Stabilization of (state, input)-disturbed CSTRs through the port-Hamiltonian systems approach
aa r X i v : . [ m a t h . O C ] J un Stabilization of (state, input)-disturbed CSTRs through theport-Hamiltonian systems approach
Yafei Lu, Zhou Fang, Chuanhou Gao
Abstract
It is a universal phenomenon that the state and input of the continuous stirred tank reactor (CSTR) systems are both disturbed.This paper proposes a (state, input)-disturbed port-Hamiltonian framework that can be used to model and further designs astochastic passivity based controller to asymptotically stabilize in probability the (state, input)-disturbed CSTR (sidCSTR)systems. The opposite entropy function and the availability function are selected as the Hamiltonian for the model andcontrol purposes, respectively. Furthermore, the proposed (state, input)-disturbed port-Hamiltonian model can simultaneouslycharacterize the first law and the second law of thermodynamics when the opposite entropy function acts as the Hamiltonian. Asimple CSTR example illustrates how the port-Hamiltonian method is utilized for modeling and controlling sidCSTR systems.
Key words:
CSTR, port-Hamiltonian, stochastic passivity, entropy, (state, input)-disturbed
Continuous stirred tank reactors (CSTRs) are a classof widely used continuous reactors in chemical indus-trial processes. The operation of a CSTR takes placewith reactants fed continuously into the tank throughports at the top while products removed continuouslyfrom the bottom, shown in Fig. 1. A basic assumptionfor this device is that it admits perfect mixing so thatthe contents, like temperature, concentrations, etc., haveuniform properties throughout the tank, and moreover,the exit stream has the same contents as in the tank.Even so, the CSTR process still exhibits highly complexbehaviors due to nonlinear coupling among irreversiblethermodynamics, reaction kinetics and hydrodynamics,characterized by multiple equilibrium points, instabil-ity, lack of precise physical interpretation, etc. (Favache& Dochain, 2009). It thus poses a great challenge to in-vestigate CSTRs for both academical research and en-gineering applications.Sustainable production concepts require the CSTRprocess to run with optimal performance, which oftenresults from the unstable operation conditions (Hoang,Couenne, Jallut, & Gorrec, 2008). This motivates thestabilization of CSTR around an unstable steady state tobecome an active research issue during the past decades. ⋆ The authors are with the School of MathematicalSciences, Zhejiang University, Hangzhou 310027, China(e-mail: [email protected]; zhou [email protected]; [email protected] (Correspondence: C. H. Gao)).
Motor Feed Cooling jacketAgitatorMixed product
Fig. 1. Schematic diagram of a CSTR.
The early stabilizing strategies include input/outputfeedback linearization (Adebekun &Schork, 1991) andnonlinear PI control algorithm (Alvarez-Ramirez & Fe-mat, 1999) while the recent studies focus on linkingthermodynamics with mathematical system theory (Yd-stie & Alonso, 1997, Alonso & Ydstie,2001, Ruszkowski,Garcia-Osorio, & Ydstie, 2005, Favache & Dochain,2009). The purpose of the latter strategy is to developa entropy-based Lyapunov function, a notion derivedfrom thermodynamics concepts, for transport reactionsystems, of which the CSTR is the most typical rep-resentative. Ydstie and his coworkers (1997,2001,2005)firstly connected the thermodynamic system with thepassivity theory of nonlinear control through taking theLegendre transform of the entropy difference between
Preprint submitted to Automatica 7 October 2018 ifferent states as the storage function, and then devel-oped the inventory (like total mass, energy or the holdupof a chemical constituent) control approach based onthe theory of dissipative systems to reach stabilization.Favache and Dochain (2009) performed a detailed studyon finding Lyapunov function candidates related to thethermodynamics, including the entropy, the entropyproduction and the internal energy. Their applicablescopes were also exhibited, such as the entropy onlysuitable for isolated systems, the entropy productionemphasized with linear phenomenological laws, etc.Another class of investigations (Eberard, Maschke, &van der Schaft, 2007, Favache, Dochain, & Maschke,2010, Ramirez, Maschke, & Sbarbaro, 2013, Ramrez,Gorrec, Maschke, & Couenne, 2016, Hoang, Couenne,Jallut, & Gorrec, 2011, Hoang, Couenne, Jallut, & Gor-rec, 2012) manage to formulate the dynamics of theCSTR into a structure of a port/quasi-port Hamilto-nian system with the main motivation of expressing thefirst and second principle of thermodynamics simulta-neously. However, this good imagination sensitively de-pends on the selection of the Hamiltonian. The negativeentropy as the Hamiltonian fails to support the imagi-nation (Hoang, Couenne, Jallut, & Gorrec, 2011) whilethe internal energy based availability function as theHamiltonian supports it (Ramrez, Gorrec, Maschke, &Couenne, 2016). The stabilization of the CSTR systemin the form of the port/quasi-port Hamiltonian struc-ture may be reached through relating the Hamiltonianstructure to the passivity control