Saturated output-feedback nonlinear control of a 3-continuous exothermic reactor train

Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd on Proceedings, 2nd IFAC IFAC Conference Conference on Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Available online at www.sciencedirect.com Modelling, Identification and Control of Nonlinear Systems Modelling, Identification and Control of Nonlinear Systems Proceedings, 2nd IFAC Conference on Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018

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IFAC PapersOnLine 51-13 (2018) 425–430

Saturated Saturated output-feedback output-feedback nonlinear nonlinear Saturated output-feedback nonlinear control of a 3-continuous exothermic control of a 3-continuous exothermic Saturated output-feedback nonlinear control of a reactor 3-continuous exothermic train train control of a reactor 3-continuous reactor train exothermic ∗ reactor train Luis A. Contreras Hugo A. Franco-de los Reyes ∗ Luis A. Contreras ∗∗ Hugo A. Franco-de los Reyes ∗∗

∗∗ ´ A. ∗ Hugo Luis A. A. Contreras ContrerasJes´ A. Franco-de Franco-de los Reyes Reyes ∗ us Alvarez Luis Hugo los ∗∗ ´ Jes´ u s Alvarez ∗∗ ´ ∗ ∗∗ Jes´ uss Alvarez Alvarez ´ A. Franco-de Luis A. ContrerasJes´ Hugo los Reyes ∗ u ∗ ∗∗ ´ de M´exico, 04510 Mexico City, ounoma Jes´ s Alvarez ∗ Universidad Nacional Aut´ o noma de M´eexico, 04510 Mexico City, ∗ Universidad Nacional Aut´ ∗ Universidad Nacional Aut´ o noma de Mexico (e-mail: {LContrerasA,HFrancoR}@iingen.unam.mx). Universidad Nacional Aut´ o noma de M´ M´exico, xico, 04510 04510 Mexico Mexico City, City, (e-mail: {LContrerasA,HFrancoR}@iingen.unam.mx). ∗∗∗ Mexico Mexico (e-mail: Aut´ o{LContrerasA,HFrancoR}@iingen.unam.mx). nomaAut´ Metropolitana, 09340 MexicoMexico City, Mexico ∗∗ Universidad Mexico (e-mail: {LContrerasA,HFrancoR}@iingen.unam.mx). Universidad Nacional o noma de M´ e xico, 04510 City, Universidad Aut´ o noma Metropolitana, 09340 Mexico City, Mexico ∗∗ ∗∗ Universidad Aut´ o noma 09340 [email protected]). Universidad Aut´ o(e-mail: noma Metropolitana, Metropolitana, 09340 Mexico Mexico City, City, Mexico Mexico Mexico (e-mail: {LContrerasA,HFrancoR}@iingen.unam.mx). (e-mail: [email protected]). ∗∗ (e-mail: [email protected]). [email protected]). Universidad Aut´ o(e-mail: noma Metropolitana, 09340 Mexico City, Mexico (e-mail: [email protected]). Abstract: The problem of controlling with saturated output-feedback (OF) temperature control Abstract: The with saturated saturated output-feedback output-feedback (OF) (OF) temperature temperature control control Abstract: The problem problem of of controlling controlling with with aAbstract: six-state (concentration-temperature) bistable trainoutput-feedback of three continuous tank reactors The problem of controlling saturated (OF)stirred temperature control aa six-state (concentration-temperature) bistable train of three continuous stirred tank reactors bistable train of three stirred tank reactors with recycle(concentration-temperature) is addressed. closed-loop system mustoutput-feedback operated about (OF) the unstable steady-state a six-state six-state (concentration-temperature) bistable train of three continuous continuous stirred tank reactors Abstract: The problem ofThe controlling with saturated temperature control with recycle is addressed. The closed-loop system must operated about the unstable steady-state with recycle is addressed. The closed-loop system must operated about the unstable steady-state (SS) by manipulating the coolant temperature on the basis of one of the tank temperatures. with recycle is addressed. The closed-loop system must operated about the unstable steady-state a six-state (concentration-temperature) bistable train of three continuous stirred tank reactors (SS) by manipulating the coolant temperature on the basis of one of the tank temperatures. (SS) by the coolant temperature the of one of the tank temperatures. The is to attain robust closed-loop stability with abasis control scheme as simple as possible. (SS) aim by manipulating manipulating theThe coolant temperature on theoperated basis of about one ofthe the tank temperatures. with recycle is addressed. closed-loop systemon must unstable steady-state The aim is to attain robust closed-loop stability with a control scheme as simple as possible. aim is to to attain attain robust closed-loop stability with abasis control scheme as simple as behavior possible. The constructive combination of advanced and on industrial control ideas yields atemperatures. the The aim is robust closed-loop stability with a control scheme as simple as possible. (SS) by manipulating the coolant temperature the of one of the tank The constructive combination of advanced and industrial control ideas yields a the behavior The constructive combination of advanced and industrial control ideas yields aa the behavior of a aim passive nonlinear state-feedback (NLSF) stabilizing controller can as be recovered with an constructive combination of advanced and industrial control ideas yields the behavior The is to attain robust closed-loop stability with a control scheme simple as possible. of aa passive nonlinear state-feedback (NLSF) stabilizing controller can be recovered with an of passive nonlinear state-feedback (NLSF) stabilizing controller can be recovered with industrial-type proportional-integral (PI) control with anti-windup protection plus criteria to of a passive nonlinear state-feedback (NLSF) stabilizing controller can be recovered with an an The constructive combination of advanced and industrial control ideas yields a the behavior industrial-type proportional-integral (PI) control with anti-windup protection plus criteria to industrial-type proportional-integral (PI) control with anti-windup protection plus criteria to choose the sensor location as well as control gains and limits. industrial-type proportional-integral (PI) control with anti-windup plus criteria to of a passive nonlinear state-feedback (NLSF) stabilizing controller protection can be recovered with an choose the sensor location as well as control gains and limits. choose the the sensor sensor location as as well well as as control control gains with and limits. limits. choose location gains and industrial-type proportional-integral (PI) control anti-windup protection plus criteria to © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: reactors, cstr,as passive control, saturated choose the cascaded sensor location as well control gains and limits.control, sensor location Keywords: cascaded reactors, cstr, passive control, saturated control, sensor location Keywords: Keywords: cascaded cascaded reactors, reactors, cstr, cstr, passive passive control, control, saturated saturated control, control, sensor sensor location location Keywords: cascaded reactors, cstr, passive control, saturated control, sensorislocation 1. INTRODUCTION testing. When the reactor open-loop (OL) bistable and 1. INTRODUCTION INTRODUCTION testing. When the reactor is open-loop (OL) bistable and 1. testing. When the reactor is open-loop bistable must operate about the unstable SS, the(OL) control gains and 1. INTRODUCTION testing. When the reactor is open-loop (OL) bistable and must operate about the unstable SS, the control gains and must operate the unstable the control gains and must beabout carefully chosen toSS, preclude induction ofand an An important class1.ofINTRODUCTION materials in the chemical and petro- limits must operate about the unstable SS, the control gains testing. When the reactor is open-loop (OL) bistable limits must be carefully chosen to preclude induction of an An important class of materials in the chemical and petrolimits must be carefully chosen to preclude induction of an unproductive (extinction) or catastrophic (ignition) stable An important class of materials in the chemical and petrochemical industries are produced through exothermic relimits must be carefully chosen to preclude induction of an must operate about the unstable SS, the control gains and An important class of materials in through the chemical and petrounproductive (extinction) or catastrophic (ignition) stable chemical industries are produced exothermic reunproductive (extinction) or catastrophic (ignition) stable extraneous SS [Alvarez et al., 1991, Chen and Chang, chemical industries are produced through exothermic reactions in continuous tubular reactors and trains of conunproductive (extinction) or catastrophic (ignition) stable limits must be carefully chosen to preclude induction of an chemical are produced re- extraneous SS [Alvarez et al., 1991, Chen and Chang, An important class of materials in through the chemical andofpetroactions inindustries continuous tubular reactors andexothermic trains conextraneous [Alvarez al., 1991, Chen and Chang, 1985]. The SS reactor trainet control problem has been adactions in continuous tubular and trains of continuous tank reactors with heatreactors removal system [Froment extraneous SS [Alvarez et al., 1991, Chen and Chang, unproductive (extinction) or catastrophic (ignition) stable actions in continuous tubular reactors and trains of conchemical industries are produced through exothermic reThe reactor train control problem has been adtinuous tank tank reactors reactors with with heat heat removal removal system system [Froment [Froment 1985]. 1985]. reactor train problem has adwithSS model predictive [Rivotti etChen al., 2012], and tinuous et al., 2011]. In some with cases,heat these reactors have complex 1985]. The The reactor trainetcontrol control problem has been adextraneous [Alvarez al., 1991, andbeen Chang, tinuous tank reactors removal system [Froment actions in continuous tubular reactors and trains of con- dressed dressed with model predictive [Rivotti et al., 2012], and adet al., 2011]. In some cases, these reactors have complex dressed with model predictive [Rivotti et al., 2012], and advanced [Tetiker et al., 2008] control techniques. However, et al., 2011]. In some cases, these reactors have complex nonlinear dynamics, including steady-state (SS) multiplicdressed with model predictive [Rivotti et al., 2012], and ad1985]. The reactor train control problem has been et al., 2011]. In some cases, these reactors have complex tinuous tank reactors with heat removal system [Froment vanced [Tetiker et al., 2008] control techniques. However, nonlinear dynamics, dynamics, including including steady-state steady-state (SS) (SS) multiplicmultiplic- vanced [Tetiker et al., 2008] control techniques. However, industrial practitioners regard advanced techniques with nonlinear ity, limit cycling, and bifurcation phenomena [Aris and vanced [Tetiker et al., 2008] control techniques. However, dressed with model predictive [Rivotti et al., 2012], and adnonlinear dynamics, including steady-state (SS) multiplicet al., 2011]. In some these reactors have[Aris complex practitioners regard advanced techniques with ity, limit cycling, and cases, bifurcation phenomena and industrial industrial practitioners regard advanced techniques with skepticism because of 2008] concerns on reliability, complexity ity, limit cycling, and bifurcation phenomena [Aris and Amundson, 1958, Uppal et al.,steady-state 1974, Varma, 1980]. The industrial practitioners regard advanced techniques with vanced [Tetiker et al., control techniques. However, ity, limit cycling, and bifurcation phenomena [Aris and nonlinear dynamics, including (SS) multiplicbecause of concerns on reliability, complexity Amundson, 1958, 1958, Uppal Uppal et et al., al., 1974, 1974, Varma, Varma, 1980]. 1980]. The The skepticism skepticism because of on complexity costs. These considerations the present study Amundson, safe process-equipment design closed-loop operation skepticism because of concerns concerns on reliability, reliability, complexity industrial practitioners regard motivate advanced techniques with Amundson, 1958, Uppal et al.,and 1974, Varma, 1980]. The ity, limit cycling, and bifurcation phenomena [Aris and and and costs. These considerations motivate the present study safe process-equipment design and closed-loop operation and costs. These considerations motivate the present study on the saturated (OF) control of reactor trains, in the safe process-equipment design and closed-loop operation with manipulation of heat removal rate are issues of central and costs. These considerations motivate the present study skepticism because of concerns on reliability, complexity safe process-equipment design closed-loop operation Amundson, 1958, Uppal et al.,and 1974, Varma, 1980]. The on the saturated (OF) control of reactor trains, in the with manipulation of heat removal rate are issues of central on the saturated (OF) control of reactor trains, in the understanding that the approach can be extended to with manipulation of heat removal rate are issues of central importance. Tubular reactors are modeled by nonlinear on the saturated (OF) control of reactor trains, in the and costs. These considerations motivate the present study with manipulation of heat removal rate are issues of central safe process-equipment design are and modeled closed-loop operation understanding that the approach can be extended to the importance. Tubular reactors by nonlinear understanding that the approach can be extended to the case of N -unit trains and tubular reactor cases. importance. Tubular reactors are modeled by nonlinear partial differential equations (PDEs) and reactor trains by on understanding that the approach can be extended to the saturated (OF) control of reactor trains, in the importance. Tubular reactors are modeled by nonlinear with manipulation of heat removal rate are issues of central case of N -unit trains and tubular reactor cases. partial differential equations (PDEs) and reactor trains by case of N -unit trains and tubular reactor cases. partial differential equations (PDEs) and reactor trains by staged ordinary differential equations (ODEs). According case of N -unit the trains andapproach tubular reactor cases. that the can be aextended to OF the partial ordinary differential equations (PDEs) and reactor trains by understanding importance. Tubular reactors are modeled by According nonlinear In this study, problem of designing saturated staged differential equations (ODEs). In this study, the problem of designing saturated OF staged ordinary differential equations (ODEs). According to the cell modeling approach [Deansand andreactor Lapidus, 1960] of N -unitcontrol trains and tubular reactor aacases. staged ordinary differential equations (ODEs). According partial differential equations (PDEs) trains by case In this study, the problem of designing saturated OF temperature to robustly stabilize an OL bistable to the cell modeling approach [Deans and Lapidus, 1960] In this study,control the problem of designing aansaturated OF to robustly stabilize OL bistable to the modeling approach [Deans and Lapidus, 1960] tubular reactors can be described with(ODEs). a low-order staged to the cell cell modeling approach [Deans Lapidus, 1960] temperature staged ordinary differential equations According temperature control to robustly stabilize an OL bistable three-tank (six-state) exothermic reactor train is OF adtubular reactors can be be described withand a low-order low-order staged temperature control to robustly stabilize an OL bistable In this study, the problem of designing a saturated three-tank (six-state) exothermic reactor train is tubular reactors can described with a staged ODE tubular