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Passivity based controller and observer of exothermic chemical reactors*
8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Singapore, July 10-13, 2012
Passivity based controller and observer of exothermic chemical reactors H. Hoang ∗ F. Couenne ∗∗ Y. Le Gorrec ∗∗∗ Chang-Liang Chen ∗∗∗∗ B. Erik Ydstie † ∗
Faculty of Chemical Engineering, University of Technology, Vietnam National University - Ho Chi Minh City, 268 Ly Thuong Kiet Str., Dist. 10, HCM City, Vietnam (e-mail: [email protected]) ∗∗ LAGEP, University of Lyon, University of Lyon 1, UMR CNRS 5007, Villeurbanne, France (e-mail: [email protected]) ∗∗∗ ENSMM Besan¸con FEMTO-ST / AS2M, Besan¸con, France (e-mail: [email protected]) ∗∗∗∗ Department of Chemical Engineering, National Taiwan University, Taiwan (e-mail: [email protected]) † Chemical Engineering Dept., Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA (e-mail: [email protected]) Abstract: This paper focuses on non linear control and state estimation of non isothermal exothermic Continuous Stirred Tank Reactors (CSTRs). More precisely, the asymptotic stabilization of such CSTRs about any operating point (including unstable open loop stationary point) is treated using the jacket temperature as the only control input. Since state variables are used in the feedback law, a state observer is also proposed. The convergence properties of the controller coupled with the observer are shown using passivity based tools in the Hamiltonian framework. Some numerical simulations with a first order chemical reaction are given to validate our theoretical results. Keywords: Passive Hamiltonian systems, Lyapunov function, CSTR networks, State observer. 1. INTRODUCTION Continuous Stirred Tank Reactors (CSTRs) belong to a typical class of nonlinear dynamical systems described by Ordinary Differential Equations ODEs (Luyben (1990)). They can present a complex behavior, for instance, they can be operated under multiplicity because of highly nonlinear constitutive relations (reaction kinetics...) (Viel et al. (1997); Favache and Dochain (2010)). As a consequence, such CSTRs have been investigated with respect to control design for stabilization (Luyben (1990); Hoang et al. (2011, 2012); Favache et al. (2011); Alvarez et al. (2011)) and state observer synthesis in a large number of studies (Gibon-Fargeot et al. (1994); Soroush (1997); Alvarez-Ram´ırez (1995); Dochain et al. (2009)). The underlying motivation for controlling the CSTRs is that industrial chemical reactors may have to be operated at unstable operating conditions which correspond to some optimal process performances (Bruns and Bailey (1975)). Numerous strategies have been developed to control such non linear systems. Let us cite for example: input/output feedback linearization (Viel et al. (1997)) for control under constraints, nonlinear PI control (AlvarezRam´ırez and Morales (2000)), classical Lyapunov based control (Antonelli and Astolfi (2003)) and more recently thermodynamical Lyapunov based control (Hoang et al. 1
This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2012-20-22.
978-3-902823-05-2/12/$20.00 © 2012 IFAC
377
(2012)), (pseudo) Hamiltonian framework (Hangos et al. (2001); Ram´ırez et al. (2009); D¨orfler et al. (2009); Hoang et al. (2011); Alvarez et al. (2011)), power-shaping control (Favache and Dochain (2010)) and inventory control (Du et al. (2010)). Obervation/estimation strategies have been developed for industrial applications since it is often the case that online measurements of reactant concentrations are difficult and/or very expensive to obtain. Usually, the reactor temperature is the only on-line available measurement. The missing state variables are then estimated (GibonFargeot et al. (1994); Soroush (1997); Alvarez-Ram´ırez (1995); Dochain et al. (1992, 2009)) and used in the control strategy. Results given in (Alvarez-Ram´ırez (1995); Dochain et al. (1992, 2009)) are related to systems in which unfortunately, no feedback is imposed. In this paper we focus on control and state reconstruction problems. More precisely, we first propose a passivity based approach for operating CSTR networks around an open loop unstable steady state. This approach is based on passive Hamiltonian concepts defined in (Brogliato et al. (2007); van der Schaft (2000); Maschke et al. (2000)). The shaped Hamiltonian storage function is chosen such that the resulting state feedback leads to physically admissible control variable solicitations. Second, for practical implementation of the controlled input variable, an asymptotic observer based on thermodynamics invariants of chemical systems (Dochain et al. (1992, 2009)) is proposed in which 10.3182/20120710-4-SG-2026.00051
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
the reactor temperature is supposed to be the only online measurement available. The convergence of the system coupled with the observer is proven using passive properties again. This paper is organized as follows: in section 2, the passivity based approach is briefly introduced and the way to obtain the state feedback law is mentioned. In section 3 the dynamical model of the CSTR networks is presented and analyzed. This section is devoted to the design of the state feedback insuring asymptotic stability. Moreover it is shown that the resulting control is admissible. A state observer is then proposed to reconstruct the concentrations from the only temperature measurement for practical implementation in this section. Finally, some simulation results and conclusions are given in section 4 and section 5 respectively.
3. CASE STUDY: A NON ISOTHERMAL CSTR MODEL Let us consider a CSTR with one reaction involving 2 chemical species A and B: νA A → νB B (7) By convention for the dynamic representation, νA and νB are the suitable signed stoichiometric coefficients: νA < 0 and νB > 0 (see also Hoang et al. (2011)). The reactor is fed by species A and B and Inert at inlet temperature TI . The temperature of the jacket TJ is supposed to be uniform and is used as the only control input. The following assumptions are made: (A1) The fluid is supposed to be ideal, incompressible and isobaric. (A2) The heat flow exchanged with the jacket is represented by Q˙ J = λ(TJ − T ) (8) ˙ Consequently, the only control input is either QJ or TJ . (A3) The specific heat capacities cpA and cpB are assumed to be constant. (A4) The molar number of species A and B in the reactor and in the inlet molar flow are very low compared to the molar number of the Inert. Hence the reaction volume V is written as follows: (9) V vInert NInert = const and it is supposed to be constant (vInert stands for the molar volume of species Inert). As a consequence of the constraint on the volume, the inlet and outlet volume flow rates are equal. (A5) The reaction rate is described by the mass action law: |ν | rv V = k(T )NA A (10) where k(T ) is the kinetics of the liquid phase reaction.
2. PASSIVITY BASED APPROACH (PBA) Let us consider open chemical systems that are affine in the control input u and whose dynamics is given by the following set of ODE’s: dx = f (x) + g(x) u (1) dt n where x = x(t) ∈ R is the state vector, f (x) ∈ Rn represents the smooth nonlinear function with respect to x, g(x) ∈ Rn×m is the input-state map and u ∈ Rm is the control input. The purpose of the PBA is to find a static state-feedback control u = β(x) such that the closed loop dynamics is a Port Controlled Hamiltonian system (PCH system) (Maschke et al. (2000); Ortega et al. (2002)) with dissipation: ∂Hd (x) dx = Qd (x) (2) dt ∂x where the controlled Hamiltonian storage function Hd (x) has a strict local minimum at the desired equilibrium x ; and Qd (x) = [Jd (x) − Rd (x)] where Jd (x) = −Jd (x)T and Rd (x) = Rd (x)T ≥ 0 are some desired interconnection and damping matrices respectively. The system (2) is passive in the sense that the time derivative T ∂Hd (x) ∂Hd (x) dHd (x) =− Rd (x) (3) dt ∂x ∂x is always negative and the Hamiltonian Hd (x) is bounded from below (van der Schaft (2000); Brogliato et al. (2007)). Consequently, it then plays role of Lyapunov function for stabilization purpose at the desired equilibrium x . The matching equation (EDP) (4) that follows immediately from (1) and (2) has to be solved to find u = β(x): ∂Hd (x) (4) f (x) + g(x)β(x) = Qd (x) ∂x Let consider there exists a full rank left annihilator of g(x) denoted g(x)⊥ such that g(x)⊥ g(x) = 0. If Jd (x), Rd (x) and Hd (x) are chosen such that: ∂Hd (x) (5) g(x)⊥ f (x) = g(x)⊥ Qd (x) ∂x then the control variable is deduced from the state feedback β(x) given by (Ortega et al. (2002)): ∂Hd (x) T T −1 β(x) = g(x)
g(x)g(x)
Qd (x)
∂x
− f (x)
(6)
378
We also use an additional assumption as in (Viel et al. (1997); Alvarez-Ram´ırez and Morales (2000); Antonelli and Astolfi (2003)) or recently (Favache and Dochain (2010)): (A6) Let (Td , NAd , NBd ) be the steady state 2 of the system. The isothermal dynamics (T = Td ) is stable and globally asymptotically converges (e.g. NA → NAd and NB → NBd ). A sufficient condition that verifies this hypothesis (A6) is reaction kinetics k(T ) modelled by the Arrhenius law: −Ea k(T ) = k0 exp( ) (11) RT where Ea and R are the activation energy and the ideal gas constant respectively and k0 is kinetic constant. Some notations are introduced in Tables 1 and 2. 3.1 CSTR modelling As mentioned before, the system dynamics is given by material and internal energy balances. Under the Assumption (A1), the internal energy balance is written using 2
All time derivatives vanish at this state or the time becomes very large, e.g., goes to infinity.
