Saturated Output-Feedback Control for a Class of

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Saturated Output-Feedback Control for a Class of Continuous Fermenters A. Schaum ∗ J. Alvarez ∗∗ T. Lopez-Arenas ∗∗∗ ∗

Departamento de Matematicas Aplicadas y Sistemas, Universidad Autonoma Metropolitana - Cuajimalpa, Mexico. ∗∗ Departamento de Procesos e Hidraulica, Universidad Autonoma Metropolitana - Iztapalapa, Mexico. ∗∗∗ Departamento de Procesos y Tecnologia, Universidad Autonoma Metropolitana - Cuajimalpa, Mexico. Abstract: The problem of designing a saturated Output-Feedback (OF) controller for a three species continuous fermenter with substrate and product-inhibited kinetics is addressed. The reactor must be operated at maximum production rate by manipulating the feed rate on the basis of a (biomass, substrate or reaction rate) measurement. The combination of global dynamics concepts with State-Feedback (SF) and OF nonlinear control tools yields a linear dynamical OF saturated control design with: (i) robust stability conditions in terms of the controller gains and saturation limits, and (ii) simple tuning guidelines. The linear OF saturated controller: (i) recovers, in a practical stability sense, the behavior of a globally stabilizing nonlinear robust SF saturated controller, and (ii) becomes a linear PI controller in the absence of saturation. The proposed approach is illustrated and tested with a representative example through simulations. Keywords: Bioreactors, Nonlinear dynamics, Input constraints, PI Control, Optimisation. 1. INTRODUCTION

Output-Feedback (OF) control plays a key role in biological process optimization, and the nonlinear bioreactor dynamics implies that the associated joint processcontrol design is an interesting and complex problem. This is especially the case when simplicity, reliability and low development-maintenance costs are important application-oriented design specifications. The saturated SF and OF control design problem has been addressed, in the chemical reaction engineering field in general, employing: (i) ad hoc saturation handling schemes for nonlinear geometric (Kendi and Doyle, 1998; MendezAcosta et al., 2005), and geometric-MPC (Kurtz and Henson, 1997) SF controllers, (ii) geometric and bifurcationbased approaches (Alvarez et al., 1991; Alonso and Banga, 1998; Barron and Aguilar, 1998), (iii) observer-based antiwindup schemes (Kapoor and Daoutidis, 1999), and (iv) Lyapunov approaches for saturated PI and geometric controllers (Jadot et al., 1999; Rapaport and Harmand, 2002; El-Farra and Christofides, 2003). Even though valuable insight on the saturated control problem has been gained, the problem is still open. On one hand, the powerful advanced nonlinear control approach offers rigorous theoretical backup and systematization, but yields complex, model-dependent nonlinear dynamical control schemes. On the other hand, the simple conventional PI technique can effectively control nonlinear systems, but it is not clear when and why this can be done, and its implementation requires a combination of experience, process insight and testing. Copyright by the International Federation of Automatic Control (IFAC)

These considerations motivate the scope of the present study: the development of a saturated OF control scheme for robust regulation at maximum production rate of a continuous fermenter with inhibited kinetics with emphasis on simplicity, and reduced model dependency features. Exploiting the reactor open-loop dynamics and observability properties, a saturated OF controller is designed with reduced model-dependency features that recovers the behavior of a globally stabilizing nonlinear robust SF saturated controller and combines: (i) implementation as a saturated PI controller with anti-windup protection, (ii) robust stability conditions in terms of the controller gains and saturation limits, and (iii) simple tuning guidelines. The proposed approach is illustrated and tested with a representative example through simulations. 2. CONTROL PROBLEM Consider a bioreactor with substrate feed (at concentration Sf ) and conversion according to the reaction α1 S → α2 B + P , and the corresponding mass conservation-based dynamics (Baileys and Ollis, 1986) s˙ = −α1 ρ(s, p)b + θ(sf − s) b˙ = α2 ρ(s, p)b − θb p˙ = ρ(s, p)b − θp, y = h(x), P = θp, x = (s, b, p)′ .