reactors can be described withand a low-order to themodel. cell modeling approach [Deans Lapidus,staged 1960] temperature three-tank (six-state) exothermic reactor train is adaddressed. The(six-state) aim is toto obtain a robust stabilizing controller ODE model. three-tank exothermic reactor train is adcontrol robustly stabilize an OL bistable The aim is to obtain aa robust stabilizing controller ODE ODE model. model. tubular can be exothermic described with a low-order staged dressed. dressed. aim is to obtain robust stabilizing controller as simpleThe as (six-state) possible with sensor location and train control limit Most of reactors the industrial reactors are operated dressed. The aim is to obtain a robust stabilizing controller three-tank exothermic reactor is adsimple as possible with sensor location and control limit Most of the the industrial industrial exothermic exothermic reactors reactors are are operated operated as ODE proportional-integral model. as simple as possible with sensor location and control limit criteria. Most of with (PI) temperature control with selection as simple as possible with sensor location and control limit dressed. The aim is to obtain a robust stabilizing controller Most of the industrial exothermic reactors are operated criteria. with proportional-integral proportional-integral (PI) (PI) temperature temperature control control with with selection with anti-windup protection the handling ofoperated saturaselection criteria. as simple criteria. as possible with sensor location and control limit with proportional-integral (PI)for temperature control with selection Most of the (AW) industrial exothermic reactors are anti-windup (AW) protection for the handling of satura˚ anti-windup protection for the handling of saturation [proportional-integral A str¨ om(AW) and H¨ agglund, 2006, Shinskey, 1990]. The selection criteria. anti-windup (AW) protection for the handling of saturawith (PI) temperature control with ˚ 2. CONTROL PROBLEM tion [[˚ Astr¨ str¨ m and and H¨ H¨ gglund, 2006, 2006, Shinskey, Shinskey, 1990]. 1990]. The The tion ooom aaagglund, 2. CONTROL PROBLEM temperature measurement placed at the most sensi˚ anti-windup protection for theShinskey, handling of saturation [A Astr¨ m(AW) and H¨ gglund,is 2006, 1990]. The 2. temperature measurement is placed at the most sensi2. CONTROL CONTROL PROBLEM PROBLEM temperature measurement is placed at the most sensitive location before the hot spot of a tubular reactor, ˚ temperature measurement is placed at the most sensition [ A str¨ o m and H¨ a gglund, 2006, Shinskey, 1990]. The tive location before the hot spot of a tubular reactor, 2.1 Three-continuous exothermic reactor train 2. CONTROL PROBLEM tive location before the hot spot of a tubular reactor, or at the most sensitive tank reactor in a tank reactor Three-continuous exothermic reactor train tiveatlocation before the hot of at aintubular temperature measurement is spot placed the mostreactor, sensi- 2.1 or the most most sensitive tank reactor a tank tank reactor 2.1 Three-continuous exothermic reactor 2.1 Three-continuous exothermic reactor train train or at the sensitive tank reactor in a reactor train [Amrehn, 1977]. With respect to advanced industrial or at the most sensitive tank reactor in a tank reactor tive location before the hot spot of a tubular reactor, train [Amrehn, [Amrehn, 1977]. 1977]. With With respect respect to to advanced advanced industrial industrial 2.1 Consider the train of three continuous stirred-tank reacThree-continuous exothermic reactor train train (mostly model predictive) controllers, the advantages of train Withtank respect to advanced industrial or at [Amrehn, themodel most 1977]. sensitive reactor in aadvantages tank reactor the train of three continuous stirred-tank reac(mostly predictive) controllers, the of Consider Consider the train of three continuous stirred-tank reactors (CSTRs) (depicted in Fig. 1) with and recycle stream. (mostly model predictive) controllers, the advantages of PI control are 1977]. simplicity, reliability and low cost. The Consider the train of three continuous stirred-tank reac(mostly model predictive) controllers, the advantages of train [Amrehn, With respect to advanced industrial tors (CSTRs) (depicted in Fig. 1) with and recycle stream. PI control are simplicity, reliability and low cost. The tors (CSTRs) (depicted in Fig. 1) with and recycle stream. The feed stream is converted into product through of a PI control are simplicity, reliability and low cost. The drawbacks of PI control are that its implementation, tuntors (CSTRs) (depicted in Fig. 1) with and recycle stream. Consider the train of three continuous stirred-tank reacPI control are simplicity, reliability and low cost. The (mostly model predictive) controllers, the advantages of The feed stream is converted into product through of drawbacks of of PI PI control control are are that that its its implementation, implementation, tuntun- first-order The feed stream is converted into product through of a a exothermic reaction with Arrhenius temperadrawbacks ing and maintenance requires per process experience and The feed stream is converted into product through of a tors (CSTRs) (depicted in Fig. 1) with and recycle stream. drawbacks of PI control are that its implementation, tunPI control are simplicity, reliability and low cost. The first-order exothermic reaction with Arrhenius temperaing and and maintenance maintenance requires requires per per process process experience experience and and first-order exothermic reaction with Arrhenius temperature dependency. The corresponding nonlinear dynamics, ing first-order exothermic reaction with Arrhenius temperaThe feed stream is converted into product through of a ing and maintenance requires peritsprocess experience tunand ture dependency. The corresponding nonlinear dynamics, drawbacks of PI control are that implementation, 1 Hugo A. Franco-de los Reyes thanks the CONACyT under scholture dependency. The corresponding nonlinear dynamics, in dimensionless form, of the three-tank (3T) reactor are ture dependency. The corresponding nonlinear dynamics, first-order exothermic reaction with Arrhenius temperaingHugo andA.maintenance requires per process experience and 1 in dimensionless form, of the three-tank (3T) reactor are Franco-de los Reyes thanks the CONACyT under schol1 in form, of (3T) are arship CVU No. 598211. given by the set of six corresponding (ODEs) Hugo A. los 1 in dimensionless dimensionless form, of the the three-tank three-tank (3T) reactor reactor are ture dependency. The nonlinear dynamics, HugoCVU A. Franco-de Franco-de los Reyes Reyes thanks thanks the the CONACyT CONACyT under under scholscholarship No. 598211. given by the set of six (ODEs) arship CVU CVU No. No. 598211. 598211. given by the the set set of of six (ODEs) (ODEs) 1 arship given by six in dimensionless form, of the three-tank (3T) reactor are Hugo A. Franco-de los Reyes thanks the CONACyT under scholarship CVU No. 598211. given by the set of six (ODEs) Proceedings, 2nd IFAC Conference on 425 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Proceedings, 2nd IFAC Conference on 425 Modelling, and Control of Nonlinear Proceedings, 2nd Conference on 425 Peer reviewIdentification under responsibility of International Federation of Automatic Proceedings, 2nd IFAC IFAC Conference on 425 Control. Modelling, Identification and Control of Nonlinear Systems Modelling, Identification and Control of Nonlinear 10.1016/j.ifacol.2018.07.316 Modelling, Identification and Controlon of Nonlinear Proceedings, 2nd IFAC Conference 425 Systems Guadalajara, Mexico, June 20-22, 2018 Systems Systems Modelling, Identification and Control of Nonlinear Guadalajara, Mexico, June 20-22, 2018