theory (van der Schaft,2000, Alonso & Ydstie,2001, van der Schaft, 2004).Despite extensive studies made towards stabilizingthe CSTR systems in the literature, there is no uni-fied method applicable to a wide class of transport reac-tion systems. Their control as well as the link betweenthermodynamics and the Lyapunov stability theory re-mains an open problem (Favache & Dochain, 2009). Thecircumstance will get worse when the stochastic phe-nomena in the CSTR process have to be considered. Infact, it is highly possible that the CSTR process hasuncertainties, such as material compositions change, in-put/output flow fluctuation, external environment tem-perature fluctuation, reaction kinetics uncertainty, etc.Frankly speaking, the stochastic model can capture thenature of the CSTR process better. However, little atten-tion is paid to stochastic CSTR processes. Even thoughthe kinetic uncertainty is considered in stabilizing theCSTR process, which acts as an example of the stochas-tic nonlinear time-delay system (Liu, Yin, Zhang, Yin,& Yan, 2017), the input disturbance is often ignored. Forthis reason, the current work is devoted to stabilizingin probability the CSTR systems with state and inputboth disturbed. By utilizing the (state, input)-disturbedstochastic port-Hamiltonian system (sidSPHS) frame-work (Fang & Gao, 2017), we can rewrite the (state,input)-disturbed CSTR (sidCSTR) model to be a sid-SPHS with the opposite entropy function as the Hamil-tonian. A notable advantage for this rewritten version is that it can characterize both the first law and secondlaw of thermodynamics. Further, by means of the im-proved stochastic generalized canonical transformation T (Satoh & Fujimoto, 2013, Fang & Gao, 2017), therewritten model is transformed to another dynamicallyequivalent sidSPHS with the availability function as thenew Hamiltonian. The latter exhibits stochastic passiv-ity with respect to the new Hamiltonian, based on whicha stochastic passivity based controller is developed toasymptotically stabilize the transformed sidSPHS sys-tem as well as the original sidCSTR system in probabil-ity.The rest of this paper is organized as follows. Section2 reviews some basic concepts about stochastic stabilityand sidSPHS framework. This is followed by the formu-lation of sidCSTRs into a sidSPHS in Section 3. Sec-tion 4 develops a stochastic passivity based controller toasymptotically stabilize sidCSTR systems in probabil-ity. A sidCSTR system with first-order reaction acts as acase study to illustrate the proposed stabilizing strategyin Section 5. Finally, Section 6 concludes the paper. Notion: R n , R ≥ R + represent the space of n-dimensional real vectors, the set of nonnegative realnumbers and positive real numbers respectively ; R n × m represents the set of all ( n × m ) real matrices ; Z + rep-resents the set of positive integers; n and n representn-dimensional vector with all entries equal to 0 and 1,respectively; E( · ) represents mathematical expectation; | · | and k · k represent absolute value and 2-norm re-spectively; tr { A } , A · i and A j · represent the trace ofmatrix A , the i th column and the j th row of matrix A respectively; x ⊤ represents the transpose of the vector(or matrix) x ; n × n represents ( n × n ) matrix withall entries equal to 1; diag( x , · · · , x n ) represents the( n × n ) diagonal matrix whose diagonal elements are x j ( j = 1 , · · · , n ) while other elements equal to 0; ∇ ( · )represents the gradients of ( · ). In this section, some basic concepts and results aboutstability in the sense of probability and (state, input)-disturbed SPHSs are recalled.
Consider the input/output stochastic differential sys-tem in R n × R m written in the sense of Itˆo: ( d x = f ( x , u )d t + a ( x )d ω , y = b ( x , u ) , (1)where21) x ∈ R n , u , y ∈ R m ( m ≤ n ) are the state, measur-able input and output, respectively, t ∈ R ≥ is thetime;(2) f : R n × R m R n , a : R n R n × r and b : R n × R m R m are all local Lipschitz continuousfunctions such that they vanish in the equilibrium,denoted by x ∗ , usually set as the origin, in the un-forced case.(3) ω is an R r -valued standard Wiener process definedon a complete probability space.The stability of the equilibrium of the stochastic differ-ential equation (1) in the sense of probability is charac-terized by the following definition (Khasminskii, 2011). Definition 1
The equilibrium solution x ∗ of the stochas-tic differential equation (1) is(1) stable in probability if ∀ ǫ> there is lim x (0) → x ∗ P(sup t ≥ k x ( t ) − x ∗ k < ǫ ) = 1; (2) locally asymptotically stable in probability if lim x (0) → x ∗ P( lim t →∞ k x ( t ) − x ∗ k = 0) = 1; (3) globally asymptotically stable in probability if ∀ x (0) ∈ R n there is P( lim t →∞ k x ( t ) − x ∗ k = 0) = 1 , where x (0) is the initial state and P( · ) represents theprobability function. Besides definition, stochastic Lyapunov theorem is afrequently-used tool to render the stability in probabil-ity. Another highly related notation is stochastic passiv-ity (Florchinger, 1999) that plays an important role instabilizing stochastic systems, defined below.
