Continuous Sliding Mode Temperature Tracking of a ... - Science Direct

4th IFAC Conference on Modelling and Control in Agriculture, Horticulture and Post Harvest Industry August 27-30, 2013. Espoo, Finland

Continuous Sliding Mode Temperature Tracking of a Solid State Fermentation Reactor for Substrate Production A. Sánchez*, A.G. Loukianov*, O. Aroche*,  *Centro de Investigación y Estudios Avanzados (CINVESTAV), Unidad de Guadalajara, Av. del Bosque 1145, Col. El Bajío, Zapopan 45019, México

Abstract: The Agaricus bisporus “Phase II” composting is a complex Solid State Fermentation (SSF) process of great importance in mushroom production. Because production costs are highly sensitive to this SSF, it is desirable to establish suitable control strategies in order to obtain better mushroom yields in shorter operating times. This work proposes the use of a second order sliding mode controller continuous to track temperature profiles in a simplified 6 ODE’s dynamic model describing the behaviour of relevant system states. The closed-loop system is analyzed and simulation results are presented illustrating the performance of this scheme.

Keywords: solid state fermentation, A. bisporus, optimal temperature tracking control, sliding mode control 

Figueredo et al. (10)] proposed 11th ODE’s dynamic model based on mass and heat balances describing the behaviour of relevant system states. Kinetic and operation parameters were adjusted using parametric sensitivity studies. Experimental temperature trajectories of a pilot plant reactor were used to validate the model and optimal temperature trajectories were calculated using the steepest descent method with a fixed time formulation, aiming at maximizing the thermophilic fungi and actinobacteria production.

1. INTRODUCTION Solid State Fermentations (SSF) are complex processes with growing importance in food, pharma and agro industries for the preparation of added value products [Viccini et al. (03), Mitchel and Von Meien (00), Seki (02)]. The A. bisporus “Phase II” composting is a good example of such processes, in which a suitable substrate is produced for the cultivation of this mushroom. Production yields are highly dependent on this SSF. Therefore it is desirable to establish control strategies capable of obtaining substrates with reproducible “quality” in shorter operating times. “Phase II” is carried out in large fed-batch reactors controlling temperature profiles by manipulating fresh-air inflow and/or recirculation ratio guaranteeing aerobic conditions. These temperature profiles are established, in most cases, empirically with the objective of promoting microbial populations with specific substrate degradation tasks that may favor the growth of the A. bisporus. This is the case of thermophilic fungi, mostly Scytalidiun thermophilae, which degrade cellulose and some actinobacteria acting over lignocellulosic material. Competitors or inhibitors that may diminish the mushroom production must be eliminated. These broad microorganism families along with their respective roles on the mushroom culture have been identified on industrial composts, as well as on experimental set-ups [Kaiser (96)]. Besides the specific microorganisms growth, other complex phenomena take place in a “Phase II” reactor: metabolic heat generation, conductive and radiation heat transfer, heat and mass transfer through evaporation [Barrena et al. (06)], and oxygen, ammonia and carbon dioxide transfer from the reactor bed to the cooling air flow [Seki (02)].

This paper explores the use of continuous sliding mode control schemes to track temperature profiles. The following section discusses the dynamic model employed in this work. The section starts by introducing the model proposed by [Gonzalez-Figueredo et al. (10)]. The dynamic behaviour of the main state variables (i.e. substrate and microorganism concentration, temperature, oxygen and carbon dioxide concentration as well as water content) are well described and temperature trajectory values have been validated experimentally in a 100kg SSF reactor. However, this model presents a pathological numerical behaviour making it not suitable for control analysis purposes. Therefore a simplified version is introduced in this section. In section 3, the sliding mode control scheme is presented. A second order sliding mode controller is proposed to solve the robust temperature tracking problem. Stability of the closed-loop system is analysed. Section 4 shows some simulation results illustrating the performance and robustness of this scheme for the tracking of temperature trajectories. Finally, section 5 discusses research in progress whose objective is to experimentally validate the simplified model and the proposed control strategy. 2. DYNAMIC MODEL

Temperature profiles in “Phase II” commonly reach high enough values (pasteurization stage) to kill all mesophilic microorganisms, leaving only thermophilic fungi and actinobacteria as principal components of the microflora. However, these profiles may not guarantee optimal yields of the A bisporus.[ Sanchez et al. (10)] and [Gonzalez-

978-3-902823-44-1/2013 © IFAC

Details of the experimental rig can be found in [GonzalezFigueredo et al. (10)] in which an 11-ODE’s model was employed as shown in Eqn. (1).

