(, z w) u + g (4.) u = [ az $ ηnl + Kp( z z*) ](4.2) γ (, zw) The control law (4) is composed by a proportionaltegral (PI) structure given by: a) the difference between the output and the desired value with proportional constant Kp; and b) the tegral action duced by the term 0$ NL = g 2 I(z(t)! z$(t)) dt. Then, under assumption (A3), the control law (4) can reach

3 saturation values due to a sudden change the puts. In such a situation, the feedback is broken and the plant behaves as an open-loop with a constant put, allowg a possible degradation of the controller performance. Moreover, due to the tegral action, the control saturation can duce undesirable effects such as a large overshoot and settlg time (Kothare et al., 994). This phenomenon is known as reset wdup. From the implementation pot of view, reset wdup appears due to the tegral action sce the controller does not have knowledge of the control put saturation, so it contues tegratg even if the feedback is broken. On the other hand, note that for the control law (4), there eists an ternal feedback between the computed control put (4.2) and the observer used as dynamic estimator (4.). This means that the observer performance is affected by the computed control put. Then, the problem of reset wdup the controller (4) can be treated a natural way by feedg back the control put under the constrat described (2) to the observer. In this way, the controller (4) can be rewritten the followg form: z$& = az + $ ηnl + γ(, z w) usat + g (5) u = [ az $ ηnl + Kp( z z*) ] γ (, zw) where u sat is given by Eq. (2). Thus, the controller (5) presents an observer-based anti-reset wdup structure (see Figure ). 3.2 Reference-Feedforward/Output-Feedback Control With Anti-wdup Now, for the study case (C2), Méndez-Acosta et al., (2003) proposed a reference-feedforward/output feedback structure for nonlear systems of the form (3) without considerg the constrats the control put u. Such controller can be written the state-space as follows: z$& = az + $ ηl + γu + g u = az + Kp z z γ [ $ ηl ( *)] z$& = az + $ ηl + γusat + g u = [ az $ ηl + Kp( z z*) ] γ (6) Then, followg the same ideas reported the last subsection, it is easy to show that the control law (6) can be rewritten as follows: (7) where, once aga, u sat is given by Eq. (2). The control law (7) is conformed by a lear part given by Ks () = g ( a + g) 0 / γ g2 0 g / γ ( Kp a)/ γ Kp/ γ 0 which is connected with the saturation function (2). Thus, the lear controller K(s) is a transfer matri with three puts and one output respectively given by: z*, z, u sat and u. The coefficients g and g 2 are chosen such that the polynomial s 2 + g 2 s + g = 0 be Hurwitz. The block diagram of the controllers (5) and (7) is depicted the Figure. Figure. Block diagram for the implementation of the controllers (5) and (7). 4. ILLUSTRATIVE EXAMPLE In order to implement the controllers developed the last section, an anaerobic digester for wastewater treatment is used. 4. The Model Satisfies the Assumptions and Properties Consider the control of an anaerobic digester used the treatment of distillery vasses (Bernard et al., 200): & = ( µ D) & 2 = ( µ 2 D) 2 & 3 = ( 3, 3) D (8) & 4 = ( 4, 4) D kµ & 5 = ( 5, 5) D + k2µ k3µ 22 & = ( ) D qc+ k µ + k µ 6 6, where the states variables respectively stand for:, acidogenic bacteria concentration (g/l); 2, methanogenic bacteria concentration (g/l); 3, total alkality (mmol/l); 4, chemical oygen demand (COD, g/l); 5, volatile fatty acids (VFA, mmol/l) and 6, total organic carbon concentration (TIC, mmol/l). State variable 3 is given by the sum of VFA*s and bicarbonate concentrations. The terms i, represent the fluent composition. The dilution rate, D, is defed as the ratio of the fluent flow rate (Q ) over the reactor volume (V). Parameter " denotes process heterogeneity (" = 0 corresponds to an ideal fied bed reactor, whereas " = corresponds to an ideal contuous stirred tank reactor or CSTR). Parameter qc represents the CO 2 molar flow rate which can be computed usg Henry*s law (i.e., qc = k L a(co 2!K H P C ), where k L a represents the liquid-gas transfer coefficient, K H the

4 Henry*s constant and P C the CO 2 partial pressure). The model approach the specific grow of acidogenic and methanogenic bacteria by a Monod and Haldane ketic epressions; which are respectively given by the followg functions: µ ( ) = µ 4,ma µ ( ) = µ 2 5 2,ma 4 + K 4 S KS2 + KI 2 where :,ma, K S, : 2,ma, K S2, and K I2 are ketic constants. (9) When anaerobic digestion is used for wastewater treatment purposes, the control objective is the regulation of the effluent organic matter at a prescribed concentration the face of fluctuations at the fluent composition and environmental conditions. Such an objective is accomplished by manipulatg the dilution rate (Docha and Bast, 986). In this way, the control objective for the anaerobic digester (8) can be stated as follows: it is desired to achieve the COD regulation at a given value (i.e., 4 6 * 4 where * 4 represents