Definition 2
For an input-output stochastic system inthe sense of It ˆo , governed by Eq. (1), if there exists anon-negative and twice differentiable storage function V : R n R ≥ such that L [ V ( x )] ≤ y ⊤ u , ∀ x ∈ R n , (2) then the system is said to be stochastically passive withrespect to V ( · ) , where L [ · ] is the infinitesimal generatorof the stochastic process solution x ( t ) of the stochasticdifferential equation (1), defined by L [ V ( x )] = ∇ ⊤ V ( x ) f ( x , u ) + 12 tr (cid:26) ∂ V ( x ) ∂ x a ( x ) a ( x ) ⊤ (cid:27) . If the storage function V ( · ) is positive definite withrespect to x − x ∗ , it is easy to realize stabilization of thestochastic system (1) in the sense of probability throughconnecting it with a proportional controller in negativefeedback (Fang & Gao, 2017). A special case of the stochastic system (1) is thestochastic port-Hamiltonian system proposed by Satoh& Fujimoto (2013). Essentially, both of them only cap-ture the process noise, but fail to consider the inputnoise. As an extension, the sidSPHS (Fang & Gao, 2017)can capture both process noise and input noise, whichis of the form d x = (cid:2) ( J ( x ) − R ( x )) ∇ H ( x ) + g ( x ) u (cid:3) d t + a ( x )d ω + γ ( x )˜ u σ ( x )d ω , y = g ⊤ ( x ) ∇ H ( x ) + δ ( x ) u , (3)where(1) J , R : R n R n × n , with J = − J ⊤ , R = R ⊤ (cid:23) ω and ω are two mutually independent R r -valued and R r -valued standard Wiener processesdefined on a probability space, respectively;(3) H : R n R is the Hamiltonian representingthe total storied energy, g : R n R n × m de-scribes the control port, u = ( u , · · · , u m ) ⊤ isthe control action with the expectation satisfyingE( R t k u ( τ ) k dτ ) ≤ ∞ for all t ∈ R ≥ , a : R n R n × r is the process noise port, and γ ( x )˜ u σ ( x )with γ : R n R n × m , ˜ u = diag( u , · · · , u m ) and σ : R n R m × r is the input drift term;(4) δ ( x ) u , with δ : R n R m × m , δ (cid:23) k δ ( x ) k F <
1, measuresthe contribution of the input to output.In addition, the diffusion term (cid:2) ( J ( x ) − R ( x )) ∇ H ( x ) + g ( x ) u (cid:3) and the drift terms a ( x ), γ ( x )˜ u σ ( x ) also satisfylocal Lipschitz continuity and vanish at x ∗ if no drivingforce is imposed. All these conditions guarantee that thestochastic differential equation (3) admits a unique so-lution given any initial state and the stochastic versionof Lyapunov theorem can be applied.For the structure of sidSPHSs in Eq. (3), there are twopoints that need to be noted. One is that the Hamilto-nian H is assumed to be neither positive semi-definitenor bounded from below, the other is that there is anoutput feedthrough so that this equation looks not be-longing to the class of port-Hamiltonian systems. In fact,the form of Eq. (3) is indeed a port-Hamiltonian system,3hich can be verified by interconnecting two sidSPHSsin negative feedback to yield a new sidSPHS. Proposition 1
The negative feedback interconnectionof any two sidSPHSs given by Eq. (3) yields anothersidSPHS.
Proof.
We use subscripts “1” and “2” to identify twosidSPHSs, respectively. For simplicity but without lossof generalization, the state, input and output of thesetwo systems are assumed to satisfy dim( x ) = dim( x )and dim( u ) = dim( u ) = dim( y ) = dim( y ). The in-terconnection of these two systems in negative feedbackfollows u u ! = − II ! y y ! = − II ! g ( x ) ⊤ ∂H ( x ) ∂ x + δ ( x ) u g ( x ) ⊤ ∂H ( x ) ∂ x + δ ( x ) u ! , where I is the identity matrix with suitable dimension.Since k δ ( x ) k F < k δ ( x ) k F <
1, we have k δ ( x ) δ ( x ) k F ≤ k δ ( x ) k F k δ ( x ) k F < . Hence, I + δ ( x ) δ ( x ) and I + δ ( x ) δ ( x ) are bothnonsingular. The interconnection equation changes to be u u ! = − δ (cid:0) I + δ δ (cid:1) − − (cid:0) I + δ δ (cid:1) − (cid:0) I + δ δ (cid:1) − − δ (cid:0) I + δ δ (cid:1) − ! × g ( x ) ⊤ g ( x ) ⊤ ! ∂H ( x ) ∂ x ∂H ( x ) ∂ x ! . Combining this equation with Eq.(3) yieldsd x x ! = h ˜ J − ˜ R i ∂H ∂ x ∂H ∂ x ! d t + a ( x ) a ( x ) ! d ω + γ ( x ) 00 γ ( x ) ! ˜ u
00 ˜ u ! σ ( x ) σ ( x ) ! d ω with˜ J = g ( x ) g ( x ) ! − (cid:0) I + δ δ (cid:1) − (cid:0) I + δ δ (cid:1) − ! × g ⊤ ( x ) g ⊤ ( x ) ! + J ( x ) J ( x ) ! and˜ R = g ( x ) g ( x ) ! δ (cid:0) I + δ δ (cid:1) − δ (cid:0) I + δ δ (cid:1) − ! × g ⊤ ( x ) g ⊤ ( x ) ! + R ( x ) R ( x ) ! . Note that δ ( x ) = δ ⊤ ( x ) and δ ( x ) = δ ⊤ ( x ), we get˜ J = − ˜ J ⊤ , ˜ R = ˜ R ⊤ (cid:23)
0. Namely, the above intercon-nection yields another sidSPHS with the Hamiltonian tobe H ( x ) + H ( x ). ✷ As for the first point related to the Hamiltonian, if itis bounded from below, the sidSPHS in the form of Eq.(3) can reach power balance, or be stochastically passivewith respect to the Hamiltonian with some moderateconditions added.