(1)

Here, s, b and p are the dimensionless substrate, biomass and product concentration, referred to the nominal substrate feed concentration Sf (g/l), ρ(s, p) is the Haldane growth rate function (Agrawal et al., 1989; Henson and Seborg, 1992)

7126


k0 s(1 − p/ps ) , (2) s2 /ki + s + ks with mass action (k0 ), substrate inhibition (ki ), substrate saturation (ks ), and product saturation (ps ) coefficients, which represents two inhibition effects with respect to (i) the limiting substrate s, and (ii) the product p, ρ(s, p)b is the biomass production by bioreaction, θ = q/V is the dilution rate, α1 (or α2 ) is the substrate (or biomass) yield coefficient, sf is the substrate feed concentration, y = h(x) is the (substrate, biomass or reaction rate) measurement signal, and P is the production rate. ρ(s, p) =

In (2), the product depending inhibition accounts for inhibitory effects due to the accumulation of non-cellular product (e.g. ethanol) in the culture medium. An example of such a process is the fermentation of whey (Yamane et al., 1977). In compact vector notation the reactor model (1) is written as x˙ = f (x, θ, d), x ∈ X, θ ∈ Θ, y = h(x) (3) with exogenous input d, input (or state) space Θ (or X) d = (k0 , ki , ks , ps , sf )′ , Θ = [θ− , θ+ ], (4) X = {x ∈ R3+ : s ≤ sf , b ≤ α2 /α1 sf , p ≤ ps }, The consideration of X as a natural candidate set for control design is motivated from mass-balance considerations.

to take the reactor (1) into the form b˙ = ρ[α1 /α2 (m − b), α1 (n − (m − b)/α2 )]b − θb, m ˙ = θ(me − m), (8) n˙ = θ(ne − n), me = α2 /α1 sf , ne = sf /α1 , implying that: (i) the stoichiometric set M is invariant, (ii) M is a locally asymptotically stable (LAS) attractor in X, (iii) no limit cycles are possible, and (iv) the dynamics of m and n are asymptotically stable with SSs determined by the biomass dynamics restricted to M . As a consequence, the reactor SS multiplicity and stability in the threedimensional set X can be completely characterized on the basis of the one-dimensional dynamics of the projection of M into the biomass set Γ b˙ = [r(b) − θ]b, b(0) = b0 , b ∈ Γ = [0, be ] (9) s = sf − b/γ, p = b/α2 , be = α2 /α1 sf . where r(b) = ρ(sf − b/γ, b/α2). (10) ∗ The growth function r attains its maximum r over the compact set M at the biomass value b∗ where r′ (b∗ ) = 0. From the application of standard bifurcation tools, and the definition of production rate P (1), the next proposition follows (where GAS denotes globally asymptotically stable, and U means unstable).

Without restricting the approach, we shall consider the case of biomass measurement (y = b), in the understanding that the proposed methodology can be applied to the cases of substrate and reaction rate measurements.

α1 s α2 f b*

LAS

GAS

Our saturated SF/ OF control problem consists in designing (i) a saturated SF controller that globally and robustly (with robustness in the sense of structural stability) stabilizes in X the steady-state (SS) x ¯ of maximum production rate P, and (ii) a saturated OF controller that recovers, in a practical stability sense (La Salle and Lefschetz, 1961), the behavior of the globally stabilizing saturated SF controller. In particular we are interested in drawing closedloop (CL) stability conditions for the reactor in terms of the controller gains and saturation limits.

U

3. OPEN-LOOP ANALYSIS

0

In this section the reactor open-loop behavior is characterized on the basis of the notion of invariant sets (Seibert, 1969).