the desired COD concentration at the effluent composition) agast fluctuations at the fluent composition and modelg errors by takg D as a control put. Under normal operation conditions, the control put u = D is constraed by: Dma, D Dma sat( D) = D, Dm < D < Dma Dm, D Dm where D sat = sat(d). D ma = 0.05 hr! and D m = hr! respectively correspond to the value of wash-out condition and the mimum permissible value which may be used to avoid emptyg the digester (Steyer et al., 2002a). Note that the saturation function is not symmetric; as a consequence, it is epected that such property has a strong fluence the controller performance. Now, let us rewrite the anaerobic digestion model (8) the affe form for SISO nonlear systems: 0 = f() + g()u, where is the vector of states, 0 ú 6, u = D and f(), g() are smooth vector fields given by µ µ ( 3,, 3) f ( ) =, g ( ) = k (,, ) µ 4 4 k k (,, ) 2µ 3µ k µ k µ qc (,, ) The system output is given by the COD concentration, i.e., y = h() = 4. By computg the Lie derivative of the system output h() along the vectors fields f() and g(), it is obtaed that L g L 0 fh() = ( 4,! 4 ). Sce for normal operatg conditions ( 4,! 4 ) > 0, it is easy to shown that the system (8) has a relative degree r = which is well-defed out of the wash-out condition. In this way, such a system can be written the canonical form () as follows: z& = ( 4, z) D kµν 5( 4, z) (0.) & ν = kµν ν5( 4, z) νν 2 5 & ν2 = ( 4, z) [ k2µν 4 k3µ 2] ν4 νν (0.2) 3 5 & ν3 = ( 4, z) [ k4µν 4 k5µ 2 qc ν5] ν4 & ν4 = ν 4( µ µ 2) & ν = µν [ k ν ( z) ] , It has been proved that subsystem (0.2) is globally stable under normal operation conditions (Méndez- Acosta et al., 2002). In addition, the system output y = 4 is available for feedback (Steyer et al., 2002b). Then, the anaerobic digestion model (8) satisfies the properties (P)-(P3). In consequence, system (0) can be rewritten the form (3) as follows: z& = kµν 5( 4, z) + ( 4, z) D () & ν = ξ(, z ν, w) where a = 0, n(z, w) =!k : < 5 ( 4,! z) ", w = 4, and ((z, w) = ( 4,!z)D. 4.2 Anti-wdup Scheme for Substrate Regulation Once have been proved that model (8) satisfies the assumptions (A)-(A3) and the properties (P)-(P3), it is easy to shown that for the case (C) the followg control laws is obtaed: z$& = $ η + ( 4, z) Dsat + g $& η (2) D = [ $ η + Kp( z 4*) ] ( z) Ks () = 4, where ) 4, = 6.0 g/l is the nomal value of the perturbation for the vasses feedg to the reactor used to validate model (8) (see Steyer et al., 2002a). Now, by considerg the case (C2) and assumg without lost of generality that ) = 6.0 g/l a control law of the form (7) can be obtaed. Such a control law is given by the followg lear controller g g2 0 0 / g 0 / g2 0 0 Kp/ Kp/ 0 (3) conjunction with the saturated control put, D sat.

5 4.3 Numerical Simulations controller performance. In this subsection, the performance and robustness of the control laws (2) and (3) are evaluated via numerical simulations. In order to observe the performance of such control laws the servo-control and regulation problems, four different set-pot changes were made on the effluent COD concentration. The model parameters used the simulations are the followg: k = 42.4, K H = 6.0, k 2 = 6.5, " = 0.5, k 3 = 268.0, :,ma = 0.05, k 4 = 50.6, : 2,ma = 0.03, k 5 = 343.6, K S = 7., k 6 = 435.0, K S2 = 9.28, k La = 0.825, K I2 = 6.0 and P T = While the states itial conditions used are given by: (0) = 0.358, 2 (0) = , 3 (0) = 76., 4 (0) = , 5 (0) = and 6 (0) = Such values of the plant parameters have been identified for a pilot scale digester (Bernard et al., 200). The control parameters used along the simulation were the same for both control laws and these are given by: g =.5, g 2 = 0.5 and Kp =!2.0, which were used by Méndez-Acosta et al., (2003) is unconstraed control. Figure 3.Performance of the nonlear control (2) (solid le) and (4) (dotted le). Figure 2. COD fluent concentration, 4,. In order to consider realistic operatg conditions, the COD fluent concentration was varied randomly around a mean value, as is depicted Figure 2. In addition, the presence of noise the measurement was takg to account. For such purpose, uniformly random noise (±0. g/l) was added to the measured variable ( 4 ). Figure 3 shows the dynamical behavior of the system output, i.e., the COD effluent concentration, 4, and the time-response of the controller. The dotted le corresponds to the nonlear control with no anti-reset wdup (which was reported by Méndez-Acosta et al., 2003) whereas the solid le shows the performance of controller (2). Note that performance degradation is dimished by the anti-wdup scheme. In analogous manner, the Figure 4 shows the closed-loop performance. The solid le corresponds to the lear control law (which was reported by Méndez-Acosta et al., (2003) whereas the solid le shows the performance of the controller K(s) Eq. (3). Here, the anti-wdup scheme also avoids the degradation of the Figure 4. Performance of the controller from Eq. (3) (solid le) and Eq. (7) (dotted le).