Theorem 1 (Fang & Gao, 2017) A sidSPHS describedby Eq. (3) is stochastically passive with respect to H ( x ) ifand only if ∀ x the Hamiltonian satisfies H ( x ) ≥ H ( n ) =0 ,
12 tr (cid:26) ∂ H ( x ) ∂ x a ( x ) a ⊤ ( x ) (cid:27) ≤ ∂H ( x ) ∂ x ⊤ R ( x ) ∂H ( x ) ∂ x (4) and δ ( x ) − σ ( x ) σ ⊤ ( x ) ◦ γ ⊤ ( x ) ∂ H ( x ) ∂ x γ ( x ) (cid:23) , (5) where “ ◦ ” is the Hadamard product. Based on
Theorem 1 , if the Hamiltonian is furtherassumed to be positive definite, then a simple unity feed-back controller u = − y is able to stabilize the systemin probability at the origin. However, the Hamiltonianusually cannot behave as a Lyapunov function, and evennot bounded from below. To address this issue, Fang &Gao (2017), based on the stochastic generalized canoni-cal transformation (Satoh & Fujimoto, 2013), proposedthe following transformation T : ¯ x = ϕ ( x ) , ¯ H ( ¯ x ) = H ( x ) + H ′ ( x ) | x = ϕ − (¯ x ) , ¯ y d = y d + α ( x ) | x = ϕ − (¯ x ) , ¯ u = u + β ( x ) | x = ϕ − (¯ x ) (6)to change the Hamiltonian, where ϕ : R n R n , H ′ : R n R and α , β : R n R m are all differentiablefunctions, α ( x ) = g ⊤ ( x ) ∂H ′ ( x ) ∂ x , y d = g ⊤ ( x ) ∂H ( x ) ∂ x andmoreover, ϕ ( · ) is a diffeomorphism and can preserve theoriginal point. A lower bounded and even positive defi-nite Hamiltonian may be generated through T .4 Stochastic Hamiltonian formulation of (state,input)-disturbed CSTRs
In this section, we use the sidSPHS framework tomodel CSTRs with state and input disturbed.Consider a CSTR involving in p chemical components X , · · · , X p and l reactions R , · · · , R l with the i th re-action written as R i : p X j =1 z ji X j ⇆ p X j =1 z ′ ji X j , where z ji , z ′ ji ∈ Z ≥ are the stoichiometric coefficients.To control the temperature T , the CSTR is equippedwith a surrounding jacket, in which the cooling/heatingfluid flows for heat exchange. The dynamics can be de-duced from the mass and energy balances by consideringthe following assumptions: A.1:
The reaction mixture is ideal and incompressible;and moreover, the reaction volume V and pressure P aresupposed to be constants. A.2:
At the inlet the reactor, the pure components X j are fed at temperature T in j and molar concentration c in j . A.3:
The reaction kinetics obey the Arrhenius law,with which the forward and backward reaction rate of R i are given by r f i = k f i exp (cid:16) − E fi RT (cid:17) Π pj =1 c z ji j ,r b i = k b i exp (cid:16) − E bi RT (cid:17) Π pj =1 c z ′ ji j , where E f i , E b i are reaction activation energy, k f i , k b i are reaction rate constants, and c j is the molar concen-tration of component X j , evaluated by the molar mass N j according to c j = N j V . A.4:
The heat exchange with the jacket is propor-tional to the difference between the jacket temperature T w and the mixture temperature T in the reactor, i.e,˙ Q = λ ( T w − T ) with λ the heat transfer coefficient. A.5:
There are slight fluctuations on chemical reac-tions, inlet/outlet volume flows and heat exchange. Allfluctuations are supposed to be mutually independentstandard Wiener processes, labeled by ω , ω , ω ∈ R , respectively. They act on the corresponding objectsin proportion to their respective standard deviations ρ , ρ , ρ ∈ R + , which are assumed to be sufficientlysmall in comparison to the above mentioned macroscopicvariables. The mass balances in the Itˆo form are then given byd N = ( z − z ′ )( r b − r f ) V (d t + ρ d ω )+ (cid:18) q c in − q N V (cid:19) (d t + ρ d ω ) , (7)where N = ( N , · · · , N p ) ⊤ , c in = ( c in1 , · · · , c in p ) ⊤ , z =[ z ji ] p × l , z ′ = [ z ′ ji ] p × l , r f = ( r f , · · · , r f l ) ⊤ and r b =( r b , · · · , r b l ) ⊤ . The energy balance, evaluated by theinternal energy U , in the sense of Itˆo is of the formd U = (cid:18) q U in V − q UV (cid:19) (d t + ρ d ω ) + ˙ Q (d t + ρ d ω ) , (8)where U in is the inlet internal energy.A key point to formulate the sidCSTR system intoa sidSPHS structure is to select the Hamiltonian. Sim-ilar to the Hamiltonian formulation of the determinis-tic CSTRs (Hoang, Couenne, Jallut, & Gorrec, 2011),the opposite entropy − S is chosen as the Hamiltonian,which follows the famous Gibbs’ equationd S = 1 T d U + PT d V − µ ⊤ T d N , (9)where µ = ( µ , · · · , µ p ) ⊤ is the chemical potential vec-tor. The entropy function is set to