GAS

0 θ*

3.1 SS multiplicity, stability and optimal operation From (1) it follows that the static reactor behavior is described by the algebraic equation triplet α2 sf − s b= (sf − s), p = , [α2 ρ(s, p) − θ]b = 0. (5) α1 α1 The first two equations establish that all the reactor SSs lie in the stoichiometric set M (Aris, 1969) (a diagonal line in X) M = {x ∈ X : s ≤ sf , b = α1 /α1 (sf − s), p = (sf − s)/α1 }, (6) and the latter equation characterizes the (possibly multiple) SSs. Motivated by the presence of M , introduce the coordinate change α2 s, n = p + s/α1 , (7) b = b, m = b + α1

LAS

U

θ

θ*

Fig. 1. Open-loop bifurcation map with lower (or upper) bifurcation dilution rate θ∗ (or θ∗ ). Proposition 1. (Proof in Appendix A) (A) Let θ∗ = r∗ and θ∗ = r(0) = ρ(sf , 0). The open-loop bioreactor (9) in M has (see Figure 1): (i) For θ ∈ (0, θ∗ ), two SSs: ¯bh (GAS node), and ¯bw (U saddle) (ii) For θ = θ∗ , transcritical bifurcation and two SSs: ¯bh (GAS node), and ¯bw (U saddle) (iii) For θ ∈ (θ∗ , θ∗ ), three SSs: ¯bh (LAS node), ¯bI (U node), and ¯bw (LAS node) (iv) For θ = θ∗ , saddle-node bifurcation and two SSs: ¯b∗ (U saddle),and ¯bw (LAS node). (v) For θ > θ∗ , one SS: ¯bw (GAS node).

7127


¯ ¯b) has the following (B) The optimal operation point (θ, properties: ¯ is unique and located either in the 2(i) The pair (¯b, θ) SS region (0 < θ¯ ≤ θ∗ ) of the bifurcation diagram, or in the 3-SS region (θ∗ < θ¯ < θ∗ ) of the bifurcation diagram (see Figure 1). (ii) When ¯b is in the 2-SS (or 3-SS) region of the bifurcation diagram, ¯b is GAS (or LAS). 2

θ+ 1.25

4. GLOBALLY STABILIZING SATURATED SF CONTROL Having as point of departure the open-loop characterization of the reactor (1), in this section, based on Seibert’s Reduction Principle (Seibert, 1969), a globally stabilizing saturated SF controller is designed.

(i) x¯ is the only GAS attractor in M . (ii) M is a GAS attractor in X. (iii) M and X are compact invariant sets.

2

4.2 Global Saturated State-Feedback control in M Recall the dynamics in M (9) and enforce the linear error dynamics (11) to obtain the nonlinear SF (12) b˙ =[r(b) − θ]b = −k(b − ¯b) (11) ¯ ⇒ θ =r(b) + k(b − b)/b =: ̟(b), b > 0, (12) and its saturated version  + ̟ ≥ θ+  θ , − (13) θ = ̟s (b) = ̟(b), θ < ̟ < θ+ .  − θ , ̟ ≤< θ− The corresponding SSs of the CL dynamics are characterized by the algebraic equation pair [r(b) − θ]b = 0, θ = ̟s (b). (14) Geometrically speaking (see Figure 2, for the application example), and for two control gain values: k = 3, and 20), these equations represent two curves in the controlbiomass space Γ × Θ. The first equation is simply the graph in Γ × Θ of the open-loop biomass-to-dilution rate (rotated) bifurcation map, and the second equation is the graph of the saturated controller ̟s (b). Equations (14) only have meaningful solutions (i.e. intersections) for sufficiently large gain values, this is, k > k ∗ > 0. (15) ¯ ¯b), with ¯b being The two curves intersect themselves at (θ, LAS, and other intersections may exist. If there is another intersection with b stable (or unstable), b is (or is not) a straneous attractor (Alvarez et al., 1991). Thus, ¯b is the unique GAS attractor in M (without straneous at¯ ¯b), tractors) if and only if there is one intersection at (θ, and either: (i) no more intersections, or (ii) additional intersections with biomass repulsors. This geometric interpretation leads to the following proposition.