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10.3182/20130828-2-SF-3019.00037

IFAC AGRICONTROL 2013 August 27-30, 2013. Espoo, Finland

Bx 6a Cx 7a Gx 1a2 Ex 6a F Bx 6a Cx 7a Gx 1ax 2a < N2a (x 5a , x 8a , x 9a )  M2a (x 9a ) > x 2a Ex 6a F Bx 6a Cx 7a Gx1ax 2a < N3a (x 5a , x 8a , x 9a )  M3a (x 9a ) > x 3a Ex 6a F Bx 6a Cx 7a Gx1ax 2a < N4a (x 5a , x 8a , x 9a )  M4a (x 9a ) > x 4a Ex 6a F Bx 6a Cx 7a Gx 1ax 5a Ex 6a F

Table 1. State Variables

x1a   x2a  x 3a  x 4a  x5a 

4

  1 ¯ œ ¡  Nia(x 5a , x 8a , x 9a )  Mi (x 9a ) ° x ia H ¡ ° ¢ ± i i 2 x6a 

State Variable

x 1a (m)

(1)

4

4

0.45

x 2a kg m-3)

Actinobacteria concentration

0.1

x 3a (kg m-3)

Thermophilics concentration

0.1

x 4a (kg m-3)

Inhibitors concentration

0.1

x 5a (kg m-3)

Substrate concentration

100.39

x 6a (m3 m-3)

Volumetric solid mass content

0.1175

x 7a (m3 m-3)

Volumetric moisture content

0.3958

Oxygen concentration

0.3

x 9a (°C)

Temperature

28.7

a (kg H20 kg-DA-1) x 10

Air humidity

0.016

a (kg m-3) x 11

Bx 6a Cx 7a 1 x7a  Gx1ax 7a œ < Ei Nia(x 5a , x 8a , x 9a ) > x ia C i 2 Ex 6a F

Initial Value

Compost height

x 8a (g m-3)

  ¯ 1 1¬ Bx 6a Cx 7a Gx 1ax 6a œ ¡ žž 1  ­­­ Nia(x 5a , x 8a , x 9a ) ° x ia B i 2 ¡¢ Ÿ Hi ® °± Ex 6a F