2018 IFAC MICNON 426 Guadalajara, Mexico, June 20-22, 2018 Luis A. Contreras et al. / IFAC PapersOnLine 51-13 (2018) 425–430

State c¯1 τ¯1 c¯2 τ¯2 c¯3 τ¯3

¯E x 0.986 1.851 0.983 1.778 0.982 1.731

¯U x 0.415 2.228 0.271 2.183 0.227 2.102

¯I x 0.088 2.461 0.034 2.297 0.025 2.167

Table 1. Open-loop SSs of the model (1) Fig. 1. Reactor train

¯ I ) is the extinction (or ignition) stable SS, ¯ E (or x where x ¯ U is the unstable saddle SS. and x

= θce − λc (τ1 )c1 + ϑb c2 , c1 (0) = c1o = θτe − λτ τ1 + c1 α(τ1 ) + ϑb τ2 + υτc , τ1 (0) = τ1o = ϑc1 − [λc (τ1 ) + ϑb ] c2 + ϑb c3 , c2 (0) = c2o = ϑτ1 − (λτ + ϑb ) τ2 + c2 α(τ2 ) + ϑb τ3 + υτc , τ2 (0) = τ2o c˙3 = ϑc2 − λc (τ3 )c3 , c3 (0) = c3o τ˙3 = ϑτ2 − λτ τ3 + c3 α(τ3 ) + υτc , τ3 (0) = τ3o y = τm , m ∈ {1, 2, 3} := M where ϑ = (1 + )θ, ϑb = θ, λτ = ϑ + υ c˙1 τ˙1 c˙2 τ˙2

(1a) (1b) (1c) (1d) (1e) (1f) (1g)

λc (τi ) = ϑ + α(τi ), α(τ ) = ear −ε/τ ci (or τi ) is the i-th reactant concentration (or temperature), ce (or τe ) is the inlet unmeasured (or measured) concentration (or temperature), τc is the coolant adjustable temperature, θ is the dimensionless measured volumetric flow rate,  is the recycle fraction, υ is the Stanton heat transfer number, ar is the frequency-Damk¨ ohler number, and ε is the activation energy. From mass-energy conservation, the concentrations and temperatures are bounded as 0≤(c1 +c2 +c3 )/3≤c+ =ce , ci , τi > 0 (2a) − min(τe , τc )=τ ≤(τ1 +τ2 +τ3 )/3≤τ + =max(τe , τc )+τa (2b) where τa = 1 is the adiabatic temperature rise. In vector form, the 3T reactor dynamics (1) are given by x˙ = f (x, d) + bu, x(0) = xo , (3a) + y = cy x, x ∈ X, u ∈ U = [u− , u ], (3b) where x = (xT1 , xT2 , xT3 )T , xi = (ci , τi )T , i = 1, 2, 3 f = (f T1 , xT2 , xT3 )T ,