θ=η−1(b) θ=ϖ(b)

θ*

k=3

θ− 0.1 0 0

4.1 Reduction-Principle-based geometric solution approach Given the invariance and local attractivity of the stoichiometric set M , the following result is a direct consequence of Seibert’s Reduction Principle (RP) (Seibert, 1969). Proposition 2. (RP-based solvability) The SS x ¯ is GAS in X if for the CL bioreactor (1) with saturated SF controller θ = µs (x):

1

dilution rate

θ*

biomass b

k=20

b

α1s α2 f

Fig. 2. Open-loop bifurcation map η −1 (θ) and saturated SF controller θ = ̟s (b) (13) for two gain values: k = 3 and 20. Proposition 3. (Proof in Appendix B.) ¯ ¯b) denote the optimal dilution rate-biomass pair, Let (θ, and θ∗ (or θ∗ ) denote the lower (or upper) bifurcation value of θ. Then ¯b is the unique robust GAS attractor in M if and only if: (i) The control gain k satisfies (15). (ii) The control limits (θ− and θ+ ) satisfy 0 < θ− < θ¯ < θ+ , θ− , θ+ ∈ / [θ∗ − ǫ∗ , θ∗ + ǫ∗ ]. (16) 4.3 Global Saturated State-Feedback control in X The enforcement of (11) in a neighbourhood Nx ⊂ X of x ¯ yields the local nonlinear SF controller θ = ρ(s, p) + k(b − ¯b)/b =: µ(x), b > 0. (17) The corresponding global saturated SF controller is given by  +  θ if µ(x) ≥ θ+ θ = µs (x) := µ(x) if θ− < µ(x) < θ+ , x ∈ X, (18)  − θ if µ(x) ≤ θ− whose restriction to M yields the saturated SF controller θ = ̟s (b) (13) in M . The reactor dynamics with the saturated SF controller are given by x˙ = fs (x, d), x ∈ X, fs = f (x, µs (x), d). (19) Departing from Propositions 2 and 3, it is sufficient to show that (i) X is an invariant for the CL reactor, and (ii) that M is a global attractor in X, to conclude that x ¯ is globally stable in X. Proposition 4. (Proof in Appendix C) Consider the CL fermentor (1) with the nonlinear saturated SF controller (18). The optimal SS x ¯ is the only robust GAS attractor in the invariant set X if the conditions of Proposition 3 are satisfied. Note that (18) is a passive controller (relative degree one and corresponding GAS zero dynamics). Accordingly, the derived robust stability result can be understood as a

7128


specification of the general connection between passivity, robustness and optimality (Sepulchre et al., 1997). 5. SATURATED OUTPUT-FEEDBACK CONTROL In this section an observer-based dynamic linear OF saturated controller is designed that recovers the behavior of the globally stabilizing SF saturated controller (18), following recent unconstrained control studies (Gonzalez and Alvarez, 2005; Diaz-Salgado et al., 2007; CastellanosSahagun and Alvarez, 2006; ?). 5.1 Output-feedback design

(i) The control gain k and limit pair (θ− , θ+ ) meet the conditions of Proposition 6. (ii) The estimator-controller gain pair (ω, k) is chosen so that ω ∈ [̟− (k, Z), ̟+ (k, Z)]. (28)

Introduce the deviation coordinate change ¯ β = b − ¯b, σ = s − s¯, π = p − p¯, u = θ − θ, ˜ ¯ ¯ d = d − d, a ≈ b > 0, δ = (ρ − θ)b + aθ = ι(σ, β, u), ¯ d¯ , ϕs = f (x, θ, d) − f x¯, θ,

and rewrite the reactor (8) in the form β˙ = −au + δ, δ = ι(σ, β, π, u), ˜ σ˙ = ϕs (σ, β, π, u, d) ˜ π˙ = ϕp (σ, β, π, u, d).