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
Table 1. Notation of the variables of the model Notation hiI hi NAI NBI NInertI NA NB NInert d rv V
unit Jmol−1 Jmol−1 mol mol mol mol mol mol s−1 mols−1
Inlet molar enthalpy of i (i = A, B, Inert) Partial molar enthalpy of i (i = A, B, Inert) Inlet mole number of species A Inlet mole number of species B Inlet mole number of species Inert Mole number of species A Mole number of species B Mole number of species Inert dilution rate Reaction rate
the enthalpy H instead of internal energy U . The balance equations are (Favache et al. (2011)): ⎧ dNA ⎪ ⎪ = d(NAI − NA ) + νA rv V ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ dNB (12) = d(NBI − NB ) + νB rv V ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dH ⎪ ⎩ = d (HI − H) + Q˙ J dt Remark 1. Since we suppose ideality of the mixture, on one side the enthalpy of species A in the mixture can be expressed as: hA (T ) = cpA (T − Tref ) + hAref (a similar expression for species B and Inert) where Tref and hAref are the reference temperature and the reference molar enthalpy respectively of species A. The total enthalpy H of the reaction system can be expressed by (13) H = NA hA (T ) + NB hB (T ) + NInert hInert (T ) Finally the constraint on the volume (see the Assumption (A4)), (9) leads to: dNInert = d(NInertI − NInert ) = 0 (14) dt Thanks to the local equilibrium hypothesis, the energy balance can be written in terms of temperature. This is done by using the expression for the enthalpy H as given in (13), we obtain (see Hoang et al. (2012)): dT Cp (15) = − ∆r H rv V + d(TI − T )CpI + Q˙ J dt where ∆r H = (νB hB (T ) + νA hA (T )) < 0 is the enthalpy of the exothermic reaction and Cp = cpA NA + cpB NB + cpInert NInert is the total heat capacity, respectively. Remark 2. One can use the dynamics of states variables either (NA , NB , H) (12) or (NA , NB , T ) ((12),(15)) to represent completely the behavior of the system. 3.2 Controller synthesis In what follows, we use (NA , NB , T ) to represent the dynamics of the system. We have from ((15), (12)): ⎧ dN A ⎪ = d(NAI − NA ) + νA rv V ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dNB = d(NBI − NB ) + νB rv V (16) dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ∆r H ⎪ dT CpI 1 ˙ ⎪ ⎩ QJ = rv V + d(TI − T ) + dt Cp Cp Cp 379
Writing (16) into form (1) with u = Q˙ J , we obtain: ⎛ ⎞ d(NAI − NA ) + νA rv V ⎛ 0 ⎞ ⎛ ⎞ NA ⎜ ⎟ ⎟ ⎜ ⎟ d(NBI − NB ) + νB rv V ⎟ ⎜ d ⎜ ⎟ ⎜ 0 ⎟ ⎜ NB ⎟ = ⎜ + ⎜ ⎟ ⎜ ⎟u ⎠ ⎜ dt ⎝ ⎟ ⎝ ⎠ ⎝ − ∆r H ⎠ 1 CpI T rv V + d(TI − T ) Cp Cp Cp x g(x) f (x)
(17)
The proposition 1 proposes a nonlinear state feedback for the jacket temperature to stabilize the reactor at a desired operating point. Proposition 1. Under the Assumption (A2), the system (17) is stabilized at a desired state xd = (NAd , NBd , Td ) with the following equivalent feedback on the jacket temperature TJ : ∂Hd dNB 1 ∂Hd −1 ∂Hd dNA TJ = T + + Cp − λ ∂T ∂N dt ∂NB dt A ∂Hd −KT − − ∆r H rv V + d(TI − T )CpI ∂T (18) where KT > 0 is a tuning parameter. Furthermore, the closed loop dynamics is a passive Hamiltonian system: dx ∂Hd (x) = [Jd (x) − Rd (x)] (19) dt ∂x where: ⎛ ⎞ −1 ⎜ ⎜ ⎜ ⎜ Jd (x) = ⎜ ⎜ ⎜ ⎝ −
0
∂Hd ∂T
0
0 ∂Hd ∂T
−1
0
dNA ∂Hd − dt ∂T
⎛
−1
0 0 0
dNB dt
⎞
∂Hd ∂T
dNA dt
−1
dNB dt
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0 (20)
⎟ ⎜ ⎟ ⎜ Rd (x) = ⎜ 0 0 0 ⎟ (21) ⎠ ⎝ 0 0 KT and Hd (x) is chosen under the following form: Hd (x) = Hd (T, NA , NB ) (T − Td )2 1 + KA (NA − NAd )2 + KB (NB − NBd )2 = 2 (22) where KA ≥ 0 and KB ≥ 0. Proof. We apply the passivity based tools (see Section 2), the full rank left annihilator⎛of g(x)⎞ (17) denoted 1 0 0 ⎟ ⎜ ⎟ ⎜ by g ⊥ (x) is given by: g ⊥ (x) = ⎜ 0 1 0 ⎟. Let us take ⎠ ⎝ 0 0 0 ⎞ ⎛ q11 (x) q12 (x) q13 (x) ⎟ ⎜ ⎟ ⎜ Qd (x) = Jd (x) − Rd (x) = ⎜ q21 (x) q22 (x) q23 (x) ⎟, we ⎠ ⎝ q31 (x) q32 (x) q33 (x) obtain using the matching equation (5):
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
⎛
∂Hd ∂Hd ∂Hd ⎜ q11 (x) ∂NA + q12 (x) ∂NB + q13 (x) ∂T ⎜ ⎜ ⎜ ∂Hd ∂Hd ∂Hd ⎜ + q22 (x) + q23 (x) ⎜ q21 (x) ⎜ ∂NA ∂NB ∂T ⎝ 0
⎞
⎛ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟ ⎝ ⎠
⎞ dNA dt ⎟ ⎟ ⎟ dNB ⎟ ⎟ dt ⎟ ⎠ 0
(23) Due to the negative definiteness of the matrix Qd (x), we choose q11 (x) = q12 (x) = q21 (x) = q22 (x) = 0, −q13 (x) = q31 (x), −q23 (x) = q32 (x) and q33 (x) = −KT < 0. We obtain for solutions: ∂H −1 dN ∂H −1 dN d A d B q13 (x) = , q23 (x) = ∂T dt ∂T dt d q13 (x) and q23 (x) is well defined at the limit ∂H ∂T → 0 (see Remark 3). Finally, the feedback law is derived from (6): ∂Hd −1 ∂Hd dNA ∂Hd −1 ∂Hd dNB − Cp − ∂T ∂T ∂NB dt ∂NA dt ∂Hd − ∆r H rv V + d(TI − T )CpI = u −KT − ∂T Using the Assumption (A2) with u = Q˙ J , it leads to the feedback law (18). Let us notice that a possible choice for Hd (x) for which the feedback law (18) well-defined is given in (22). The function Hd (x) is positive and its time derivative, ∂H 2 dHd (x) d = −KT <0 (24) dt ∂T The latter ends the proof. 2 Remark 3. The closed loop dynamics for the temperature T with the feedback law (18) can be written as follows: −1 −1 dT =− dt
∂Hd ∂T
dNA ∂Hd − dt ∂NA
∂Hd ∂T
the feedback law may present a different shape in terms of the amplitude and variation rate. Remark 5. Let us consider the CSTR networks with n species Sj (j = 1 . . . n) and r chemical reactions taking place in the reactor. We then have: n νij Sj = 0, i = 1 . . . r (26) j=1
where νij is the signed stoichiometric coefficient of species j as it enters in reaction i (Hoang et al. (2011)). For such a system and under the isobaric conditions (A1), the mathematical modeling of dynamics (17) can be generally rewritten (Hoang et al. (2012); Favache and Dochain (2010)): ⎛ ⎞ r d(N − N ) + ν r V 1 j1 vj 1I ⎜ ⎟ ⎜ ⎟ j=1 ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎜ ⎟ r N1 ⎜ ⎟ ⎜ ⎟ d(N − N ) + ν r V 2 j2 vj 2I ⎜ ⎟ ⎜ ⎟ ⎜ N2 ⎟ ⎜ ⎟ j=1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ d ⎜ ⎜ ⎟ . ⎜ .. ⎟ = ⎜ . ⎟ . ⎟ . dt ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r ⎜ Nn ⎟ ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ d(NnI − Nn ) + νjn rvj V ⎜ ⎟ ⎜ ⎟ T j=1 ⎟ ⎜ ⎜ ⎟ x ⎜ r ⎟ ⎝ j=1 − ∆rj H rvj V ⎠ C
Cp
+ d(TI − T )
f (x)
dNB ∂Hd ∂Hd − KT dt ∂NB ∂T (25)
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ +⎜ ⎜ ⎜ ⎜ ⎜ ⎝
First let us note that from Hd (x) (22), ∂Hd → 0 ⇔ T → Td ∂T ∂Hd ∂Hd and lim ∂N → 0, lim ∂N → 0. Second we have A B T →Td
T →Td
from (22) Hd (T = Td , NA , NB ) = 0 and from (24) dHd = 0. It is shown that Hd (x) is a dt
T →Td
∂Hd →0 ∂T
⎧ ∂H −1 dN d A ⎪ ⎪ lim <∞ ⎪ ⎨ ∂Hd →0 ∂T dt ∂T ∂H −1 dN ⇒ d B ⎪ ⎪ lim <∞ ⎪ ⎩ ∂Hd ∂T dt →0
0 0 .. . 0 1 Cp
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ˙ QJ ⎟ ⎟ ⎟ u ⎟ ⎟ ⎠
g(x)
(27)
where rvj V and ∆rj H represent the reaction rate and the enthalpy of the chemical reaction j (j = 1 . . . r). By using the same procedure as previous sim in the and ple case, let us note Q(x) = qij (x) i,j=1...(n+1)
∂T
Remark 4. If we take KA = KB = 0 in Hd (x) (22), the feedback law (18) is then similar to the one proposed by Viel (Viel et al. (1997)) in the case without constraint on the control input (see the equation (3) in (Viel et al. (1997)) for feedback law and adapt it to the system under consideration). The stabilization obtained by these strategies is dominated by the regulation of the thermal part inside the reactor in accordance with the Assumption (A6). In other cases (KA × KB = 0), due to presence of the additional material contributions (see (18) and (22)), 380
(T =Td ,NA ,NB )
Lyapunov function for the stabilization of the temperature T inside the reactor. As a consequence, we have to obtain dT lim dT dt = lim dt = 0 and thus we deduce from (25):
⎛
pI
Cp
g ⊥ (x) = diag ( 1, . . . , 1, 0 ) ∈ R(n+1)×(n+1) . The following matching equation which generalizes (23) can be derived from (5) : ⎧ dN1 ∂Hd ∂Hd ∂Hd ⎪ ⎪ q (x) = + . . . + q1n (x) + q1(n+1) (x) ⎪ ⎪ 11 ∂N1 ∂Nn ∂T dt ⎨ .. . ⎪ ⎪ ⎪ dNn ∂H ∂H ∂Hd d d ⎪ ⎩ qn1 (x) = + . . . + qnn (x) + qn(n+1) (x) ∂N1 ∂Nn ∂T dt The number of equations equals n with n×(n+1) unknown variables qij (x). Hence this system has an infinite number of solutions and a simple solution can be found as follows: • qij (x) = qji (x) = 0 for i, j = 1 . . . n
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
−1 dNi d • then qi(n+1) (x) = −q(n+1)i (x) = ∂H ∂T dt for i = 1 . . . n, • and let we choose q(n+1)(n+1) (x) = −KT From this, we obtain for the matrix Q(x) ≡ Jd (x) − Rd (x) , where: ⎛ ∂Hd −1 dN1 ⎞ 0
⎜ ⎜ ⎜ . . ⎜ . ⎜ Jd (x) = ⎜ ⎜ ⎜ 0 ⎜ ⎝ ∂Hd −1 dN1 −
and
∂T
dt
...