f i = (fic , fiτ )T

b=υ(0, 1, 0, 1, 0, 1)T , u=τc , cy x=τm , m ∈ M = {1, 2, 3}  N   X = x ∈ R6 | 0≤(c1 +c2 +c3 )/3≤c+ , τ − ≤ τi /3≤τ + i=1 τm is the temperature measurement at the reactor, u is the control input in the set U , and X is the bounded state set where the OL system motions x(t) = τ i [t, xo , d(·), u(·)] ∈ X, xo ∈ Xo , t ∈ [0, ∞) evolve. Due to mass-heat conservation, the set X is invariant (each motion born in X stays in X). For given ¯ u nominal input-pair (d, ¯) the corresponding statics, with ns steady-states, are: ¯ + b¯ u = 0, y¯i = cy xi , u = 1, . . . , ns (5) f (¯ xi , d) A bistable reactor train with parameters θ = 3, ρ = 0.6, ce = υ = 1, τe = 2, α = 23.719, ε = 50

and SS set (listed in Table 1) ¯ U , x¯I } , x ¯ U := x ¯, σ(¯ u) = {¯ xE , x

¯ y¯ = cy = x

(6)

(7) 426

2.2 Statement of the problem Our problem consists of designing the function pair (fc , hc ) of the saturated nonlinear dynamic OF controller x˙ c = f c (xc , u, m, kc ), u = hc (xc , y, m, u− , u+ , kc ) (8) to robustly (in a practical sense [La Salle and Lefschetz, ¯ =x ¯U 2012]) stabilize the prescribed (OL unstable) SS x (Table 1). The controller has as adjustable parameters: (i) the sensor location m ∈ {1, 2, 3}, (ii) the vector gain kc , and (iii) the limit pair (u− , u+ ). The controller must be as simple as possible in terms of dimensionality, nonlinearity and coupling. 3. SATURATED NLSF CONTROL Here, the saturated robust nonlinear state feedback (NLSF) stabilizing control problem is addressed on the basis of passivity. 3.1 Zero dynamics Consider that the temperature sensor is in the first tank (9a) and the control is in the bounded subset Uz (to be specified): y¯ = τ¯1 ; u ∈ Uz ⊂ R (9a-b) y = τ1 , where y¯ is the nominal value of y determined by the nominal OL unstable SS x ¯ (7).The enforcement of (9a-b) on the OL dynamics (1) yields the 5-dimensional dynamical inverse [Hirschorn, 1979] c˙1 = θce − λc (¯ τ1 )c1 + ϑb c2 , c1 (0) = c1o (10a) (10b) c˙2 = ϑc1 − [λc (τ1 ) + ϑb ] c2 + ϑb c3 , c2 (0) = c2o (10c) τ˙2 = ϑ1 τ¯1 − (λτ + ϑb ) τ2 + c2 τ2 + ϑb τ3 + υu c˙3 = ϑc2 − λc (τ3 )c3 , c3 (0) = c3o , τ2 (0) = τ2o (10d) (10e) τ˙3 = ϑτ2 − λτ τ3 + c3 α(τ3 ) + υu, τ3 (0) = τ3o u = [λτ y¯−θτe −c1 α(¯ y )−ϑb τ2 ]/υ :=µz (¯ y , xz , d) ∈ Uz (10f) where xz = (c1 , c2 , τ2 , c3 , τ3 ) ,

+ Uz = [u− y , Xz , d), z , uz , ] = µz (¯

Xz = {x ∈ X|τ1 = y¯}

(10a-e) is the saturated version of the standard zero dynamics (ZD) [Isidori, 2013], and (10f) is the associated NLSF saturated controller. In vector form and for general sensor location, the dynamical inverse (Eq. (10) with y = τm ) becomes x˙ z = f z (xz , u, d, m), xz (0) = xzo , y¯ = τ¯m (11a) u = µz (xz , d, m)∈Uz , xz ∈Xz ={x∈X|τm =y¯} ⊂ X (11b) m ∈ M, rd(u, ym )=1, fz (¯ xz , u, d, m)=0 (11c) where rd(u, ym ) = 1 means that, because of the reactorjacket wall diathermicity (υ > 0), the reactor train system

2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018 Luis A. Contreras et al. / IFAC PapersOnLine 51-13 (2018) 425–430

States c¯1 τ¯1 c¯2 τ¯2 c¯3 τ¯3

ZD1 SS1 0.415 2.228 0.271 2.183 0.227 2.102

SS1 0.746 2.012 0.458 2.183 0.267 2.224

ZD2 SS2 0.415 2.228 0.271 2.183 0.227 2.102

SS3 0.136 2.403 0.090 2.183 0.079 2.075

SS1 0.939 1.894 0.872 1.900 0.730 2.102

ZD3 SS2 0.415 2.228 0.271 2.183 0.227 2.102

427

SS3 0.095 2.451 0.042 2.275 0.035 2.102

Table 2. Steady-states (SS) of the zerodynamics (ZD) (10) has relative degree equal to one for any sensor location m ∈ M.