In the absence of modeling and measurement errors the stability of x¯ is asymptotic. 2

ψ=β (20)

In terms of the parameter-function pair (a, δ) the SF controller (18) is written as follows kβ + δ u = sat , δ = ι(σ, β, u). (21) a From the reactor model realization (20) we have that the value δ of ι is instantaneously observable (Hermann and Krener, 1997), and consequently a quickly convergent estimate δˆ of δ can be obtained with the reduced-order observer (Gonzalez and Alvarez, 2005) χ˙ = −ωχ + ω(au − ωψ), δˆ = χ + ωψ. (22) From the combination of (21) with (22) the dynamic linear OF controller is obtained χ˙ = −ωχ + ω(au − ωψ), u = sat {[(k + ω)ψ + χ]/a} . (23) In the absence of saturation, this controller has the PI form Z 1 t k+ω k+ω u=κ ψ+ ,τ= ψ(τ ′ )dτ ′ , κ = − τ 0 a kω (24) with proportional gain κ and reset time τ . 5.2 Closed-loop stability & tuning The application of the OF controller (23) to the bioreactor (1) yields the CL dynamics ˜ δ, ˜ y˜, d, ˜ d), ˜˙ e˙ = φs (e) + φ(e; e∈E (25) ˙˜ ˙ ˜ ˜ ˜ ˜ ˙ δ = −ω δ + g˜(e; δ, y˜, y˜, d, d), δ˜ ∈ ∆, e is the deviated state, δ˜ the estimation, y˜ the measurement, and d˜ the exogeneous input error, where e = x − x¯, δ˜ = δˆ − δ, y˜ = y − b, Z := E × ∆, ¯ φ˜ = f − φs (e), u˜ = u − us , φs (e) = fs (¯ x + e, θ¯ + u, d), ¯ us = θ − µs (¯ x + e), g˜ := δ˙ Introduce the definition of effective estimator convergence rate (Alvarez and Fernandez, 2009) λω = ω − λd (ω, k, Z),

λd = Lgδ˜˜ + Lδϕ˜˜ Lge˜ /µs .

positive real roots (ω − and ω + ) which determine the gain interval [ω − , ω + ] where the CL reactor is practically stable (λω > 0), according to the expressions k > 0, ω ∈ [ω − , ω + ], ω − = ̟− (k, Z) < ω + = ̟+ (k, Z) (27) − + where [ω , ω ] grows with the decrease of k, and decreases with the size of Z. Proposition 5. The optimal SS x¯ of the CL reactor (1) with the linear dynamic OF saturated controller (23) is practically stable in Z ⊂ R4 if:

(26)

For k sufficiently small (Alvarez and Fernandez, 2009), the treshold condition ω = λd (ω, k, Z) has two distinct

From the CL stability assessment associated with Proposition 7, the next tuning guidelines follow, adapted from (Gonzalez and Alvarez, 2005): (0) Set the control limits θ− and θ+ according to Proposition 6. (i) Set k and ω conservatively at k ≈ 1 − 2 and ω ≈ 3 − 5. (ii) Increase ω until the response becomes oscillatory at ω + , and back-off at ω = ω + /(2 − 3). (iii) Increase k untill the response becomes oscillatory at k = k + , and back-off at k = k + /(2 − 3). (iv) If necessary, adjust (k, ω) and (θ− , θ+ ) to improve the behavior. Occasional off-line substrate measurements, that are usually taken for product quality and operation assessment purposes, can be used to adjust the optimal value ¯b of the biomass set point. 6. APPLICATION EXAMPLE In this section, the proposed OF saturated controller is tested for a representative case example with typical parameters recalled from the literature (Henson and Seborg, 1992) (k0 , ks , ki , ps , α1 , α2 ) = (57, 1, 0.01, 5, 1/0.88, 0.5). 6.1 Open-loop behavior The corresponding bifurcation map is illustrated in Figure 1, with lower und upper bifurcation dilution rates θ∗ ≈ 0.2, θ∗ ≈ 1. The corresponding optimal SS-dilution rate ¯ is given by (¯ ¯ = ((0.17, 0.36, 0.73)′, 1). The x, θ) pair (¯ x, θ) optimal SS is close to the upper bifurcation (see Figure 1), and consequently x ¯ is LAS but not robustly stable. In Figure 3 this feature is illustrated for the case that the ¯ and is subjected to a reactor operates initially at (¯ x, θ) +2% step increase in sf at t = 0. The reactor motion x(t) moves towards the undesired washout SS x ¯w . 6.2 CL behavior with saturated State-Feedback controller In Figure 4 is presented the CL reactor response with the saturated SF controller θ = µs (x) (18) with controller gain and saturation limits k = 3, θ− = 0.1, θ+ = 1.25 (29) which satisfy Proposition 4. Initially, the reactor state x0 = (0.4, 0.2, 0.1) ∈ X − is in the vecinity of the optimal