0
..
.
. . .
...
0
... −
⎛
0 ⎜ .. . Rd (x) = ⎜ ⎝0 0
∂T . . .
∂Hd −1 dNn
∂Hd −1 dNn ∂T
... .. . ... ...
dt
∂T
dt
dt
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0
⎞ 0 0 .. .. ⎟ . . ⎟ 0 0 ⎠ 0 KT
(28)
(29)
The feedback law for u (27) can be found using (6) with Jd (x) (28), Rd (x) (29) and Hd (x) is chosen to be proportional to: n (T − Td )2 (30) 1+ Ki (Ni − Nid )2 Hd (x) = 2 i=1
where Ki ≥ 0. Finally, let us notice that the proposed passive controller stabilizes the reactor as soon as the Assumption (A6) holds 3 (see also Antonelli and Astolfi (2003)). 3.3 A state observer We now consider that only T is measured. As a consequence we have to build an observer to reconstruct the mixture composition. The main feature of the proposed estimator is that it is independent of the knowledge of the system kinetics and is called asymptotic observers. These asymptotic observers are first proposed in (Dochain et al. (1992, 2009)) for a nonlinear class of CSTR networks without considering feedback law. In what follows, we shall show that with or without feedback law, the convergence of estimated state variables to their exact values is exponential. Let us consider the original system (12): ⎧ dH ⎪ ⎪ = Q˙ + d(HI − H) ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ dNA Σ = d(NAI − NA ) + νA rv V ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ dN ⎪ B ⎪ ⎩ = d(NBI − NB ) + νB rv V dt
(31)
3
In general, (A6) verifies for a large range of non linear reaction systems in which the (high-order) mass action laws take place, rvj V = kj (T )
n
α
Nk jk , j = 1 . . . r
k=1
where Arrhenius based reaction kinetics kj (T ) is considered.
381
Doing a variable transformation N = νB N A − νA N B (32) and using the second and third equations in (31), we obtain: dN = d(NI − N ) dt It is clear that the dynamics of new variable N does not depend on the reaction kinetics. The same situation holds for the dynamics of the enthalpy H (see the first equation in (31)). As a consequence, we then an asymptotic derive ˆ (33) for the system Σ (31): observer Σ ⎧ ˆ ⎪ ⎪ dH = Q˙ + d(HI − H) ˆ ⎨ dt ⎪ ˆ Σ (33) ⎪ ⎪ ˆ ⎪ d N ⎩ ˆ) = d(NI − N dt ˆ = hA (T )N ˆ A + hB (T )N ˆ B + hInert (T )N ˆInert and where H ˆ ˆ ˆ N = νB NA − νA NB are estimated values of the enthalpy H and the new variable N , respectively. We assume here that the reactor temperature T is the only available on-line measurement. Remark 6. The Inert doesn’t participate to the reaction (7) but its presence in the reactor should be considered (see (13) for total enthalpy for example). The following ˆInert of equation presents the dynamics of the estimate N NInertI ˆInert dN ˆInert ) = d(NInertI − N (34) dt Proposition 2. Closed loop stability using observer. ˆ ˆ The estimations NA of NA and NB of NB defined from ˆ Σ , converges to NA and NB respectively with time 1 constant τ = 2d . Furthermore the system Σ controlled by the feedback law (18) coupled with the state observer ˆ is exponentially stable. Σ
1
⎛
ˆ −H H
⎞
⎠. By = ⎝ ˆ −N 2 N subtracting (33) to (31), one can get: d 0 1 d =− (35) dt 0 d 2 The dynamics of is then presented in a port Hamiltonian format (2), d = [J () − R ()] ∂H∂ () where J () = 0, dt d 0 R () = and the Hamiltonian storage function 0 d H () = 12 T ≥ 0. The last one plays a role of a Lyapunov function for the stability of the zero dynamics of because: ∂H () T d 0 ∂H () dH =− ≤0 dt ∂ ∂ 0 d Furthermore, it can be rewritten as follows: − 1t dH = −d T = −2d H ⇒ H (t) = H (0) exp 2d dt Because of d > 0, H (t) exponentially converges to 0 with 1 the time constant τ = 2d . As a consequence, (t) → 0 at Proof. Let us define (t) =
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
Details on analysis of the steady states can be found in (Hoang et al. (2012)). Its shows that the system has three stationary operating points. Simulations are given in Fig. 1 with the initial conditions (C1) and (C2). 0.18
P
2
0.12 0.1 0.08 0.06 0.04
ˆ B → NB ⎩ lim N
P3
0.02
t<+∞
0
It is important to notice that the convergence does not depend on the feedback strategy. The latter ends the proof. 2 4. SIMULATION In this section, the exothermic reaction under consideration (7) is treated for simulation with νA = −1 and νB = 1 (e.g. the first order reaction, this type of reaction is often appeared in the literature (Viel et al. (1997); Favache and Dochain (2010)). The reaction rate is then rewritten as follows (see Assumption (A5)): −Ea rv V = k0 exp( )NA RT In Table (2), we give numerical values used for simulation.
300
320
340
360
380
400
T (K)
Fig. 1. The representation of the open loop phase plane In the next section, we propose to operate the system around the unstable middle point P2 using the feedback law for the jacket temperature TJ (18). 4.2 Closed loop simulation In a first instance we consider the state variables are measured, e.g. the system is closed using the proposed state feedback law without observer. We take KT = 0.001, KA = 0 and KB = 0 for the shaped function Hd (x) (22). In Figure 2 the closed loop phase plane is represented. We can see that for all the considered initial conditions the system converges to the desired operating point P2 . Furthermore Figure 2 also shows that the control variable solicitations TJ (18) are physically admissible in terms of amplitude and dynamics.
Table 2. Parameters of CSTR
Heat capacity of species A Heat capacity of species B Heat capacity of Inert Activation energy Reference enthalpy of A Reference enthalpy of B Reference enthalpy of Inert kinetic constant gas constant Reference temperature heat transfer coefficient Dilution rate
0.18
P
with (C1) with (C2)
1
0.16 0.14
P2
0.12
NA (mol)
Numerical value 221.9 (JK −1 mol−1 ) 128.464 (JK −1 mol−1 ) 21.694 (JK −1 mol−1 ) 73.35 (KJmol−1 ) −5.8085 105 (Jmol−1 ) −6.6884 105 (Jmol−1 ) −3.3 105 (Jmol−1 ) 2.58 109 (s−1 ) 8.314 (JK −1 mol−1 ) 298 (K) 0.75 (W K −1 ) 0.0070 (s−1 )
with (C1) with (C2)
0.14
t<+∞
cpA cpB cpInert Ea hAref hBref hInertref k0 R Tref λ d
P1
0.16
NA (mol)
the finite time (t > 3τ ) and at that time we obtain (with Remark 6): ⎞ ⎛ ˆ A − NA N hA (T ) hB (T ) ⎠=0 ⎝ lim t<+∞ ˆ νB −νA NB − NB hA (T ) hB (T ) Because Det = −νA hA (T )−νB hB (T ) = νB −νA −∆r H > 0 (for exothermic reaction under consideration), the matrix is full rank. As a consequence, ⎧ ˆ A → NA ⎨ lim N
0.1 0.08 0.06 0.04 0.02
4.1 Open loop simulation First of all let us consider open loop simulation with input defined by (36) ! TI = TJ = 298 (K), NAI = 0.18 (mol), (36) NBI = 0 (mol), NInertI = 3.57 (mol) 382
0 290
300
310
320
330
340
T (K) 306 with (C1) with (C2)
304
J
Jacket temperature T (K)
The goal of this section is to illustrate the efficacy of the aforementioned control strategy with or without the state observer for stabilization purpose. The open and closed loop simulations are carried out with respect to two different initial conditions, (C1) with T0 = 340 (K), NA0 = 0.04 (mol), NB0 = 0.001 (mol) and (C2) with T0 = 300 (K), NA0 = 0.15 (mol), NB0 = 0.03 (mol) .