Description

Carbon dioxide concentration

3.8 ·10-4

The open loop temperature profile approximates adequately the experimental profile deviations no bigger than 2 ºC i 2 during the first 150 hours (duration of the industrial process). However equations for calculating water content in the Bx 6a Cx 7a * a ) R(x 9a  x 10 x9a  2 substrate as well as humidity in the void fraction of substrate H Bx 6a Cx 7a Bx 6a as well as in air contains exponential expression making them 4 extremely sensitive. Detailed numerical simulations were œ < 7 i Nia(x 5a , x 8a , x 9a ) >x ia carried out to establish the degree of drifting from steady i 2 state when doing stability calculations (for instance with Bx 6a Cx 7a Matlab). These calculations showed that convergence to 2 H Bx 6a Cx 7a Bx 6a steady state values for the equations of compost (substrate) height, volumetric solid and moisture contents, air humidity * *   ¯ a a a a a  I ¢ (1 Jx10  x10 )K (Lx10F x10  x10 ) ± u and carbon dioxide concentration was unachievable. In order 1 to avoid this numerical pathology, the corresponding * a  a F  x 10  3x10 a ) x10 (Lx10 equations were not considered in the model. Pseudo-steady M state values for these values are used in the restricted model a a a mco  N N2 N3 * a  1 a )u when required [Aroche, 10].  x11 x11 (x1a x 2a x 3a ) (x11 a yo x x5 Therefore, new state variables were defined as follows:  b1 j ­¬ x 1  x 2a , x 2  x 3a , x 3  x 4a , x 4  x 5a ¸ x 5  x 8a , x 6  x 9a ­­ a1 j exp žžž and adding the actuator simplified dynamics u  b1v the žŸ x 9a 273 ­­® x 5a x 8a , system (1) reduces to Nja (x 5a , x 8a , x 9a )   b1 j b2 j ¬­ K S x 5a KO x 8a a1 j ­­ x1  < N1 (x 4 , x 5 , x 6 )  M1(x 6 ) > x1 1 exp žžž a2i Ÿž x 9a 273 ®­­ x2  < N2 (x 4 , x 5 , x 6 )  M2 (x 6 ) > x 2 Mj (x 9a )  adj exp bdj (x 9a 273) , j  2, 3, 4 ; B , C , R , x 3  < N3 (x 4 , x 5 , x 6 )  M3 (x 6 ) > x 3 I , E, F, G, H, K, L, M are constant parameters. 3   1 ¯  (2) x  4 œ ¡¢¡  Hi Ni (x 4 , x 5 , x 6 )  Mi (x 6 ) °±° xi The corresponding state variables are described in Table 1. i 1 The air inflow is employed as the control variable u. The 3 1 model contains a total of 45 parameters classified as x5  œ < Vi Ni (x 4 , x 5 , x 6 )x i > (x 5*  x 5 )u operation and kinetic adjusted to reproduce the optimal M i 1 growth temperature range and maximum death temperatures, x6  f6 (x )  b6u K6 (x , t ) [Gonzalez-Figueredo et al.10]. Operation parameters, such u  b1v as the reactor dimensions, compost and cooling air properties and heat transfer coefficients are based on our experimental where v is the actuator input, M (x )  M a (x a ) , i 6 i 1 9 rig or previous experimental data. Since no reliable methods a a a a Ni (x 4 , x 5 , x 6 )  Ni 1 (x 5 , x 8 , x 9 ) , i  1, 2, 3 ; are available in industry for direct online biomass

x 8a 

4

œ < Vi Ni (x 5a , x 8a , x 9a ) >xia

1 (x a*  x 8a )u M 8

concentration measuring or substrate consumption rates, reactor temperature profiles were used for model validation.

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IFAC AGRICONTROL 2013 August 27-30, 2013. Espoo, Finland

f6 (x ) 

Bx 6a* Cx 7a* H

Bx 6a*

2 Cx 7a*



4

œ < 7 i Nia(x 5a , x 8a , x 9a ) >x ia i 2



Bx 6a*

sliding mode motion on the manifold s0  0 is described by reduced first order system e6  k6e6 . (8) Under condition k6  0 a solution of the system (8) is asymptotically stable, thus * lim x 6 (t )  Tref (t ) t p ts .

a* ) R(x 9a  x10

Bx 6a* Cx 7a*

, 2 H Bx 6a* Cx 7a* Bx 6a*

b6  IK , b1  0 and K6 (x , t ) is an uncertainty term.

t ld

On the other hand, the sliding mode control (6) yields invariant

3. REFERENCE TEMPERATURE PROFILE TRACKING The variable to be controlled is temperature x 6  T . The time-varying

temperature

Calculation of optimal temperature trajectories for this reactor has been studied elsewhere [Sanchez et al. 10]. Defining the tracking error as * (3) e6  x 6  Tref (t )

u  b1v

s0  0 (7)

(4)

* replaced by Tref (t ) yielding

* * x1   ¢ N1 (x 4 , x 5 ,Tref (t ))  M1(Tref (t )) ¯± x1 * * x2   ¢ N2 (x 4 , x 5 ,Tref (t ))  M2 (Tref (t )) ¯± x 2 * * x 3   ¢ N3 (x 4 , x 5 ,Tref (t ))  M3 (Tref (t )) ¯± x 3

To stabilise the system (5) we apply the block control feedback linearization technique combined with sliding mode [Loukianov, 98]. On the first step, calculating the desired value ud of the air flow u of the form ud (x , t )  b61 K6 (x , t )  k6e6 , k6  0

s0  f0 (x , t ) b0v

(5)

v 

sign(s0 ) k3s0 v0

v0  k2sign(s0 ) that yields the closed-loop system of the form e6  k6e6 e0 s0  f0 (x, t )  k1b0 s0