7129


π/ π

π/ π

1 0.5

1 0.5 0 1

0

s

s

1 0.5

0.5 0 1 b

0 1 b

0.5

0 1 p

0.5

θ

0 1 p

0.5

0 1.5 1 0.5 0

0.5 0 1.04

1 1.1 1 0.9 0 10 time t 5 Fig. 4. CL reactor behavior with SF (continuous line) and OF (discontinuous line) under the intial condition and feed concentration perturbations. se

se

1.02

0.98 0

5 10 time t Fig. 3. Open-loop reactor response to a +2% step perturbation in the feed concentration sf , initial condition ¯ x0 = x ¯, and θ = θ. LAS SS x ¯, in the time interval [0, 5] the feed concentration is at its nominal value sf = 1, and at time t = 5 there is a +10% step disturbance in sf . As it can be seen in Figure 4: (i) the optimal SS x ¯ is reached in about 5 residence times, and (ii) after the step input disturbance (at t = 5), the motion x(t) reaches a deviated SS x¯d in the neighbourhood of x ¯. This test verifies that x ¯ is a CL robust GAS SS. 6.3 CL behavior with saturated Output-Feedback controller The CL fermentor response with the linear dynamic OF saturated controller (23) is presented in Figure 4 (discontinuous lines). The fermentor was subjected to the same initial value and feed perturbations as in the test of the SF saturated controller (continuous lines), and additionally set with an about +3% deviated biomass setpoint and measurement noise with 6% amplitude and about 20 times larger frequency than the natural frecuency of the fermentor. The saturation limits (29) were chosen satisfying the conditions of Proposition 4, and the application of the tuning guidelines (Subsection 5.2) yielded the controllerestimator gain pair (k, ω) = (2, 6). The CL response is presented in Figure 4 (discontinuous plots), showing: (i) that the behavior of the exact model based nonlinear SF saturated controller (18) is adequately recovered in a practical stability sense, (ii) that the presence of straneous attractors is effectively ruled out, and (iii) the effectiveness of the anti-windup scheme contained in the observercontroller realization (23). 7. CONCLUSION The problem of regulating the biomass growth rate to its maximal production rate for a class of continuous

fermenters with substrate and product inhibited kinetics, biomass (or substrate, or reaction rate) measurement and manipulation of substrate exchange rate has been addressed. Following a geometric approach and exploiting the presence of a stoichiometric attractive invariant in the state space, a linear dynamic OF saturated controller was designed with: (i) CL practical stability conditions in terms of controller gains and limits, (ii) easy-to-imply tuning guidelines, (iii) reduced model dependency features, and (iii) recovery of the CL behavior attained with a saturated nonlinear passive SF controller. In the absence of saturation, the OF controller is equivalent to a conventional PI controller. The controller performance features were illustrated via a representative case example. ACKNOWLEDGEMENTS Alexander Schaum gratefully acknowledges the support of UAM-Iztapalapa. REFERENCES P. Agrawal, G. Koshy and M. Ramseier, 1989, An algorithm for poperating a fed-batch fermentor at optimum specific-growth rate, Biotech. & Bioeng., 33(1), p. 115125. A.A. Alonso and J.R. Banga, 1998, Design of a class of stabilizing nonlinear state-feedback controllers with bounded inputs, Ind. Eng. Chem. Res., 37, p. 131-144. J. Alvarez, J. Alvarez and R. Suarez, 1991, Nonlinear bounded control for a class of continuous agitated tank reactors, Chem. Eng. Sci., 46 (12), p. 3235-3249.