302 300 298 296 294 292 290 288 0
500
1000
1500 time (s)
2000
2500
3000
Fig. 2. Representation of the closed loop phase plane (the point P3 outside the frame) and the feedback law TJ For any initial conditions, Hd (x) (22) plays the role of Lyapunov function and consequently it converges to 0 as shown in Figure 3.
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
300 with (C1) with (C2)
Controlled function Hd
250
200
150
100
50
0 0
500
1000
1500 time (s)
2000
2500
Fig. 5. System in closed loop with the asymptotic observer
3000
Fig. 3. The dynamics of Hd Remark 7. The convergence speed of the controlled system goes faster by increasing the tuning parameter KT .
In fact, the convergence of the controlled system coupled with asymptotic observer is more interesting. For the ˆ initial conditions (C2) and (C1), the simulations are given in Figure 6. (a)
4.3 Simulation with the asymptotic observer
NA exact
0.16
NA observed
0.14 0.12 0.1 0
500
1000
1500 time (s)
2000
2500
3000
0.08
NB (mol)
For the sake of simplicity, the initial condition (C2) is reused for the system. The initial conditions of the asymp ˆ ˆ totic observer are (C1) with H(0) = 0.98H(T0, NA0 , NA0 ), ˆ (0) = 0.75N (NA0, NB0 ) and (C2) ˆ ˆ N with H(0) = ˆ (0) = 0.95N (NA0 , NB0 ) where H(0.85T0 , NA0 , NB0 ), N the numerical values of T0 , NA0 and NB0 are given with the initial condition (C2).
NA (mol)
0.18
0.06 0.04
0 0
In open loop, the convergence of the system coupled with asymptotic observer is illustrated in Fig. 4. With the initial condition (C2), the system normally converges to the stable point P1 .
NB exact
0.02
NB observed 500
1000
1500 time (s)
2000
2500
3000
(b) 304
303
302
T (K)
(a) 301
0.16 N 0.14 0.12 0.1 0
exact
300
NA observed
299
A
A
N (mol)
J
0.18
200
400
600 time (s)
800
1000
298 0
1200
1000
2000 3000 time (s)
4000
5000
NB (mol)
0.03 N
exact
N
observed
B
0.025
B
0.02
Fig. 6. NA and its estimate with the asymptotic observer in the closed loop case -(a) for the initial conditions ˆ (C2) and (C1) -(b) the control input TJ with the asymptotic observer
0.015 0.01 0
200
400
600 time (s)
800
1000
1200
(b)
NA (mol)
0.2
ˆ For the initial conditions (C2) and (C2), the simulations are illustrated in Fig. 7. The simulations (Figures 6(a) and 7(a)) show that the stabilization at the unstable point P2 of the controlled system via the state observer is insured. Furthermore, Tw is still admissible (see Fig. 6(b) and Fig. 7(b)).
0.15 NA exact
0.1
N
0.05 0 0
A
200
400
600 time (s)
800
observed
1000
1200
NB (mol)
0.2 N
exact
N
observed
B
0.15
B
0.1
5. CONCLUSION
0.05 0 0
200
400
600 time (s)
800
1000
1200
Fig. 4. NA , NB and their estimates in the open loop case ˆ -(b) for -(a) for the initial conditions (C2) and (C1) ˆ initial conditions (C2) and (C2) In closed loop, the controlled system Σ and the asymp " are coupled as in Fig. 5. L represents totic observer Σ the feedback law (18). 383
In this work, we have shown by means of passivity based method: (1) how to stabilize the CSTR networks about a desired operating point. The obtained feedback law covers the one proposed by (Viel et al. (1997)) in the case without constraint on the control input for a simple first order chemical reaction. (2) Afterwards for practical implementation of the controlled system, we have coupled the controller with a state observer.
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
(a)
NA (mol)
0.2 0.15 0.1
N exact A
0.05 0 0
N observed A
500
1000
1500 time (s)
2000
2500
3000
NB (mol)
0.2 NB exact
0.15
NB observed
0.1 0.05 0 0
500
1000
1500 time (s)
2000
2500
3000
(b) 310
308
J
T (K)
306
304
302
300
298 0
1000
2000 3000 time (s)
4000
5000
Fig. 7. NA and its estimate with the asymptotic observer in the closed loop case -(a) for the initial conditions ˆ -(b) the control input TJ with the (C2) and and (C2) asymptotic observer The closed loop convergence of the system with/without the observer is theoretically guaranteed. The simulation results showed that convergence objective is satisfied and that the state feedback law is physically implementable since jacket temperature TJ remains in some physical domain with admissible rate of variation. It remains now: • to evaluate performances and robustness of the obtained results in terms of perturbations and parameters uncertainty. • to introduce the dynamical modeling of the jacket into the problem. • to extend the proposed method to inventory control (Farschman et al. (1998)). REFERENCES D¨orfler, F. and Johnsen, J. K. and Allg¨ower, F. (2009). An introduction to interconnection and damping assignment passivity-based control in process engineering. J. Proc. Control, 19, 1413-1426. Favache, A. and Dochain, D. (2010). Power-shaping of reaction systems : the CSTR case study. Automatica, 46(11), 1877-1883. Favache, A. and Dochain, D. and Winkin, J. (2011). Power-shaping control: writing the system dynamics into the Brayton-Moser form. Syst. Cont. Let., 60(8), 618-624. Hoang, H. and Couenne, F. and Jallut, C. and Le Gorrec, Y. (2011). The Port Hamiltonian approach to modeling and control of Continuous Stirred Tank Reactors. J. Proc. Control, 21(10), 1449-1458. Hoang, H. and Couenne, F. and Jallut, C. and Le Gorrec, Y. (2012). Lyapunov-based control of non isothermal continuous stirred tank reactors using irreversible thermodynamics. J. Proc. Control, 22(2):412-422. 384
Alvarez J., J. Alvarez-Ram´ırez, G. Espinosa-Perez and A. Schaum (2011). Energy shaping plus damping injection control for a class of chemical reactors. Chem. Eng. Sci., 66(23), 6280-6286. Luyben, W. L. (1990). Process modeling, simulation, and control for chemical engineers. McGraw-Hill, Singapore. Maschke, B. and Ortega, R. and van der Schaft, A. (2000). Energy based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans. on Autom. Control, 45(8), 1498-1502. Ortega, R. and van der Schaft, A. and Maschke, B. and Escobar, G. (2002). Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian Systems. Automatica, 38, 585-596. Ram´ırez, H. and Sbarbaro, D. and Ortega, R. (2009). On the control of non-linear processes: An IDA-PBC approach. J. Proc. Control, 19, 405-414. van der Schaft, A. (2000). L2 -gain and passivity techniques in nonlinear control. Springer-Verlag, London, 2nd edt. Brogliato, B. and Lozano, R. and Maschke, B. and Egeland, O. (2007). Dissipative systems analysis and control. Springer, London, 2nd edition. Alvarez-Ram´ırez, J. and Morales, A. (2000). PI control of continuously stirred tank reactors: stability and performance, Chem. Eng. Sci., 55(22):5497-5507. Antonelli, R. and Astolfi, A. (2003). Continuous stirred tank reactors: easy to stabilise?. Automatica, Vol. 39, pp. 1817-1827. Viel, F. and Jadot, F. and Bastin, G. (1997). Global stabilization of exothermic chemical reactors under input constraints. Automatica, Vol. 33, No. 8, pp. 1437-1448. Bruns, D. D. and Bailey, J. E (1975). Process operation near an unstable steady state using nonlinear feedback control. Chem. Eng. Sci., Vol. 30, pp. 755-762. Dochain, D. and Perrier, M. and Ydstie, B. E. (1992). Asymptotic observers for stirred tank reactors. Chem. Eng. Sci., Vol. 47, No. 15/16, pp. 4167-4117. Alvarez-Ram´ırez, J. (1995). Observers for a class of continuous tank reactors via temperature measurement. Chem. Eng. Sci., Vol. 55, No. 9, pp. 1393-1399. Dochain, D. and Couenne, F. and Jallut, C. (2009). Enthalpy Based Modelling and Design of Asymptotic Observers for Chemical Reactors. International Journal of Control, Vol. 82, No. 8, pp. 1389-1403. Soroush, M. (1997). Nonlinear state-observer design with application to reactors. Chem. Eng. Sci., Vol. 52, No. 3, pp. 387-404. Gibon-Fargeot, A. M. and Hammouri, H. and Celle, F. (1994). Nonlinear observers for chemical reactors. Chem. Eng. Sci., Vol. 49, No. 14, pp. 2287-2300. Du, J. and Laird, C. M. and Ydstie, B. E. (2010). The measurement selection of inventory control. American Control Conference, Marriott Waterfront, Baltimore, MD, USA June 30-July 02. Hangos, K. M. and Bokor, J. and Szederk´enyi, G. (2001). Hamiltonian view on process systems, AIChE journal, vol. 47, no 8, pp. 1819-1831. Farschman, C. A. and Viswanath, K. P. and Ydstie, B. E. (1998). Process systems and inventory control. AIChE journal, 44:8, pp. 1841-1857.