1

2

sign(s0 ) k3s0 v0

3

 

¯

1

x 4 

œ ¡¡¢  Hi Ni (x 4 , x 5,Tref* (t )) Mi (Tref* (t )) °°± xi

x5 

œ  ¢ Vi Ni (x 4 , x 5,Tref* (t ))xi ¯± M (x 5*  x 5 )ueq (x, t )

i 1 3

(9)

1

i 1

* x 6  Tref (t ), s 0  0,

Now we assume that in the admissible region D the following condition satisfies * * Mi (Tref (t))  Ni (x4, x5,Tref (t))  0, x ‰ D, t p ts , i  1,2,3.

where f0 (x , t ) is a continuous function and b0  b1b6 . Therefore, to robustly stabilize the system (5), we propose to apply the quasi continuous SM super-twisting algorithm [Moreno 2008] 1 k1 e6 2

ueq (x , t )  b01 f0 (x , t )

and substituted in the system (3). Then, the variable x 6 is

* where K6 (x , t )  f6 (x ) K6 (x , t )  Tref (t ) .

The sliding variable s 0 is defined as s 0  u  ud (x , t ) . Then, the system (4) becomes e6  k6e6 s 0

,

and (6). The dynamics of I on this invariant subspace is zero dynamics. It can be noted that the proposed direct tracking output feedback (6) can be implemented if the system is minimum phase, that is, the zero dynamics are asymptotically stable. In order to derive the zero dynamics, first, the equivalent control ueq (x, t ) is calculated as a solution for

* trajectoryTref (t ) .

and using the system (2) results in e6  b6u K6 (x , t )

\ x6  Tref* , I ‰ D ‡ R5 ^

I  (x 1,..., x 5 )T in the state space of closed-loop system (2)

control objective is to track of the output signal y  x 6 a reference

subspace

Under this condition, the solutions of the first three timevarying equations of the system (9) represented as (10) xi  ai (t )x i , i  1, 2, 3

(6)

* (t )), i  1,2, 3 , is with ai (t )  Ni (x 4 (t ), x 5 (t ), t )  Mi (Tref

asymptotically tends to zero lim x i (t ), i  1,2, 3 .

(7)

t ld

v0  k2sign(s0 )

This implies in turn that the solutions x 4 (t ) and x 5 (t ) of

Assume now that the term f6 (x , t ) considered in system (7) as a perturbation, is bounded in the admissible region D by f0 (x , t ) b E1 e6 E2 x ‰ D

the zero dynamics (9) converge to a steady state x 4*  0 and x 5*  0.3 where

for constants E1  0 and E2  0 . Then, under the following conditions: k1b0 E2 1 8 E22 3 k1  0 , k2  k1b0 and k3  E1 k1b0  2E2

4 the state vector of the closed-loop system (7) reaches the manifold e6  0 in finite time ts , [Moreno 2008]. The

x 4*

(11)

is the steady state of substrate for the substrate

when all this has been consumed, and x 5* is the steady state for the oxygen at the end of the reaction when microorganisms ceased to grow, implying that the oxygen to the input of the reactor is the same as the output of the same.

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IFAC AGRICONTROL 2013 August 27-30, 2013. Espoo, Finland

4. SIMULATION RESULTS REFERENCES

Simulation experiments were carried out to evaluate the performance of the designed controller and the closed-loop system response. For simulation testing purposes, an optimal temperature trajectory [Sanchez et al. 10] was employed. Since temperature is a very important state variable that is frequently used in practice to predict or track the reactor performance, control robustness to temperature was studied in detail. [Aroche, 10]. Two results are shown in this work in order to illustrate the control performance. Other initial conditions were established based on experimental data or estimations already validated in previous experiments as is the case of micro-organism concentrations within the compost to perform validation of mathematical model of reactor [Gonzalez-Figueredo et al., 10]. A lower bound control

value

( u  0.35 m 3 hour )