Definition. The reactor train (1), with temperature measurement at the m-th reactor, is robustly passive if the ZD (11) have the prescribed SS x ¯z (associated with the OL SS ¯ ) as unique-robust asymptotic attractor in Xz . x The application of numerical continuation [Dhooge et al., 2003] to the reactor example (eqpara) yields the dependency of the SS set on the sensor location m listed in Table 3, meaning that the ZD (11) are robustly monostable only if the sensor is in the first reactor. Otherwise, when the sensor is in reactors 2 and 3, the ZD are bistable, with the prescribed nominal SS as unstable SS accompanied by two undesired stable SSs. Consequently, the 3T reactor train is passive, or equivalently, the NLSF control problem is solvable only when the temperature of the first reactor is the regulated output. This is in agreement with industrial practice [Amrehn, 1977]. 3.2 Controller The enforcement of the prescribed linear output regulation dynamics, with bounded control set U , y˙ = −k(y − y¯), k > 0, y = τ1 , u ∈ U ⊆ Uz (12) on the OL system (3), with sensor at the first reactor, yields the NLSF saturated controller u = µs (x, d) ∈ U = [u− , u+ ] (13a)  + + if µ(x, d) > u u µs = sat[µ(x, d)] = µ(x, d) if u− ≤µ(x, d)≤u+ (13b)  u− if µ(x, d) < u− y , xz , d) (13c) µ(x, d) = −(kυ)(y − y¯) + µz (¯ and the application of this controller to the reactor train (6) yields the closed-loop dynamics x˙ =f (x, d)+bµs (x, d) := f c (x, d), (14a) y=cy x, x(0)=x0 , x∈X, u∈U (14b) with statics (15) f (x, d) + bµs (x, d) = 0 which in staged form are written as f 1 (x, d)+bu µs (x, d)=0, f 2 (x, d)+bu µs (x, d)=0 (16a-b) (16c-d) f 3 (x, d)+bu µs (x, d)=0, µs (x, d) = u In teared form the staged CL statics (16) are written as: m(x1 , d, u) = 0, h(x1 , d, u) = 0, (17a-b) u = µc (x1 , d, u), (xT2 , xT3 )T = s(x1 , d, u) (17c-d) where (17d) is the unique and robust analytic solution for (x1 , x2 ) of (16a-b). According to (17) the closed-loop statics (16) has as unique-robust solution the prescribed state-control pair if and only if the OL [O1 :(18b)] and [C1 :(18c)] have one robust intersection. 427

Fig. 2. Cosed-loop statics for the case study (1): (i) C1g ensures a unique closed-loop SS, and (ii) C1w produce three closed-loop SSs (¯ x1 , u ¯ ) = O1 ∩ C 1 ∈ X 1 × U

(18a)

where (18b) O1 = {(x1 , u) | γo (x1 , d, u) = 0} C1 = {(x1 , u) | γc (x1 , d, u) = 0} , C1 = O1 ⇔ k = 0 (18c) γo (x1 , d, u) = [m(x1 , d, u), h(x1 , d, u)]T γc (x1 , d, u) = [m(x1 , d, u), µc (x1 , d, u)]T To restate, in the next proposition, (18) in terms of control gains and limits, define the control bifurcation values in the OL curve O1 (18-b) u∗ = max O1 at x∗1 , u∗ = min O1 at x1∗ (19) u≤¯ u

u≥¯ u

Proposition 1. The CL reactor train with saturated NLSF control (13a) has as unique robust SS the prescribed ¯ if and only if: (i) the control (open-loop unstable) one x gain k is sufficiently large and (ii) the lower (or upper) control limit u− (or u+ ) is sufficiently below (or above) the lower (or upper) bifurcation value u∗ (or u∗ ) (19). This is, k > εk , u− < u∗ − ε∗ , u+ > u∗ + ε∗ (20) ∗ where εk , ε∗ and ε are tolerance values for robustness. The gain condition (anterior-a) ensures that C1 (18-c) is sufficiently different from O1 (18-b), and the limit conditions (20b-c) preclude extraneous SS attractors. The possibility of limit cycling is rather rare event (even in a single reactor [Alvarez et al., 1991]). For this to happen, there must be a saddle unstable limit cycle or nested limit cycles. Conditions to rule out such possibilities are given in the next proposition. Proposition 2. The CL reactor train (6) with saturated NLSF control (13a) has as unique robust SS the prescribed ¯ if and only if: (i) the conditions (open-loop unstable) one x of Proposition 1 are met, and the lower (or upper) is sufficiently small (or large) in the control set Uz of the ZD (10), according to the expressions u − ≥ u− u+ ≤ u+ (21) z , z Proof. When the control set U = Uz is the one of the ZD ¯ is a unique (10), by construction: (i) the nominal SS x robust asymptotically stable attractor SS for Xz (10),

2018 IFAC MICNON 428 Guadalajara, Mexico, June 20-22, 2018 Luis A. Contreras et al. / IFAC PapersOnLine 51-13 (2018) 425–430

(ii) Xz disconnects the invariant set X (11) and (iii) Xz is a robust attractive hypersurface for X. By Seibert’s ¯ is the unique robust Reduction Principle [Seibert, 1970], x asymptotically stable attractor for X. The same property can be attained by choosing the control set U ⊆ Uz sufficiently close (21) to Uz . In Fig. 2 is presented the curve pair (O1 , C1 ) (18) of the bistable reactor (3) with control gain k = 2, for the cases where: (i) the control limit conditions (20) are met (bluecontinuous and green-dashed plots) with (22) (u− , u+ ) = (1.3, 1, 8) and (ii) when the lower limit condition in (20) is violated (blue-continuous and red-dotted plots) with (u− , u+ ) = (1.45, 1, 8) (23) In the former case (22), the CL system has a unique SS (green dot). In the latter case (23) the reactor has three SSs: the prescribed SS is unstable, accompanied by two a extraneous SSs (red dots) (one stable and one unstable). 4. SATURATED OUTPUT-FEEDBACK CONTROLLER Following our previous studies on single-tank reactors [Gonz´ alez and Alvarez, 2005, Schaum et al., 2015],and staged distillation columns [Castellanos-Sahag´ un et al., 2005], here the behavior of the NLSF controller (8) is recovered with a saturated linear dynamic OF controller. 4.1 Simplified model for control design Recall the nonlinear reactor system (6) and rewrite it in the ι-parametric form τ˙1 =au + ι, τ1 (0)=τ1o , y =τ1 , ι=fι (τ1 , xz , u, d) (24a) (24b) x˙ z = f z (τ1 , xz , d, u), xz (0) = xzo where xz = (c1 , c2 , τ2 , c3 , τ3 ) (25a) fι (τ1 , xz , d, u) = f1τ (τ1 , xz , d, u) − au, a = υˆ (25b) |fι (τ1 , xz , d, u)| ≤ ι+ , |x˙ z |/|xz | ≤ λz (25c) λz is the dominant time of the ”internal” dynamics (24b), ι+ is the upper bound of fι , and a is an estimate of the heat exchange number υ. From the assumption that the input signal ι is in a slow varying regime, with respect to the convergence rate ω of an observer (to be constructed), i.e., ι˙ ≈ 0, |˙ι/ι| := λι ≈ λz << ω followed by the omission of the internal dynamics (24b), the reactor train nonlinear system (24) becomes the simplified linear model τ˙1 = au + ι, τ1 (0) = τ1o , y = τ1 (26) with unmeasured bounded signal ι (25): The solution for ι of (26) followed by the substitution of y = τ1 yields ι = au − y˙ meaning that ι is observable from the known signal pair (y, u). Thus, the signal ι can be reconstructed arbitrarily fast (up to measurement error) with a dynamics observer. 4.2 Observer-based output-feedback controller Assuming, that the input signal ι of the model (26) is known, the enforcement of the output regulation dynamics 428