6 5. CONCLUSIONS An anti-reset wdup scheme for a class of nonlear systems has been proposed this paper. The antiwdup strategy is based on an observer structure. The degradation of the controller performance due to saturation is dimished by the proposed strategy. An anaerobic digester was used as illustrative eample. The dynamical model of the digester satisfies all properties of the class system and the assumptions made for the anti-wdup design. Simulations from an identified model were carried out to illustrate the results. Eperimental implementation is a pilot scale digester is epected. Results this direction will be reported as soon as available. REFERENCES Alonso, A.A. and Banga J.R. (998). Design of a class of stabilizg nonlear state feedback controller with bounded puts. Ind. Eng. Chem. Res., 37, Aström, K.J. and Wittenmark, B. (984). Computer Controlled Systems Theory and Design. Prentice Hall, Englewood Cliffs, New Jersey. Bernard, O., Hadj-Sadok, Z., Docha, D., Genovesi, A., and Steyer, J.P. (200). Dynamical model development and parameter identification for anaerobic wastewater treatment process. Biotech. & Bioeng., 75, Bohn C. and Atherton D.P. (995). An analysis package comparg PID anti-wdup strategies. IEEE Control Systems, 5, Docha, D. and Bast, B., (986), Modellg and Control of Biotechnological Process, Pergamont Press, Oford U.K. Doyle, J.C., Smith, R.S. and Enns D.F. (987). Control of plants with put saturation nonlearities. Proc. of Amer. Contr. Conf., 034. Doyle, F.J. (999). An anti-wdup put-output learization scheme for SISO systems. J. Process Control, 9, Glattfelder A.H. and Schaufelberger W. (983). Stability analysis of sgle loop control systems with saturation and anti-reset wdup circuits. IEEE Trans. Automatic Control, 28, Isidori, A. and Byrnes, C. (990). Output Regulation of Non-lear Systems, IEEE Trans. Automatic Control, 35, Kappor, N. and Daoutidis, P. (999). An observerbased anti-wdup scheme for nonlear systems with put constrats. Int. J. Control, 72, Kendi, T.A. and Doyle, F.J. (998). Nonlear ternal model control for systems with measured disturbances and put constrats. Ind. Eng. Chem. Res., 37, Kothare, M.V., Campo, P.J. Morari, M. and Nett, C.N. (994). A unified framework for the study of anti-wdup designs. Automatica, 30, Kurtz, M.J. and Henson, M.A. (997). Input-output learizg control of constraed nonlear processes. J. Process Control, 7, 3-7. Méndez-Acosta, H.O., Femat, R., González-Alvarez V. and Steyer, J.P. (2002). Substrate regulation an anaerobic digester based on geometric control, Proce. of Lat-American Congress of Automatic Control Guadalajara, Meico. Méndez-Acosta, H.O., Campos-Delgado, D.U., Femat, R., and González-Alvarez V. (2003). Robust feedforward/feedback control for a class of nonlear systems, accepted Robust Control Design Congress Milan, Italy. Steyer, J.P., Bouvier, J.C., Conte, T., Gras, P., and Sousbie, P. (2002a). Evaluation of a four year eperience with a fully strumented anaerobic digestion process. Wat. Sci. Tech., 45, Steyer, J.P., Bouvier, J.C., Conte, T., Gras, P., Harmand, J., and Delgenes, J.P. (2002b). Onle measurements of COD, TOC, VFA, total and partial alkality anaerobic digestion processes usg fra-red spectrometry. Wat. Sci. Tech., 45, Valluri, S., Soroush, M. and Nikravesh, M. (998). Shortest-prediction-horizon nonlear model predictive control. Chem. Eng. Sci., 53, Walgama, K.S. and Sternby, J. (990). Inherent observer property a class of anti-wdup compensators. Int. J. Control, 52, Wu, W. (2002). Anti-wdup schemes for a constraed contuous stirred tank reactor process. Ind. Eng. Chem. Res., 4,