Passivity based controller and observer of exothermic chemical reactors H. Hoang ∗ F. Couenne ∗∗ Y. Le Gorrec ∗∗∗ Chang-Liang Chen ∗∗∗∗ B. Erik Ydstie † ∗
Faculty of Chemical Engineering, University of Technology, Vietnam National University - Ho Chi Minh City, 268 Ly Thuong Kiet Str., Dist. 10, HCM City, Vietnam (e-mail: [email protected]) ∗∗ LAGEP, University of Lyon, University of Lyon 1, UMR CNRS 5007, Villeurbanne, France (e-mail: [email protected]) ∗∗∗ ENSMM Besan¸con FEMTO-ST / AS2M, Besan¸con, France (e-mail: [email protected]) ∗∗∗∗ Department of Chemical Engineering, National Taiwan University, Taiwan (e-mail: [email protected]) † Chemical Engineering Dept., Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA (e-mail: [email protected]) Abstract: This paper focuses on non linear control and state estimation of non isothermal exothermic Continuous Stirred Tank Reactors (CSTRs). More precisely, the asymptotic stabilization of such CSTRs about any operating point (including unstable open loop stationary point) is treated using the jacket temperature as the only control input. Since state variables are used in the feedback law, a state observer is also proposed. The convergence properties of the controller coupled with the observer are shown using passivity based tools in the Hamiltonian framework. Some numerical simulations with a first order chemical reaction are given to validate our theoretical results. Keywords: Passive Hamiltonian systems, Lyapunov function, CSTR networks, State observer. 1. INTRODUCTION Continuous Stirred Tank Reactors (CSTRs) belong to a typical class of nonlinear dynamical systems described by Ordinary Differential Equations ODEs (Luyben (1990)). They can present a complex behavior, for instance, they can be operated under multiplicity because of highly nonlinear constitutive relations (reaction kinetics...) (Viel et al. (1997); Favache and Dochain (2010)). As a consequence, such CSTRs have been investigated with respect to control design for stabilization (Luyben (1990); Hoang et al. (2011, 2012); Favache et al. (2011); Alvarez et al. (2011)) and state observer synthesis in a large number of studies (Gibon-Fargeot et al. (1994); Soroush (1997); Alvarez-Ram´ırez (1995); Dochain et al. (2009)). The underlying motivation for controlling the CSTRs is that industrial chemical reactors may have to be operated at unstable operating conditions which correspond to some optimal process performances (Bruns and Bailey (1975)). Numerous strategies have been developed to control such non linear systems. Let us cite for example: input/output feedback linearization (Viel et al. (1997)) for control under constraints, nonlinear PI control (AlvarezRam´ırez and Morales (2000)), classical Lyapunov based control (Antonelli and Astolfi (2003)) and more recently thermodynamical Lyapunov based control (Hoang et al. 1
This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number B2012-20-22.
978-3-902823-05-2/12/$20.00 © 2012 IFAC
377
(2012)), (pseudo) Hamiltonian framework (Hangos et al. (2001); Ram´ırez et al. (2009); D¨orfler et al. (2009); Hoang et al. (2011); Alvarez et al. (2011)), power-shaping control (Favache and Dochain (2010)) and inventory control (Du et al. (2010)). Obervation/estimation strategies have been developed for industrial applications since it is often the case that online measurements of reactant concentrations are difficult and/or very expensive to obtain. Usually, the reactor temperature is the only on-line available measurement. The missing state variables are then estimated (GibonFargeot et al. (1994); Soroush (1997); Alvarez-Ram´ırez (1995); Dochain et al. (1992, 2009)) and used in the control strategy. Results given in (Alvarez-Ram´ırez (1995); Dochain et al. (1992, 2009)) are related to systems in which unfortunately, no feedback is imposed. In this paper we focus on control and state reconstruction problems. More precisely, we first propose a passivity based approach for operating CSTR networks around an open loop unstable steady state. This approach is based on passive Hamiltonian concepts defined in (Brogliato et al. (2007); van der Schaft (2000); Maschke et al. (2000)). The shaped Hamiltonian storage function is chosen such that the resulting state feedback leads to physically admissible control variable solicitations. Second, for practical implementation of the controlled input variable, an asymptotic observer based on thermodynamics invariants of chemical systems (Dochain et al. (1992, 2009)) is proposed in which 10.3182/20120710-4-SG-2026.00051
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
the reactor temperature is supposed to be the only online measurement available. The convergence of the system coupled with the observer is proven using passive properties again. This paper is organized as follows: in section 2, the passivity based approach is briefly introduced and the way to obtain the state feedback law is mentioned. In section 3 the dynamical model of the CSTR networks is presented and analyzed. This section is devoted to the design of the state feedback insuring asymptotic stability. Moreover it is shown that the resulting control is admissible. A state observer is then proposed to reconstruct the concentrations from the only temperature measurement for practical implementation in this section. Finally, some simulation results and conclusions are given in section 4 and section 5 respectively.
3. CASE STUDY: A NON ISOTHERMAL CSTR MODEL Let us consider a CSTR with one reaction involving 2 chemical species A and B: νA A → νB B (7) By convention for the dynamic representation, νA and νB are the suitable signed stoichiometric coefficients: νA < 0 and νB > 0 (see also Hoang et al. (2011)). The reactor is fed by species A and B and Inert at inlet temperature TI . The temperature of the jacket TJ is supposed to be uniform and is used as the only control input. The following assumptions are made: (A1) The fluid is supposed to be ideal, incompressible and isobaric. (A2) The heat flow exchanged with the jacket is represented by Q˙ J = λ(TJ − T ) (8) ˙ Consequently, the only control input is either QJ or TJ . (A3) The specific heat capacities cpA and cpB are assumed to be constant. (A4) The molar number of species A and B in the reactor and in the inlet molar flow are very low compared to the molar number of the Inert. Hence the reaction volume V is written as follows: (9) V vInert NInert = const and it is supposed to be constant (vInert stands for the molar volume of species Inert). As a consequence of the constraint on the volume, the inlet and outlet volume flow rates are equal. (A5) The reaction rate is described by the mass action law: |ν | rv V = k(T )NA A (10) where k(T ) is the kinetics of the liquid phase reaction.
2. PASSIVITY BASED APPROACH (PBA) Let us consider open chemical systems that are affine in the control input u and whose dynamics is given by the following set of ODE’s: dx = f (x) + g(x) u (1) dt n where x = x(t) ∈ R is the state vector, f (x) ∈ Rn represents the smooth nonlinear function with respect to x, g(x) ∈ Rn×m is the input-state map and u ∈ Rm is the control input. The purpose of the PBA is to find a static state-feedback control u = β(x) such that the closed loop dynamics is a Port Controlled Hamiltonian system (PCH system) (Maschke et al. (2000); Ortega et al. (2002)) with dissipation: ∂Hd (x) dx = Qd (x) (2) dt ∂x where the controlled Hamiltonian storage function Hd (x) has a strict local minimum at the desired equilibrium x ; and Qd (x) = [Jd (x) − Rd (x)] where Jd (x) = −Jd (x)T and Rd (x) = Rd (x)T ≥ 0 are some desired interconnection and damping matrices respectively. The system (2) is passive in the sense that the time derivative T ∂Hd (x) ∂Hd (x) dHd (x) =− Rd (x) (3) dt ∂x ∂x is always negative and the Hamiltonian Hd (x) is bounded from below (van der Schaft (2000); Brogliato et al. (2007)). Consequently, it then plays role of Lyapunov function for stabilization purpose at the desired equilibrium x . The matching equation (EDP) (4) that follows immediately from (1) and (2) has to be solved to find u = β(x): ∂Hd (x) (4) f (x) + g(x)β(x) = Qd (x) ∂x Let consider there exists a full rank left annihilator of g(x) denoted g(x)⊥ such that g(x)⊥ g(x) = 0. If Jd (x), Rd (x) and Hd (x) are chosen such that: ∂Hd (x) (5) g(x)⊥ f (x) = g(x)⊥ Qd (x) ∂x then the control variable is deduced from the state feedback β(x) given by (Ortega et al. (2002)): ∂Hd (x) T T −1 β(x) = g(x)
g(x)g(x)
Qd (x)
∂x
− f (x)
(6)
378
We also use an additional assumption as in (Viel et al. (1997); Alvarez-Ram´ırez and Morales (2000); Antonelli and Astolfi (2003)) or recently (Favache and Dochain (2010)): (A6) Let (Td , NAd , NBd ) be the steady state 2 of the system. The isothermal dynamics (T = Td ) is stable and globally asymptotically converges (e.g. NA → NAd and NB → NBd ). A sufficient condition that verifies this hypothesis (A6) is reaction kinetics k(T ) modelled by the Arrhenius law: −Ea k(T ) = k0 exp( ) (11) RT where Ea and R are the activation energy and the ideal gas constant respectively and k0 is kinetic constant. Some notations are introduced in Tables 1 and 2. 3.1 CSTR modelling As mentioned before, the system dynamics is given by material and internal energy balances. Under the Assumption (A1), the internal energy balance is written using 2
All time derivatives vanish at this state or the time becomes very large, e.g., goes to infinity.