(14a) yields the simplified model-based realization u = µos (y, ι) := sat[µo (y, ι)] (27) µo (y, ι) = [−k(y − y¯) − ι] /a of the NLSF controller (13a). The substitution of (25) in (27), followed by the saturated solution for u of the resulting equation yields the NLSF controller (13a). On the basis of the observability property of the load signal ι of the simplified model (26), set the improper observer ˆι˙ = ω[y − (au − ˆι)] (28)

and apply the coordinate change χ = ι − a¯ u − ωψ, ψ = y − y¯, v = u − u ¯ to obtain the reduced-order proper observer (29a) with estimation error dynamics (29b): χ˙ = −ωχ − ω(ωψ + av), χ(0)=χ0 , ˆι = ¯ι + χ + ωψ (29a) ˜ι˙ = −ω˜ι − f˙ι (x, d, u), ˜ι(0) = ˜ι0 , ˜ι = ˆι − ι (29b) The combination of observer (29a) with the saturation of the NLSF controller (27) yields the dynamic OF saturated controller χ˙ = −ωχ − ω(ωψ + avs ), χ(0) = χo (30a) (30b) v = − [(k + ω)ψ + χ] /a, vs = sat(v) with adjustable control (k) and observer (ω) gains. The application of the OF controller (30), with estimation error dynamics (31b), to the reactor train (6) yields the CL dynamics x˙ =f c (x, d) + υ˜ι, x(0) = x0 , x ∈ X (31a) ˙ ˜ι(0)=˜ι0 , ˜ι ∈ E ˙˜ι= − ω˜ι − ϕ(x, ˜ι, d, d), (31b) made by two input-to-state stable subsystems, where ˜ = 0 ⇒ |˜ ˜ =x−x ¯ , (32a) |˜ x0 |≤δx , d x(t)|≤ax e−λx t δx , x ˜ d) ˜˙ = −f˙ι (x, d, µs (x, d) − ˜ι), d ˜ = d − d, ¯ (32b) ϕ(˜ x, ˜ι, d, ˙ ϕ ˜ ϕ ˜ d)| ˜˙ ≤ Lϕ |˜ ˜ |ϕ(˜ x, ˜ι, d, ι| + L ϕ (32c) x x| + Lι |˜ ˙ |d| d |d| + Ld

where (32a) states that (31a) with ˜ι = 0 is the globally stable closed-loop dynamics (14) with saturated NLSF control (27), and Lϕ v is the Lipschitz, D domain dependent, constant of ϕ with respect to argument v. From the application [Gonz´alez and Alvarez, 2005, Schaum et al., 2015] of the Small Gain theorem [Isidori, 2013], the next proposition follows. For this aim recall, from the proof of Proposition 2, and rewrite the threshold stability condition γT = 1 of (36b) as ϑ ω = Lϕ (33) ι + υLx (k, ω, d) := (k, ω, ϕ), ϕ = X × E Since  depends linearly and nonlinearly with ω, the solution for ω of (33) is double branched: ω − = − (k, ϕ), ω + = + (k, ϕ), ω − < ω + (34) − + where  (or  ) is iso (or anti) tonic in k.

Proposition 3. The closed-loop reactor train (6) with saturated OF control (14) has as unique robust SS attractor the prescribed (open-loop unstable) one x ¯ (7) if: (i) the conditions of Proposition 2 are met, and (ii) the control (k) and observer (ω) gains are chosen as ω > k > k1 , − (k, ϕ) + − < ω < − (k, ϕ) + + (35) where k , − and + are margins for robustness.

2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018 Luis A. Contreras et al. / IFAC PapersOnLine 51-13 (2018) 425–430

Proof. The CL system (33 impreso) is IS stable if the Small Gain Theorem [Isidori, 2013] conditions are met: γιι < 1, (γιx γxι )/(1 − γιι ) := γT < 1 (36a-b) ι x ι ϕ γ ι = Lϕ ι /ω, γι = ax υ/λx , γx = Lx /ω, Since − (or + ) is the small (or large) value solution (34) of (33), the conditions (35) of Proposition 3 state that the small gain conditions (36) are met, implying that the CL dynamics (eq) has as unique robust SS attractor the prescribed one x ¯ (7).QED