7130


J. Alvarez and C. Fernandez, 2009, Geometric estimation of nonlinear process systems, J. Proc. Cont., 19 (2), p. 247-260. R. Aris, 1969, Introduction to the Analysis of Chemical Reactors, Prentice-Hall, Englewood Cliffs, N.J.. J.E. Bailey and D.F. Ollis, 1986, Biochemical Engineering Fundamentals, McGraw-Hill. M.A. Barron and R. Aguilar, 1998, Dynamic behavior of a continuous stirred bioreactor under control input saturation, J. Chem. Technol. Biotechnol., 72, p. 15–18. E. Castellanos-Sahagun and J. Alvarez, 2006, Synthesis of two-point linear controllers for binary distillation columns, Chem. Eng. Comm., 193, p. 206-232. J. Diaz-Salgado, A. Schaum, J.A. Moreno and J. Alvarez, 2007, Interlaced estimator-control design for continuous exothermic reactors with non-monotonic kinetics, 8th IFAC DYCOPS, 2007, p. 43-48. N.H. El-Farra and P.D. Christofides, 2003, Bounded robust control of constrained multivariable nonlinear processes, Chem. Eng. Sci., 58, p. 3025-3047. P. Gonzalez and J. Alvarez, 2005, Combined PI-inventory control of solution homopolymerization reactors, Ind. Eng. Chem. Res. J., 44,p. 7147-7163. M.A. Henson and D.E. Seborg, 1992, Nonlinear control strategies for continuous fermentors, Chemical Engineering Science, 47 (4), p. 821-835. R. Hermann and A.J. Krener, 1997, Nonlinear controllability and observability, IEEE Trans. Aut. Cont., 22 (5), p. 728-740. F. Jadot, G. Bastin and F. Viel, 1999, Robust global stabilization of stirred tank reactors by saturated outputfeedback, Eur. J. Cont., 5, p. 361-371. N. Kapoor and P. Daoutidis, 1999, An observer-based anti-windup scheme for nonlinear systems with input constraints, Int. J. Cont., 9, p. 18-29. T.A. Kendi and F.J. Doyle, 1998, Nonlinear internal model control for systems with measured disturbances and input constraints, Ind. Eng. Chem. Res., 37, p. 489-505. M.J. Kurtz and M.A. Henson, 1997, Input-Output linearzing control of constrained nonlinear processes, J. Proc. Cont., 7 (1), p.3-17. J. La Salle and S. Lefschetz, 1961, Stability by Lyapunov’s Direct Method, Academic Press. H.O. Mendez-Acosta, D.U. Campos-Delgado, R. Femat and V. Gonzalez-Alvarez, 2005, A robust feedforward/feedback control for an anaerobic digester, Comp. Chem. Eng., 29, p. 1613-1623. A. Rapaport and J. Harmand, 2002, Robust regulation of a class of partially observed nonlinear continuous bioreactors, J. Proc. Cont., 12, p. 291-302. P. Seibert, 1969, On stability relative to a set and to the whole space, 5th Int. Conf. on Nonlin. Oscillations, 1969, V.2, Inst. Mat. Akad. Nauk USSR, 1970, p. 448–457. R. Sepulchre, M. Jankovic and P. Kokotovic, 1997, Constructive Nonlinear Control, Springer-Verlag, London. T. Yamane, T. Kume, E. Dada and T. Takamatsu, 1977, J. Ferment. Technol., 55, 587. Appendix A. PROOF OF PROPOSITION 1 The washout vector ¯bw is always a solution of (5). Due to the non-monotonicity of r(b) (5) has additionally: one solution for θ ∈ (0, θ∗ ), two bifurcation solutions for θ = θ∗ , two solutions for θ ∈ (θ∗ , θ∗ ), one bifurcation solution for θ = θ∗ , and no solution for θ > θ∗ .