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
Table 1. Notation of the variables of the model Notation hiI hi NAI NBI NInertI NA NB NInert d rv V
unit Jmol−1 Jmol−1 mol mol mol mol mol mol s−1 mols−1
Inlet molar enthalpy of i (i = A, B, Inert) Partial molar enthalpy of i (i = A, B, Inert) Inlet mole number of species A Inlet mole number of species B Inlet mole number of species Inert Mole number of species A Mole number of species B Mole number of species Inert dilution rate Reaction rate
the enthalpy H instead of internal energy U . The balance equations are (Favache et al. (2011)): ⎧ dNA ⎪ ⎪ = d(NAI − NA ) + νA rv V ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ dNB (12) = d(NBI − NB ) + νB rv V ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dH ⎪ ⎩ = d (HI − H) + Q˙ J dt Remark 1. Since we suppose ideality of the mixture, on one side the enthalpy of species A in the mixture can be expressed as: hA (T ) = cpA (T − Tref ) + hAref (a similar expression for species B and Inert) where Tref and hAref are the reference temperature and the reference molar enthalpy respectively of species A. The total enthalpy H of the reaction system can be expressed by (13) H = NA hA (T ) + NB hB (T ) + NInert hInert (T ) Finally the constraint on the volume (see the Assumption (A4)), (9) leads to: dNInert = d(NInertI − NInert ) = 0 (14) dt Thanks to the local equilibrium hypothesis, the energy balance can be written in terms of temperature. This is done by using the expression for the enthalpy H as given in (13), we obtain (see Hoang et al. (2012)): dT Cp (15) = − ∆r H rv V + d(TI − T )CpI + Q˙ J dt where ∆r H = (νB hB (T ) + νA hA (T )) < 0 is the enthalpy of the exothermic reaction and Cp = cpA NA + cpB NB + cpInert NInert is the total heat capacity, respectively. Remark 2. One can use the dynamics of states variables either (NA , NB , H) (12) or (NA , NB , T ) ((12),(15)) to represent completely the behavior of the system. 3.2 Controller synthesis In what follows, we use (NA , NB , T ) to represent the dynamics of the system. We have from ((15), (12)): ⎧ dN A ⎪ = d(NAI − NA ) + νA rv V ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ dNB = d(NBI − NB ) + νB rv V (16) dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ∆r H ⎪ dT CpI 1 ˙ ⎪ ⎩ QJ = rv V + d(TI − T ) + dt Cp Cp Cp 379
Writing (16) into form (1) with u = Q˙ J , we obtain: ⎛ ⎞ d(NAI − NA ) + νA rv V ⎛ 0 ⎞ ⎛ ⎞ NA ⎜ ⎟ ⎟ ⎜ ⎟ d(NBI − NB ) + νB rv V ⎟ ⎜ d ⎜ ⎟ ⎜ 0 ⎟ ⎜ NB ⎟ = ⎜ + ⎜ ⎟ ⎜ ⎟u ⎠ ⎜ dt ⎝ ⎟ ⎝ ⎠ ⎝ − ∆r H ⎠ 1 CpI T rv V + d(TI − T ) Cp Cp Cp x g(x) f (x)
(17)
The proposition 1 proposes a nonlinear state feedback for the jacket temperature to stabilize the reactor at a desired operating point. Proposition 1. Under the Assumption (A2), the system (17) is stabilized at a desired state xd = (NAd , NBd , Td ) with the following equivalent feedback on the jacket temperature TJ : ∂Hd dNB 1 ∂Hd −1 ∂Hd dNA TJ = T + + Cp − λ ∂T ∂N dt ∂NB dt A ∂Hd −KT − − ∆r H rv V + d(TI − T )CpI ∂T (18) where KT > 0 is a tuning parameter. Furthermore, the closed loop dynamics is a passive Hamiltonian system: dx ∂Hd (x) = [Jd (x) − Rd (x)] (19) dt ∂x where: ⎛ ⎞ −1 ⎜ ⎜ ⎜ ⎜ Jd (x) = ⎜ ⎜ ⎜ ⎝ −
0
∂Hd ∂T
0
0 ∂Hd ∂T
−1
0
dNA ∂Hd − dt ∂T
⎛
−1
0 0 0
dNB dt
⎞
∂Hd ∂T
dNA dt
−1
dNB dt
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0 (20)
⎟ ⎜ ⎟ ⎜ Rd (x) = ⎜ 0 0 0 ⎟ (21) ⎠ ⎝ 0 0 KT and Hd (x) is chosen under the following form: Hd (x) = Hd (T, NA , NB ) (T − Td )2 1 + KA (NA − NAd )2 + KB (NB − NBd )2 = 2 (22) where KA ≥ 0 and KB ≥ 0. Proof. We apply the passivity based tools (see Section 2), the full rank left annihilator⎛of g(x)⎞ (17) denoted 1 0 0 ⎟ ⎜ ⎟ ⎜ by g ⊥ (x) is given by: g ⊥ (x) = ⎜ 0 1 0 ⎟. Let us take ⎠ ⎝ 0 0 0 ⎞ ⎛ q11 (x) q12 (x) q13 (x) ⎟ ⎜ ⎟ ⎜ Qd (x) = Jd (x) − Rd (x) = ⎜ q21 (x) q22 (x) q23 (x) ⎟, we ⎠ ⎝ q31 (x) q32 (x) q33 (x) obtain using the matching equation (5):
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
⎛
∂Hd ∂Hd ∂Hd ⎜ q11 (x) ∂NA + q12 (x) ∂NB + q13 (x) ∂T ⎜ ⎜ ⎜ ∂Hd ∂Hd ∂Hd ⎜ + q22 (x) + q23 (x) ⎜ q21 (x) ⎜ ∂NA ∂NB ∂T ⎝ 0
⎞
⎛ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟ ⎝ ⎠
⎞ dNA dt ⎟ ⎟ ⎟ dNB ⎟ ⎟ dt ⎟ ⎠ 0
(23) Due to the negative definiteness of the matrix Qd (x), we choose q11 (x) = q12 (x) = q21 (x) = q22 (x) = 0, −q13 (x) = q31 (x), −q23 (x) = q32 (x) and q33 (x) = −KT < 0. We obtain for solutions: ∂H −1 dN ∂H −1 dN d A d B q13 (x) = , q23 (x) = ∂T dt ∂T dt d q13 (x) and q23 (x) is well defined at the limit ∂H ∂T → 0 (see Remark 3). Finally, the feedback law is derived from (6): ∂Hd −1 ∂Hd dNA ∂Hd −1 ∂Hd dNB − Cp − ∂T ∂T ∂NB dt ∂NA dt ∂Hd − ∆r H rv V + d(TI − T )CpI = u −KT − ∂T Using the Assumption (A2) with u = Q˙ J , it leads to the feedback law (18). Let us notice that a possible choice for Hd (x) for which the feedback law (18) well-defined is given in (22). The function Hd (x) is positive and its time derivative, ∂H 2 dHd (x) d = −KT <0 (24) dt ∂T The latter ends the proof. 2 Remark 3. The closed loop dynamics for the temperature T with the feedback law (18) can be written as follows: −1 −1 dT =− dt
∂Hd ∂T
dNA ∂Hd − dt ∂NA
∂Hd ∂T
the feedback law may present a different shape in terms of the amplitude and variation rate. Remark 5. Let us consider the CSTR networks with n species Sj (j = 1 . . . n) and r chemical reactions taking place in the reactor. We then have: n νij Sj = 0, i = 1 . . . r (26) j=1
where νij is the signed stoichiometric coefficient of species j as it enters in reaction i (Hoang et al. (2011)). For such a system and under the isobaric conditions (A1), the mathematical modeling of dynamics (17) can be generally rewritten (Hoang et al. (2012); Favache and Dochain (2010)): ⎛ ⎞ r d(N − N ) + ν r V 1 j1 vj 1I ⎜ ⎟ ⎜ ⎟ j=1 ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ ⎜ ⎟ r N1 ⎜ ⎟ ⎜ ⎟ d(N − N ) + ν r V 2 j2 vj 2I ⎜ ⎟ ⎜ ⎟ ⎜ N2 ⎟ ⎜ ⎟ j=1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ d ⎜ ⎜ ⎟ . ⎜ .. ⎟ = ⎜ . ⎟ . ⎟ . dt ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ r ⎜ Nn ⎟ ⎜ ⎟ ⎝ ⎠ ⎜ ⎟ d(NnI − Nn ) + νjn rvj V ⎜ ⎟ ⎜ ⎟ T j=1 ⎟ ⎜ ⎜ ⎟ x ⎜ r ⎟ ⎝ j=1 − ∆rj H rvj V ⎠ C
Cp
+ d(TI − T )
f (x)
dNB ∂Hd ∂Hd − KT dt ∂NB ∂T (25)
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ +⎜ ⎜ ⎜ ⎜ ⎜ ⎝
First let us note that from Hd (x) (22), ∂Hd → 0 ⇔ T → Td ∂T ∂Hd ∂Hd and lim ∂N → 0, lim ∂N → 0. Second we have A B T →Td
T →Td
from (22) Hd (T = Td , NA , NB ) = 0 and from (24) dHd = 0. It is shown that Hd (x) is a dt
T →Td
∂Hd →0 ∂T
⎧ ∂H −1 dN d A ⎪ ⎪ lim <∞ ⎪ ⎨ ∂Hd →0 ∂T dt ∂T ∂H −1 dN ⇒ d B ⎪ ⎪ lim <∞ ⎪ ⎩ ∂Hd ∂T dt →0
0 0 .. . 0 1 Cp
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ˙ QJ ⎟ ⎟ ⎟ u ⎟ ⎟ ⎠
g(x)
(27)
where rvj V and ∆rj H represent the reaction rate and the enthalpy of the chemical reaction j (j = 1 . . . r). By using the same procedure as previous sim in the and ple case, let us note Q(x) = qij (x) i,j=1...(n+1)
∂T
Remark 4. If we take KA = KB = 0 in Hd (x) (22), the feedback law (18) is then similar to the one proposed by Viel (Viel et al. (1997)) in the case without constraint on the control input (see the equation (3) in (Viel et al. (1997)) for feedback law and adapt it to the system under consideration). The stabilization obtained by these strategies is dominated by the regulation of the thermal part inside the reactor in accordance with the Assumption (A6). In other cases (KA × KB = 0), due to presence of the additional material contributions (see (18) and (22)), 380
(T =Td ,NA ,NB )
Lyapunov function for the stabilization of the temperature T inside the reactor. As a consequence, we have to obtain dT lim dT dt = lim dt = 0 and thus we deduce from (25):
⎛
pI
Cp
g ⊥ (x) = diag ( 1, . . . , 1, 0 ) ∈ R(n+1)×(n+1) . The following matching equation which generalizes (23) can be derived from (5) : ⎧ dN1 ∂Hd ∂Hd ∂Hd ⎪ ⎪ q (x) = + . . . + q1n (x) + q1(n+1) (x) ⎪ ⎪ 11 ∂N1 ∂Nn ∂T dt ⎨ .. . ⎪ ⎪ ⎪ dNn ∂H ∂H ∂Hd d d ⎪ ⎩ qn1 (x) = + . . . + qnn (x) + qn(n+1) (x) ∂N1 ∂Nn ∂T dt The number of equations equals n with n×(n+1) unknown variables qij (x). Hence this system has an infinite number of solutions and a simple solution can be found as follows: • qij (x) = qji (x) = 0 for i, j = 1 . . . n
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
−1 dNi d • then qi(n+1) (x) = −q(n+1)i (x) = ∂H ∂T dt for i = 1 . . . n, • and let we choose q(n+1)(n+1) (x) = −KT From this, we obtain for the matrix Q(x) ≡ Jd (x) − Rd (x) , where: ⎛ ∂Hd −1 dN1 ⎞ 0
⎜ ⎜ ⎜ . . ⎜ . ⎜ Jd (x) = ⎜ ⎜ ⎜ 0 ⎜ ⎝ ∂Hd −1 dN1 −
and
∂T
dt
...