abide by some basic as-sumptions (Herbert, 1985): i) principle of maximum en-tropy; ii) positive homogeneousness of degree one. Fromthese two assumptions, it is easy to obtain that the en-tropy is concave. We further assume that the entropyfunction is twice continuously differentiable, so the Hes-sian matrix of − S is positive semi-definite. For the sidC-STRs under consideration, note that d V = 0 in Eq. (9),then by letting x = ( U, N ⊤ ) ⊤ we have ∂ ( − S ) ∂ x = (cid:18) − T , µ ⊤ T (cid:19) ⊤ . (10)The calculation of the second derivative of the negativeentropy requires the definitions of the heat capacity andchemical potential in the case of the ideal and incom-pressible mixture (Hoang, Couenne, Jallut, & Gorrec,2011), i.e., U = N ⊤ [ C P ( T − T ref ) + h ref ] − P V, µ T = − C P ln TT ref + R Ln (cid:16) NN ⊤ p (cid:17) − s ref + h T , (11)where C P = ( C P , · · · , C P p ) ⊤ is the isobaric heat ca-pacity vector, T ref the reference temperature, h ref =( h ref , · · · , h p ref ) ⊤ the reference molar enthalpy vector, s ref = ( s ref , · · · , s p ref ) ⊤ the reference molar entropy5ector and h = C P ( T − T ref ) + h ref represents the molarenthalpy vector. Combining Eqs. (10) and (11), we get ∂ ( − S ) ∂ x = θ , − h ⊤ θ − h θ , hh ⊤ θ − R N ⊤ p p × p + diag (cid:16) RN j (cid:17) (12)with θ = T N ⊤ C P , j = 1 , · · · , p . Throughout the pa-per, we sometimes drop “( x )” for notational simplicity.The following task is to define J , R , u and others sothat Eqs. (7), (8) are rewritten according to the sidSPHSstructure of Eq. (3). Proposition 2
For any sidCSTR governed by Eqs. (7)and (8), assume that the disturbances originating fromthe inlet/outlet volume flows and heat exchange are weakenough so that ρ M + ρ < θ , (13) where M = ∆ ⊤ c hh ⊤ − θR N ⊤ p p × p + diag (cid:18) θRN j (cid:19)! ∆ c − U in − UV h ⊤ ∆ c + (cid:18) U in − UV (cid:19) , (14) j = 1 , · · · , p and ∆ c = c in − N V . Then the sidCSTR isa sidSPHS, written by Eq. (12) with J = ( p +1) × ( p +1) , H = − S and R = , × p p × , V T P li =1 ( r fi − r bi )∆ ⊤ z · i µ ∆ z · i ∆ ⊤ z · i ! , (15) where ∆ z · i = z · i − z ′· i . Proof.
The proof may be implemented in two separateprocedures. The first one is to verify Eq. (16) is equiva-lent to Eqs. (7) and (8) while the second one is to prove that the variables defined in Eq. (16) satisfies the con-straints made in Eq. (3).(i) By inserting Eqs. (10) and (15) into Eq. (16), it iseasy to verify that Eq. (16) is completely equivalent toEqs. (7) and (8).(ii) The sidSPHS structure requests R ( x ), δ ( x ) to bepositive semi-definite and k δ ( x ) k F < λ = ( λ , λ ⊤ p ) ⊤ be any R p +1 vector, then we have λ ⊤ Rλ = l X i =1 ( r f i − r b i ) V T ∆ ⊤ z · i µ λ ⊤ p ∆ z · i ∆ ⊤ z · i λ p = l X i =1 ( r f i − r b i ) V T ∆ ⊤ z · i µ (∆ ⊤ z · i λ p ) . Note that ( r fi − r bi ) V T ∆ ⊤ z · i µ is a well defined nonnegative func-tion since it has the same sign as the entropy creation, ( r fi − r bi ) VT ∆ ⊤ z · i µ ≥
0, of the i th chemical reaction (Pri-gogine, Defay, & Everett, 1954), so λ ⊤ Rλ ≥
0. Clearly, R = R ⊤ . We thus get that R is positive semi-definite.The positive semi-definiteness of δ ( x ) may be easilyverified from the facts that θ> M = θ γ ⊤· ∂ ( − S ) ∂ x γ · ≥ δ ( x ) = δ ⊤ ( x ). Also, since k δ ( x ) k F = 14 ρ M θ + 14 ρ θ , which combines with Eq. (13), we obtain k δ ( x ) k F <
1. Inaddition, the noise ports in Eq. (16) will vanish in theequilibrium with zero input. ✷ Remark 1
Under the conditions of no disturbances, thesidCSTR of Eq. (16) will degenerate to the deterministicCSTR. If no environmental exchanges occur again, the Σ s : d x = ( J ( x ) − R ( x )) ∂ ( − S ) ∂ x d t + U in V − UV c in − N V !| {z } g ( x ) q ˙ Q !| {z } u d t + z − z ′ )( r b − r f ) ρ V !| {z } a ( x ) d ω + U in V − UV c in − N V !| {z } γ ( x ) q
00 ˙ Q !| {z } ˜ u ρ ρ !| {z } σ ( x ) d ω d ω ! y = U in V − UV ( c in − N V ) ⊤ !| {z } g ⊤ ( x ) ∂ ( − S ) ∂ x + ρ Mθ ρ
23 1 θ !| {z } δ ( x ) q ˙ Q !| {z } u (16)6 eterministic model simplifies to be ˙ x = ( J ( x ) − R ( x )) ∂ ( − S ) ∂ x . This structure can represent the first law and the secondlaw of thermodynamics simultaneously since ( d U d t = 0 , d S d t = (cid:0) ∂S∂ x (cid:1) ⊤ R ( x ) ∂S∂ x ≥ . Remark 2