Regarding stability, when θ < θ∗ , fb > 0 ∀ b0 > 0, implying that the washout SS ¯bw is unstable. When θ = θ∗ , fb = 0 for s = sf , meaning that ¯bw is marginaly stable. When θ > θ∗ , fb < 0 about b = 0, implying that ¯bw is LAS. When 0 < θ < θ∗ , b0 > ¯bh (or b0 < ¯bh and close to ¯bh ), implying that fb < 0 (or fb > 0), and consequently, ¯bh is LAS. When θ = θ∗ fb < 0 about ¯b∗ , implying that ¯b∗ is an unstable saddle. When θ∗ < θ < θ∗ , fb > 0 for b0 > ¯bI (or fb < 0 for b0 < ¯bI ), signifying that the intermediate SS ¯bI is an unstable node. This proves the stability assessments stated in Proposition 2. Substituting (5) in P (1), enforce the maximum production rate condition P(b) = r(b)b, P ′ (b) = 0, and conclude ¯ satisfies the that the optimal state-dilution rate pair (¯b, θ) two algebraic equation set r(b) = −b, θ¯ = r(b). (A.1) r′ (b) The growth function r is isotonic (i.e., r′ > 0) for b ∈ [0, b∗ ) and is antitonic (i.e., r′ < 0) for b > b∗ , and the right side of (A.1) is antitonic. Consequently, there is a unique solution b∗ < ¯b < α2 /α1 sf in the antitonic branch of the growth function r. Stability follows from the above assessment. Q.E.D. Appendix B. PROOF OF PROPOSITION 3 ¯ ¯b) is the only solution of (14) for By construction (θ, − + θ ∈ (θ , θ ). Thus it suffices to characterize the solutions of (14) for the case θ = θ− and θ = θ+ . Assume that ¯b is the unique GAS attractor with θ− (or θ+ ) ∈ [θ∗ , θ∗ ], i.e. condition (16) is not necessary. It follows from Proposition 2 that there are three solutions for the equation pair (14), two of them being LAS. This contradicts the assumption that ¯b is the only GAS solution of (14). This proves the necessity of conditions (16). Assume condition (16) holds (i.e. θ− , θ+ ∈ / [θ∗ , θ∗ ]). From Proposition 2 it follows: (i) when θ = θ− , there are two ¯ ¯b) and (θ, ¯ ¯bw ), and since ¯bw is a solutions for (14), (θ, − ¯ repulsor for θ = θ < θ∗ , b is the only GAS solution of (14), and (ii) when θ = θ+ > θ∗ the equations (14) does not have a solution. Thus the sufficiency of conditions (16) follows. Q.E.D. Appendix C. PROOF OF PROPOSITION 4 Define the six-side boundary Ξ of X + − + − + Ξ = Ξ− b ∪ Ξb ∪ Ξp ∪ Ξp ∪ Ξs ∪ Ξs , − + Ξb = {x ∈ X : b = 0}, Ξb = {x ∈ X : b = α2 /α1 sf }, + Ξ− p = {x ∈ X : p = 0}, Ξp = {x ∈ X : p = ps }, − + Ξs = {x ∈ X : s = 0}, Ξs = {x ∈ X : s = sf }. + At Ξ− b , fb = 0. For x0 ∈ Ξb : (i) when θ = µs (x), ¯ x → C = {x ∈ X : b = b}, or (ii) when the controller θ = µs (x) is saturated at θ = θ+ (x ∈ X + ), b˙ = [ρ(s, p) − θ+ ]α2 /α1 sf < 0 because θ+ > ρ(s, p) ∀ (s, p) ∈ [0, sf ] × + [0, sf /α1 ]. For x0 ∈ Ξ− p (or Ξp ), fp > 0 (or fp < 0). For − + x0 ∈ Ξs (or Ξs ), fs > 0 (or fs < 0). Thus the compact set X is invariant under the CL vector field fs (19).

From (8) with θ = µs (x) > θ− the GAS property of M in X is concluded. The result follows from the combination of Proposition 2 with Proposition 3. Q.E.D.

7131