0
..
.
. . .
...
0
... −
⎛
0 ⎜ .. . Rd (x) = ⎜ ⎝0 0
∂T . . .
∂Hd −1 dNn
∂Hd −1 dNn ∂T
... .. . ... ...
dt
∂T
dt
dt
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
0
⎞ 0 0 .. .. ⎟ . . ⎟ 0 0 ⎠ 0 KT
(28)
(29)
The feedback law for u (27) can be found using (6) with Jd (x) (28), Rd (x) (29) and Hd (x) is chosen to be proportional to: n (T − Td )2 (30) 1+ Ki (Ni − Nid )2 Hd (x) = 2 i=1
where Ki ≥ 0. Finally, let us notice that the proposed passive controller stabilizes the reactor as soon as the Assumption (A6) holds 3 (see also Antonelli and Astolfi (2003)). 3.3 A state observer We now consider that only T is measured. As a consequence we have to build an observer to reconstruct the mixture composition. The main feature of the proposed estimator is that it is independent of the knowledge of the system kinetics and is called asymptotic observers. These asymptotic observers are first proposed in (Dochain et al. (1992, 2009)) for a nonlinear class of CSTR networks without considering feedback law. In what follows, we shall show that with or without feedback law, the convergence of estimated state variables to their exact values is exponential. Let us consider the original system (12): ⎧ dH ⎪ ⎪ = Q˙ + d(HI − H) ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎨ dNA Σ = d(NAI − NA ) + νA rv V ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ dN ⎪ B ⎪ ⎩ = d(NBI − NB ) + νB rv V dt
(31)
3
In general, (A6) verifies for a large range of non linear reaction systems in which the (high-order) mass action laws take place, rvj V = kj (T )
n
α
Nk jk , j = 1 . . . r
k=1
where Arrhenius based reaction kinetics kj (T ) is considered.
381
Doing a variable transformation N = νB N A − νA N B (32) and using the second and third equations in (31), we obtain: dN = d(NI − N ) dt It is clear that the dynamics of new variable N does not depend on the reaction kinetics. The same situation holds for the dynamics of the enthalpy H (see the first equation in (31)). As a consequence, we then an asymptotic derive ˆ (33) for the system Σ (31): observer Σ ⎧ ˆ ⎪ ⎪ dH = Q˙ + d(HI − H) ˆ ⎨ dt ⎪ ˆ Σ (33) ⎪ ⎪ ˆ ⎪ d N ⎩ ˆ) = d(NI − N dt ˆ = hA (T )N ˆ A + hB (T )N ˆ B + hInert (T )N ˆInert and where H ˆ ˆ ˆ N = νB NA − νA NB are estimated values of the enthalpy H and the new variable N , respectively. We assume here that the reactor temperature T is the only available on-line measurement. Remark 6. The Inert doesn’t participate to the reaction (7) but its presence in the reactor should be considered (see (13) for total enthalpy for example). The following ˆInert of equation presents the dynamics of the estimate N NInertI ˆInert dN ˆInert ) = d(NInertI − N (34) dt Proposition 2. Closed loop stability using observer. ˆ ˆ The estimations NA of NA and NB of NB defined from ˆ Σ , converges to NA and NB respectively with time 1 constant τ = 2d . Furthermore the system Σ controlled by the feedback law (18) coupled with the state observer ˆ is exponentially stable. Σ
1
⎛
ˆ −H H
⎞
⎠. By = ⎝ ˆ −N 2 N subtracting (33) to (31), one can get: d 0 1 d =− (35) dt 0 d 2 The dynamics of is then presented in a port Hamiltonian format (2), d = [J () − R ()] ∂H∂ () where J () = 0, dt d 0 R () = and the Hamiltonian storage function 0 d H () = 12 T ≥ 0. The last one plays a role of a Lyapunov function for the stability of the zero dynamics of because: ∂H () T d 0 ∂H () dH =− ≤0 dt ∂ ∂ 0 d Furthermore, it can be rewritten as follows: − 1t dH = −d T = −2d H ⇒ H (t) = H (0) exp 2d dt Because of d > 0, H (t) exponentially converges to 0 with 1 the time constant τ = 2d . As a consequence, (t) → 0 at Proof. Let us define (t) =
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
Details on analysis of the steady states can be found in (Hoang et al. (2012)). Its shows that the system has three stationary operating points. Simulations are given in Fig. 1 with the initial conditions (C1) and (C2). 0.18
P
2
0.12 0.1 0.08 0.06 0.04
ˆ B → NB ⎩ lim N
P3
0.02
t<+∞
0
It is important to notice that the convergence does not depend on the feedback strategy. The latter ends the proof. 2 4. SIMULATION In this section, the exothermic reaction under consideration (7) is treated for simulation with νA = −1 and νB = 1 (e.g. the first order reaction, this type of reaction is often appeared in the literature (Viel et al. (1997); Favache and Dochain (2010)). The reaction rate is then rewritten as follows (see Assumption (A5)): −Ea rv V = k0 exp( )NA RT In Table (2), we give numerical values used for simulation.
300
320
340
360
380
400
T (K)
Fig. 1. The representation of the open loop phase plane In the next section, we propose to operate the system around the unstable middle point P2 using the feedback law for the jacket temperature TJ (18). 4.2 Closed loop simulation In a first instance we consider the state variables are measured, e.g. the system is closed using the proposed state feedback law without observer. We take KT = 0.001, KA = 0 and KB = 0 for the shaped function Hd (x) (22). In Figure 2 the closed loop phase plane is represented. We can see that for all the considered initial conditions the system converges to the desired operating point P2 . Furthermore Figure 2 also shows that the control variable solicitations TJ (18) are physically admissible in terms of amplitude and dynamics.
Table 2. Parameters of CSTR
Heat capacity of species A Heat capacity of species B Heat capacity of Inert Activation energy Reference enthalpy of A Reference enthalpy of B Reference enthalpy of Inert kinetic constant gas constant Reference temperature heat transfer coefficient Dilution rate
0.18
P
with (C1) with (C2)
1
0.16 0.14
P2
0.12
NA (mol)
Numerical value 221.9 (JK −1 mol−1 ) 128.464 (JK −1 mol−1 ) 21.694 (JK −1 mol−1 ) 73.35 (KJmol−1 ) −5.8085 105 (Jmol−1 ) −6.6884 105 (Jmol−1 ) −3.3 105 (Jmol−1 ) 2.58 109 (s−1 ) 8.314 (JK −1 mol−1 ) 298 (K) 0.75 (W K −1 ) 0.0070 (s−1 )
with (C1) with (C2)
0.14
t<+∞
cpA cpB cpInert Ea hAref hBref hInertref k0 R Tref λ d
P1
0.16
NA (mol)
the finite time (t > 3τ ) and at that time we obtain (with Remark 6): ⎞ ⎛ ˆ A − NA N hA (T ) hB (T ) ⎠=0 ⎝ lim t<+∞ ˆ νB −νA NB − NB hA (T ) hB (T ) Because Det = −νA hA (T )−νB hB (T ) = νB −νA −∆r H > 0 (for exothermic reaction under consideration), the matrix is full rank. As a consequence, ⎧ ˆ A → NA ⎨ lim N
0.1 0.08 0.06 0.04 0.02
4.1 Open loop simulation First of all let us consider open loop simulation with input defined by (36) ! TI = TJ = 298 (K), NAI = 0.18 (mol), (36) NBI = 0 (mol), NInertI = 3.57 (mol) 382
0 290
300
310
320
330
340
T (K) 306 with (C1) with (C2)
304
J
Jacket temperature T (K)
The goal of this section is to illustrate the efficacy of the aforementioned control strategy with or without the state observer for stabilization purpose. The open and closed loop simulations are carried out with respect to two different initial conditions, (C1) with T0 = 340 (K), NA0 = 0.04 (mol), NB0 = 0.001 (mol) and (C2) with T0 = 300 (K), NA0 = 0.15 (mol), NB0 = 0.03 (mol) .