For a sidCSTR, whether the condition of Eq.(13) is true mainly depends on the first term ρ M sincethe second one ρ is far less than θ according to the as-sumption A.5. Basically, as long as ρ γ ⊤· ∂ ( − S ) ∂ x γ · < ,Eq. (13) will hold almost surely. This may be easilyachieved in practice if the temperature and the concen-trations of input species are manipulated properly. The sidCSTR is a sidSPHS if the negative entropy − S is set as the Hamiltonian and a relatively weak conditionis added. However, since − S is not bounded from below,the stabilization method derived from the sidSPHS the-ory (Fang & Gao, 2017) cannot directly apply to the cur-rent sidCSTR structure of Eq. (16). For this reason, wemanage to map the negative entropy to a new Hamilto-nian, which is bounded from below and even can behaveas a Lyapunov function, and then stabilize the systemin probability in this section.We define the following transformation for the above purpose. T : ¯ x = x , ¯ H ( ¯ x ) = − S ( x ) + x ⊤ π ∗ | x =¯ x , ¯ y d = y d + α ( x ) | x =¯ x , ¯ u = u , (17)where π ∗ = ∂S∂ x | x = x ∗ = (cid:16) T ∗ , − µ ∗ T ∗ ⊤ (cid:17) ⊤ defined accordingto Eq. (10). Remark 3
The transformed Hamiltonian has explicitphysical significance, i.e., representing the availabilityfunction (Ydstie & Alonso, 1997). Based on Euler’s the-orem for homogeneous functions, the entropy functionshould satisfy S ( x ) = x ⊤ ∂S∂ x = π ⊤ x . Therefore, thetransformed Hamiltonian can be rewritten as ¯ H ( ¯ x ) = − S ( ¯ x ) + ¯ π ∗⊤ ( ¯ x − ¯ x ∗ ) + S ( ¯ x ∗ ) . Geometrically, it evaluates the distance between the en-tropy function and its tangent plane at ¯ x ∗ . This togetherwith concavity of the entropy function means ¯ H ( ¯ x ) ≥ ,and moreover, ¯ H ( ¯ x ) is convex. In the case of setting V = V ∗ , ¯ H ( ¯ x ) is strictly convex, i.e., its Hessian ma-trix is positive definite with respect to ¯ x − ¯ x ∗ (Favache& Dochain, 2009). Based on the assumption
A.1 , we fix the volume forthe sidCSTR system of Eq. (16) to be V = V ∗ , and thenapply the transformation of Eq. (17) to this system. Proposition 3
The transformation defined in Eq.(17) can transform the sidCSTR system Σ s , de-scribed by Eq. (16), into another sidSPHS in theform of Eq. (18) if we set ¯ J ( ¯ x ) = ( p +1) × ( p +1) , ¯ R ( ¯ x ) = R ( x ) , ¯ g ( ¯ x ) = g ( x ) , ¯ u = u , ¯ a ( ¯ x ) = a ( x ) , ¯ γ ( ¯ x ) = γ ( x ) , ˜¯ u = ˜ u , ¯ σ ( ¯ x ) = σ ( x ) and ¯ δ ( ¯ x ) = δ ( x ) . Moreover, these two systems are dynami-cally equivalent. T (Σ s ) : d ¯ x = ( ¯ J ( ¯ x ) − ¯ R ( ¯ x )) ∂ ¯ H (¯ x ) ∂ ¯ x d t + U in V ∗ − UV ∗ c in − N V ∗ !| {z } ¯ g (¯ x ) q ˙ Q !| {z } ¯ u d t + z − z ′ )( r b − r f ) ρ V ∗ !| {z } ¯ a (¯ x ) d ω + U in V ∗ − UV ∗ c in − N V ∗ !| {z } ¯ γ (¯ x ) q
00 ˙ Q !| {z } ˜¯ u ρ ρ !| {z } ¯ σ (¯ x ) d ω d ω ! ¯ y = U in V ∗ − UV ∗ ( c in − N V ∗ ) ⊤ !| {z } ¯ g ⊤ (¯ x ) ∂ ¯ H (¯ x ) ∂ ¯ x + ρ Mθ ρ
23 1 θ !| {z } ¯ δ (¯ x ) q ˙ Q !| {z } ¯ u (18)7 roof. Since most of variables in the transformed sys-tem T (Σ s ) of Eq.(18) are the same as in the originalsidCSTR system Σ s of Eq. (16), it is clearly that T (Σ s )is a sidSPHS.We further prove d ¯ x = d x . Since( ¯ J ( ¯ x ) − ¯ R ( ¯ x )) ∂ ¯ H ( ¯ x ) ∂ ¯ x = − R ∂ ( − S ) ∂ x − Rπ ∗ and Rπ ∗ = V ∗ TT ∗ l X i =1 ( r f i − r b i )∆ ⊤ z · i µ ∆ z · i ∆ ⊤ z · i µ ∗ = ( p +1) × , where the last equality holds due to ∆ ⊤ z · i µ ∗ = 0, wehave( ¯ J ( ¯ x ) − ¯ R ( ¯ x )) ∂ ¯ H ( ¯ x ) ∂ ¯ x = ( J ( x ) − R ( x )) ∂ ( − S ) ∂ x , i.e., d ¯ x = d x . Therefore, T (Σ s ) and Σ s are dynamicallyequivalent. ✷ Corollary 1
Let the sidCSTR systems Σ s and T (Σ s ) bedefined by Eqs. (16) and (18), respectively, and x ∗ ∈ R n be their equilibrium point. If x ∗ is stable/asymptoticallystable in probability for T (Σ s ) , then the correspondingresult also apply to Σ s . This corollary suggests a solution of stabilizing Σ s inprobability, i.e., towards stabilizing T (Σ s ) in probabil-ity. The latter purpose is easily reached using stochasticpassivity theorem (Florchinger, 1999). Theorem 2
The transformed sidCSTR system T (Σ s ) ,governed by Eq. (18), is stochastically passive with respectto ¯ H ( ¯ x ) if the reaction disturbance ρ is small enoughsuch that ρ V ∗ p X j =1 W j · p ≤ l X i =1 ( r f i − r b i )( z · i − z ′· i ) ⊤ µ T , (19) where W = ( hh ⊤ θ − R N ⊤ p p × p + diag (cid:16) RN j (cid:17) ) ◦ (∆ z , r ∆ ⊤ z , r ) , ∆ z , r = ( z − z ′ )( r b − r f ) . Proof.