is

established

Aroche, O. “Optimal Temperature Control of a SSF Reactor (in Spanish)”. M.Sc, Thesis. Unidad de Ingenieria Avanzada, Centro de Investigación y Estudios Avanzados, 2010. Barrena, R., Canovas C., and Sánchez A., “Prediction of temperature and thermal inertia Effect in the maturation stage and stockpiling of a large composting mass,” Waste Management, vol. 26, 2006, pp. 953-959. Gonzalez-Figueredo, C., L.M. De La Torre, and A. Sánchez, “Dynamic Modelling and Experimental Validation of a Solid State Fermentation Reactor,” Computer Applications In Biotechnology - CAB 2010, Leuven, Belgium: IFAC-ELSEVIER, 2010. Kaiser, J. “Modelling composting as a microbial ecosystem: a simulation approach,” Ecological modelling, vol. 91, 1996, p.25-37. Kirk, D.E. Optimal Control Theory, Dover, 2004. Mitchell. D.A. and O.F. Von Meien, “Mathematical modeling as a tool to investigate the design and operation of the Zymotis packed-bed bioreactor for solid-state fermentation,” Biotechnol Bioeng. Vol. 68, 2000, pp. 127135. Loukianov A. G., (1998), "Nonlinear block control with sliding mode", Automation and Remote Control, vol.59, no. 7, pp. 916-933. Moreno J. A., Osorio M., (2008), "A Lyapunov approach to second-order sliding mode controllers and observers", 47th IEEE Conference on Decision and Control, pp. 28562861. Sánchez A., González-Figueredo, C., Gurubel, De La Torre, J L.M and Labeaga, M. “Optimal Temperature Trajectories in Solid State Fermentation Reactors for Edible Mushroom Growing,” 20th European symposium on computer Aided process Engineering – ESCAPE20, S. Pierucci and G. BUZZI Ferraris, eds., Naples, Italy: ELSEVIER B. V.,2010. Seki, H. “A New deterministic model for forced-aeration compo sting processes with batch operation,” Transactions of the ASAE,vol. 45, 2002, pp. 1239-1250. Viccini, G., Mitchell, D.A. and Krieger, N. “A Model for Converting Solid State Fermentation Growth Profiles Between Absolute and Relative Measurement Bases” Food Technology and Biotehchnology, vol. 41, 2003, pp.191-201. Wiegant, W.M. “Growth Characteristics of the Thermophilic Fungus Scytalidium Thermophilum in relation to production of mushroom compost.” Applied and Environmental Microbiology, vol. 58, 1992, pp. 1301-1307

to

guarantee that the reactor will always have a minimum oxygen supply. This value was determined experimentally. * Fig. 1 shows the system response for Tref (0)  27o C with an

initial condition x 6 (0)  25oC , thus forcing the controller to diminish the air flow for the reactor to heat up, thus compromising the microorganism populations, As it can be seen in Figure 1.e, the tracking error is initially negative, so the control signal tries to increase the temperature of the reactor by limiting the air entry. The evolution of the trajectory takes nearly 50 hours to reach a zero error. Substrate is nearly depleted after 150 hours and the controller successfully maintains the temperature tracking with a negligible error. Final thermophilic concentration values are larger than actinomicetes and inhibitor concentrations, as expected [Figueredo et al. 10]. Fig. 2 presents a less astringent case with an initial reactor temperature larger than expected ( x 6 (0)  32oC ). Fig. 2.e shows that the tracking error reaches nearly a zero value in approximately 8 hours, which represents a time of convergence of approximately four times faster compared to Fig. 1.e. However the final value of inhibitor concentration is larger than in the previous case.

5. CONCLUSIONS The SSF model presented offers good approximations of the experimental temperature profiles observed in a pilot-plant reactor, and can also be used to determine optimal temperature profiles that promote the growth of thermophilic fungi and actinobacteria essential for the A. bisporus cultivation process. The control scheme proposed in this paper seems to be sufficient to guarantee a robust performance of the reactor. However, the tradeoff between desired microorganism concentration predicted by the optimal calculation and the controller performance must be studied further before proceeding to the implementation of this scheme in the experimental reactor.

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IFAC AGRICONTROL 2013 August 27-30, 2013. Espoo, Finland

Figure 1. Simulation results for x 6 (0)  25oC

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IFAC AGRICONTROL 2013 August 27-30, 2013. Espoo, Finland

Figure 2. Simulation results for x 6 (0)  32oC

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