302 300 298 296 294 292 290 288 0
500
1000
1500 time (s)
2000
2500
3000
Fig. 2. Representation of the closed loop phase plane (the point P3 outside the frame) and the feedback law TJ For any initial conditions, Hd (x) (22) plays the role of Lyapunov function and consequently it converges to 0 as shown in Figure 3.
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
300 with (C1) with (C2)
Controlled function Hd
250
200
150
100
50
0 0
500
1000
1500 time (s)
2000
2500
Fig. 5. System in closed loop with the asymptotic observer
3000
Fig. 3. The dynamics of Hd Remark 7. The convergence speed of the controlled system goes faster by increasing the tuning parameter KT .
In fact, the convergence of the controlled system coupled with asymptotic observer is more interesting. For the ˆ initial conditions (C2) and (C1), the simulations are given in Figure 6. (a)
4.3 Simulation with the asymptotic observer
NA exact
0.16
NA observed
0.14 0.12 0.1 0
500
1000
1500 time (s)
2000
2500
3000
0.08
NB (mol)
For the sake of simplicity, the initial condition (C2) is reused for the system. The initial conditions of the asymp ˆ ˆ totic observer are (C1) with H(0) = 0.98H(T0, NA0 , NA0 ), ˆ (0) = 0.75N (NA0, NB0 ) and (C2) ˆ ˆ N with H(0) = ˆ (0) = 0.95N (NA0 , NB0 ) where H(0.85T0 , NA0 , NB0 ), N the numerical values of T0 , NA0 and NB0 are given with the initial condition (C2).
NA (mol)
0.18
0.06 0.04
0 0
In open loop, the convergence of the system coupled with asymptotic observer is illustrated in Fig. 4. With the initial condition (C2), the system normally converges to the stable point P1 .
NB exact
0.02
NB observed 500
1000
1500 time (s)
2000
2500
3000
(b) 304
303
302
T (K)
(a) 301
0.16 N 0.14 0.12 0.1 0
exact
300
NA observed
299
A
A
N (mol)
J
0.18
200
400
600 time (s)
800
1000
298 0
1200
1000
2000 3000 time (s)
4000
5000
NB (mol)
0.03 N
exact
N
observed
B
0.025
B
0.02
Fig. 6. NA and its estimate with the asymptotic observer in the closed loop case -(a) for the initial conditions ˆ (C2) and (C1) -(b) the control input TJ with the asymptotic observer
0.015 0.01 0
200
400
600 time (s)
800
1000
1200
(b)
NA (mol)
0.2
ˆ For the initial conditions (C2) and (C2), the simulations are illustrated in Fig. 7. The simulations (Figures 6(a) and 7(a)) show that the stabilization at the unstable point P2 of the controlled system via the state observer is insured. Furthermore, Tw is still admissible (see Fig. 6(b) and Fig. 7(b)).
0.15 NA exact
0.1
N
0.05 0 0
A
200
400
600 time (s)
800
observed
1000
1200
NB (mol)
0.2 N
exact
N
observed
B
0.15
B
0.1
5. CONCLUSION
0.05 0 0
200
400
600 time (s)
800
1000
1200
Fig. 4. NA , NB and their estimates in the open loop case ˆ -(b) for -(a) for the initial conditions (C2) and (C1) ˆ initial conditions (C2) and (C2) In closed loop, the controlled system Σ and the asymp " are coupled as in Fig. 5. L represents totic observer Σ the feedback law (18). 383
In this work, we have shown by means of passivity based method: (1) how to stabilize the CSTR networks about a desired operating point. The obtained feedback law covers the one proposed by (Viel et al. (1997)) in the case without constraint on the control input for a simple first order chemical reaction. (2) Afterwards for practical implementation of the controlled system, we have coupled the controller with a state observer.
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
(a)
NA (mol)
0.2 0.15 0.1
N exact A
0.05 0 0
N observed A
500
1000
1500 time (s)
2000
2500
3000
NB (mol)
0.2 NB exact
0.15
NB observed
0.1 0.05 0 0
500
1000
1500 time (s)
2000
2500
3000
(b) 310
308
J
T (K)
306
304
302
300
298 0
1000
2000 3000 time (s)
4000
5000
Fig. 7. NA and its estimate with the asymptotic observer in the closed loop case -(a) for the initial conditions ˆ -(b) the control input TJ with the (C2) and and (C2) asymptotic observer The closed loop convergence of the system with/without the observer is theoretically guaranteed. The simulation results showed that convergence objective is satisfied and that the state feedback law is physically implementable since jacket temperature TJ remains in some physical domain with admissible rate of variation. It remains now: • to evaluate performances and robustness of the obtained results in terms of perturbations and parameters uncertainty. • to introduce the dynamical modeling of the jacket into the problem. • to extend the proposed method to inventory control (Farschman et al. (1998)). REFERENCES D¨orfler, F. and Johnsen, J. K. and Allg¨ower, F. (2009). An introduction to interconnection and damping assignment passivity-based control in process engineering. J. Proc. Control, 19, 1413-1426. Favache, A. and Dochain, D. (2010). Power-shaping of reaction systems : the CSTR case study. Automatica, 46(11), 1877-1883. Favache, A. and Dochain, D. and Winkin, J. (2011). Power-shaping control: writing the system dynamics into the Brayton-Moser form. Syst. Cont. Let., 60(8), 618-624. Hoang, H. and Couenne, F. and Jallut, C. and Le Gorrec, Y. (2011). The Port Hamiltonian approach to modeling and control of Continuous Stirred Tank Reactors. J. Proc. Control, 21(10), 1449-1458. Hoang, H. and Couenne, F. and Jallut, C. and Le Gorrec, Y. (2012). Lyapunov-based control of non isothermal continuous stirred tank reactors using irreversible thermodynamics. J. Proc. Control, 22(2):412-422. 384
Alvarez J., J. Alvarez-Ram´ırez, G. Espinosa-Perez and A. Schaum (2011). Energy shaping plus damping injection control for a class of chemical reactors. Chem. Eng. Sci., 66(23), 6280-6286. Luyben, W. L. (1990). Process modeling, simulation, and control for chemical engineers. McGraw-Hill, Singapore. Maschke, B. and Ortega, R. and van der Schaft, A. (2000). Energy based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans. on Autom. Control, 45(8), 1498-1502. Ortega, R. and van der Schaft, A. and Maschke, B. and Escobar, G. (2002). Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian Systems. Automatica, 38, 585-596. Ram´ırez, H. and Sbarbaro, D. and Ortega, R. (2009). On the control of non-linear processes: An IDA-PBC approach. J. Proc. Control, 19, 405-414. van der Schaft, A. (2000). L2 -gain and passivity techniques in nonlinear control. Springer-Verlag, London, 2nd edt. Brogliato, B. and Lozano, R. and Maschke, B. and Egeland, O. (2007). Dissipative systems analysis and control. Springer, London, 2nd edition. Alvarez-Ram´ırez, J. and Morales, A. (2000). PI control of continuously stirred tank reactors: stability and performance, Chem. Eng. Sci., 55(22):5497-5507. Antonelli, R. and Astolfi, A. (2003). Continuous stirred tank reactors: easy to stabilise?. Automatica, Vol. 39, pp. 1817-1827. Viel, F. and Jadot, F. and Bastin, G. (1997). Global stabilization of exothermic chemical reactors under input constraints. Automatica, Vol. 33, No. 8, pp. 1437-1448. Bruns, D. D. and Bailey, J. E (1975). Process operation near an unstable steady state using nonlinear feedback control. Chem. Eng. Sci., Vol. 30, pp. 755-762. Dochain, D. and Perrier, M. and Ydstie, B. E. (1992). Asymptotic observers for stirred tank reactors. Chem. Eng. Sci., Vol. 47, No. 15/16, pp. 4167-4117. Alvarez-Ram´ırez, J. (1995). Observers for a class of continuous tank reactors via temperature measurement. Chem. Eng. Sci., Vol. 55, No. 9, pp. 1393-1399. Dochain, D. and Couenne, F. and Jallut, C. (2009). Enthalpy Based Modelling and Design of Asymptotic Observers for Chemical Reactors. International Journal of Control, Vol. 82, No. 8, pp. 1389-1403. Soroush, M. (1997). Nonlinear state-observer design with application to reactors. Chem. Eng. Sci., Vol. 52, No. 3, pp. 387-404. Gibon-Fargeot, A. M. and Hammouri, H. and Celle, F. (1994). Nonlinear observers for chemical reactors. Chem. Eng. Sci., Vol. 49, No. 14, pp. 2287-2300. Du, J. and Laird, C. M. and Ydstie, B. E. (2010). The measurement selection of inventory control. American Control Conference, Marriott Waterfront, Baltimore, MD, USA June 30-July 02. Hangos, K. M. and Bokor, J. and Szederk´enyi, G. (2001). Hamiltonian view on process systems, AIChE journal, vol. 47, no 8, pp. 1819-1831. Farschman, C. A. and Viswanath, K. P. and Ydstie, B. E. (1998). Process systems and inventory control. AIChE journal, 44:8, pp. 1841-1857.