Since the transformed system T (Σ s ) in the formof Eq. (18) is a sidSPHS, there exists a necessary andsufficient condition, as Theorem 1 says, to suggest this system stochastically passive with respect to its Hamil-tonian. Consider12 ρ V ∗ p X j =1 W j · p = 12 ρ V ∗ tr ( ( hh ⊤ θ − R p × p N ⊤ p + diag( R/N j ))∆ z , r ∆ ⊤ z , r ) = 12 tr (cid:26) ∂ ¯ H ( ¯ x ) ∂ ¯ x ¯ a ( ¯ x )¯ a ⊤ ( ¯ x ) (cid:27) and l X i =1 ( r f i − r b i )( z · i − z ′· i ) ⊤ µ V ∗ T = ∂ ¯ H ( ¯ x ) ∂ ¯ x ⊤ ¯ R ( ¯ x ) ∂ ¯ H ( ¯ x ) ∂ ¯ x , which together with Eq. (19) support the condition ofEq. (4) in Theorem 1 . In addition, the current ¯ δ ( ¯ x )defined in Eq. (18) apparently supports the condition ofEq. (5) in Theorem 1 . Note that P pj =1 W j · p has thesame sign with tr n ∂ ¯ H (¯ x ) ∂ ¯ x ¯ a ( ¯ x )¯ a ⊤ ( ¯ x ) o , while the latteris nonnegative due to the positive semi-definite ∂ ¯ H (¯ x ) ∂ ¯ x and the positive semi-definite ¯ a ( ¯ x )¯ a ⊤ ( ¯ x ). Therefore, aslong as the reaction disturbance ρ is small enough, thetransformed sidCSTR system can be stochastically pas-sive with respect to ¯ H ( ¯ x ). ✷ Remark 4
The condition of Eq. (19) is not so harshfrom the viewpoint of practice since the order of magni-tude of ρ is quite low compared with other macroscopicvariables, which leads to Eq. (19) to be most likely truein the practical cases. The stochastic passivity of T (Σ s ) can render a sim-ple and effective way to stabilize it in probability, andfurther for the original sidCSTR system Σ s . Theorem 3
For any sidCSTR system Σ s given by Eq.(16), let x ∗ be one of its equilibrium points. Assume thatits transformed version T (Σ s ) modeled by Eq. (18) isstochastically passive with respect to ¯ H ( ¯ x ) , and there ex-ists a positive definite matrix K ∈ R × such that I + K ¯ δ is invertible. Then the controller in the form of ¯ u = − (cid:2) I + K ¯ δ (cid:3) − K ¯ g ⊤ ( ¯ x ) ∂ ¯ H ( ¯ x ) ∂ ¯ x (20) can locally asymptotically stabilize Σ s at x ∗ in probabilityif they are connected in negative feedback. Proof.