429

2.25 2.2 2.15

ref PI+AW NLSF

2.1 0.4

0.2

0 1.8 1.6

Condition (35) ensures robust CL stability in the non1.4 local practical sense [La Salle and Lefschetz, 2012]: bounded-admissible initial-state input disturbances pro1.2 0 1 2 3 4 5 6 duce bounded-admissible state motion deviations about the prescribed SS. Condition (36b) prevents self-destabilization of the observer dynamics (29), and Condition (36c) prevents destabilization by interconnection. The interval Fig. 3. Closed-loop reactor train behavior with saturated dynamic OF (38) and static NLSF (13a) control, [− + − , + − + ] of admissible observer gains shrinks when: the sensor location as well as control limit and (or grows) with the increase (or decrease) of control gain gain criteria are met. k. From previous industrial reactor [Gonz´ alez and Alvarez, 2005, Schaum et al., 2015] and staged distillation column [Castellanos-Sahag´ un et al., 2005, Porru et al., 2014] stud- rather well (up to quick load ι reconstruction) the behavior ies, the control (or observer) gain (36) is typically chosen of its NLSF counterpart. 1-to-5 (or 10-to-30) times faster than the process dominant To test the sensor location criterion (Section 3), the meadynamics (λx ), i.e., surement was switched to reactor 3 in the understanding k = nk λx , nk ∈ [1, 5] (37a) (Section 3) that the NLSF control problem is not solvable (37b) because the ZD (10) is not monostable. The corresponding ω = nω λx nω ∈ [10, 30] CL behavior with OF (30) and NLSF (13a) control is presented in Fig. 4, showing that, as expected, the reactor 4.3 Realization in PI with anti-windup scheme form train did not reach the prescribed SS x ¯ (Table1), but an extraneous stable limit cycle due to the conjunction of ZD The substitution of (30b) in (30a) yields, after some bistability control. manipulations, the saturated dynamic OF controller (30) in PI form To test the control limit criterion (Proposition 1), the     t 1 t 1 temperature measurement was placed in reactor 1 (with   v = − Kp ψ + ψdt + (˜ vs − v˜)dt (38a) robustly monostable ZD) and the lower control limit Ti 0 Tt 0 criterion (20) was violated as follows kω , Ti = k −1 + ω −1 , Tt = ω −1 (38b) vs =sat(v) Kp = u− = 1.45 > u∗ = 1.39 k+ω where Kp is the proportional gain, Ti (or Tt )is the inte- In Fig. 5 is presented the corresponding closed-loop behavgral (or anti-windup) reset time. The preceding PI+AW ior with OF (30) and NLSF (13a) control, showing that, controller with back-calculation scheme (as well as control as expected: the state motion did not reach the prescribed limits chosen according to Proposition 1, and gains tuned SS, but an ignition (catastrophic) extraneous attractor according to Proposition 2) recovers (with the speed of induced by low limit saturation. the reset time Ti ) the behavior of the robust globally stabilizing NLSF controller (13a). 6. CONCLUSIONS 5. CONTROL FUNCTIONING A saturated OF control scheme for multistable 3-tank The application of the sensor location (Section 3) con- reactor train has been developed, including: (i) systematic trol limit [(20) of Proposition 1] and gain tuning [(37) construction and tuning, (ii) robust nonlocal closed-loop of Proposition 2] criteria yield (after some simulation- stable functioning, (iii) sensor location criterion, and (iv) based refinement with initial-state, input disturbance and control limit conditions. The proposed saturated lineal dymeasurement noise) the following control limits and gains: namic single-input single-output OF controller: (i) recovers (up to quick load observer convergence) the behavior of a y = τ1 u− = 1.3 < u∗ = 1.39, u+ = 1.8 > u∗ = 1.77 saturated robust NLSF stabilizing controller, and (ii) can k = 2 = 2λx , ω = 6 = 3λx be realized as an industrial-type PI controller with antiIn Fig. 3 is presented the closed-loop reactor behavior windup protection. with saturated OF (30) and NLSF (13a) control, for the case of initial state deviation xo = [0.400, 2.150, 0.261, The present study can be a point of departure to address 2.107, 0.219, 2.029]T , showing that, as expected from Sec- the N-tank and tubular reactor cases, with the cell modtions 2 to 4: (i) both controllers stabilize the reactor train eling approach for the tubular case [Badillo-Hernandez in spite of saturation, and (ii) the OF controller recovers et al., 2013, N´ajera et al., 2016]. 429

2018 IFAC MICNON 430 Guadalajara, Mexico, June 20-22, 2018 Luis A. Contreras et al. / IFAC PapersOnLine 51-13 (2018) 425–430

3 ref OBLC NLSF

2.5

2 0.6 0.4 0.2 0 1.8 1.6 1.4 1.2 0

1

2

3

4

5

6

Fig. 4. Closed-loop reactor train behavior with saturated dynamic OF (38) and static NLSF (13a) control, when:(i) the control limit and gain conditions are met, and (ii) the sensor location criterion is violated. 3 ref OBLC NLSF

2.5

2 0.4

0.2

0 1.8

1.6

1.4 0

1

2

3

4

5

6

Fig. 5. Closed-loop reactor train behavior with saturated dynamic OF (38) and static NLSF (13a) control, when: (i) the sensor location and upper control limit criteria are met, and (ii) the lower control limit criterion is violated. REFERENCES Alvarez, J., Alvarez, J., and Su´ arez, R. (1991). Nonlinear bounded control for a class of continuous agitated tank reactors. Chemical engineering science, 46(12), 3235– 3249. Amrehn, H. (1977). Computer control in the polymerization industry. Automatica, 13(5), 533–545. Aris, R. and Amundson, N.R. (1958). An analysis of chemical reactor stability and control—i: The possibility of local control, with perfect or imperfect control mechanisms. Chemical Engineering Science, 7(3), 121–131. ˚ Astr¨ om, K.J. and H¨ agglund, T. (2006). Advanced PID control. ISA-The Instrumentation, Systems and Automation Society. Badillo-Hernandez, U., Alvarez-Icaza, L., and Alvarez, J. (2013). Model design of a class of moving-bed tubular gasification reactors. Chemical Engineering Science, 101, 674–685. Castellanos-Sahag´ un, E., Alvarez-Ram´ırez, J., and Alvarez, J. (2005). Two-point temperature control 430

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