Since the transformed system T (Σ s ) modeled byEq. (18) is stochastically passive with respect to ¯ H ( ¯ x ),we have d ¯ H (¯ x )d t ≤ ¯ y ⊤ ¯ u . Combining Eqs. (18) and (20)8 d x = − ( k f N A e − EfRT − k b N B e − EbRT ) Tµ A − µ B − − | {z } R ( x ) ∂ ( − S ) ∂ x d t + c in A h in − ( N A V h A + N B V h B ) 1 c in A − N A V − N B V | {z } g ( x ) q ˙ Q !| {z } u d t + − ρ ( k f N A e − EfRT − k b N B e − EbRT ) ρ ( k f N A e − EfRT − k b N B e − EbRT ) | {z } a ( x ) d ω + c in A h in − ( N A V h A + N B V h B ) 1 c in A − N A V − N B V | {z } γ ( x ) q
00 ˙ Q !| {z } ˜ u ρ ρ !| {z } σ ( x ) d ω d ω ! y = c in A h in − ( N A V h A + N B V h B ) c in A − N A V − N B V !| {z } g ⊤ ( x ) ∂ ( − S ) ∂ x + ρ Mθ ρ
23 1 θ !| {z } δ ( x ) q ˙ Q !| {z } u (21)yields ¯ u = − K (¯ g ⊤ ( ¯ x ) ∂ ¯ H ( ¯ x ) ∂ ¯ x + ¯ δ ¯ u ) = − K ¯ y . Hence, we obtain d ¯ H (¯ x )d t ≤ ¯ y ⊤ ¯ u = − ¯ y ⊤ K ¯ y ≤ y = , i.e., ¯ x = ¯ x ∗ . Since¯ H ( ¯ x ) is positive definite with respect to ¯ x − ¯ x ∗ , thecontroller can locally asymptotically stabilize T (Σ s ) at¯ x ∗ in probability. This together with Corollary 1 sug-gests that the equilibrium point x ∗ for Σ s can be locallyasymptotically stabilized in probability by the proposedcontroller. ✷ In this section, the proposed port-Hamiltonianmethod is applied to stabilizing in probability a CSTRsystem (Hoang, Couenne, Jallut, & Gorrec, 2011) withstate and input both disturbed, in which a first-orderreversible chemical reaction A ⇄ B takes place, andonly the pure A component is fed into the reactor. Thesystem follows all assumptions A. − A. s may bewritten in the sense of Itˆo to be d U = q [ c in A h in − ( N A V h A + N B V h B )](d t + ρ d ω )+ ˙ Q (d t + ρ d ω ) , d N A = ( k b N B e − EbRT − k f N A e − EfRT )(d t + ρ d ω ) q ( c in A − N A V )(d t + ρ d ω ) , d N B = ( k f N A e − EfRT − k b N B e − EbRT )(d t + ρ d ω ) − q N B V (d t + ρ d ω ) , where some related parameters values are listed in Ta-ble 1. For this system, there are up to three equilibriawith the middle one unstable while other two stable.The current task focuses on stabilizing in probabilitythe middle unstable equilibrium point, which is denotedby x ∗ = ( U ∗ , N ∗ A , N ∗ B ) ⊤ . The reactor works under theinitial conditions given in Table 2, based on which it iseasy to calculate the concerning equilibrium point to be(1157 . , . , . ⊤ , and the corresponding V ∗ = 0 . and T ∗ = 331 . Table 1. Parameters of the sidCSTR system: A ⇄ B Symbol Numerical value Variable name C PA AC PB Bh A ref Ah B ref -4575J/mol Reference enthalpy of Bk f . × /s Forward kinetic constant E f k b . × /s Backward kinetic constant E b P Pa Pressure R .
314 J/K/mol Molar gas constant T ref s A ref . As B ref . Bρ , ρ . , .
05 Reaction and heat exchange disturbance ρ × − Inlet/outlet flow disturbance λ . Table 2. Initial conditions and setpoint values
Initial Value Setpoint Value T in , T (0) , T w (0) 310 , , . U ∗ , T ∗ . . q (0) 9 . × − m /s V ∗ . N A (0) , N B (0) 1 , N ∗ A , N ∗ B .
3, 0 . c in A /s q ∗ . × − m /s By setting H ( x ) = − S and then applying Proposi-tion 2 , we rewrite the above dynamical equation as the9orm of sidSPHS, shown in Eq. (21), where Mθ = c in A R ( N A + N B ) N A N B + ( c in A h A − c in A h in ) θ ,θ = ( N A C P A + N B C P B ) T . (22)It is clear that at given ρ , ρ Eq. (13) is true tosupport Eq. (21) to be a sidSPHS. Further apply-ing
Proposition 3 will produce a transformed butequivalent sidSPHS to the original one of Eq. (21).Moreover, at the given ρ , the condition of Eq.(19) in Theorem 2 holds, which means the trans-formed sidCSTR system stochastically passive withrespect to its Hamiltonian. Finally, based on
Theo-rem 3 , we set the proportional gain matrix K to be K = diag( K , K ) = diag(1 . × − , I + K ¯ δ to be invertible, and then design thefollowing controller utilizing Eq. (20) q = − K (1 + ρ Mθ K ) − [ − N B V ( − µ ∗ B T ∗ + µ B T )+( c in A h in − N A V h A − N B V h B )( T ∗ − T )+( c in A − N A V )( − µ ∗ A T ∗ + µ A T )] , ˙ Q = − K (1 + ρ
32 1 θ K ) − ( T ∗ − T )to locally asymptotically stabilize the considered sidC-STR system in probability. The manipulated variable T w may be further calculated by combining the abovesecond equation with the expression ˙ Q = λ ( T w − T ) as T w = − K (1 + ρ
32 1 θ K ) − ( T ∗ − T ) λ + T. To observe the state converging behaviors for this sys-tem, Figs. 2 and 3 exhibit the response of the tempera-ture and molar mass to time, respectively. As one can ex-pect, the state vector (
T, N A , N B ) ⊤ can asymptoticallyconverge to the equilibrium state ( T ∗ , N ∗ A , N ∗ B ) ⊤ as thecontroller is put into force. Basically, from t = 3s theconverging behaviors happen. These phenomena suggestthat the proposed stochastic passivity based controlleris qualified for stabilizing the sidCSTR system in proba-bility. We also present the evolution of the manipulatedvariables q and T w in Figs. 4 and 5. In like manner, theywill asymptotically approaching the stable values q ≈ T w ≈ T ∗ as the state tends to the equilibrium point. In this paper, a port-Hamiltonian based method isproposed to model and stabilize in probability the CSTRsystems with state and input both disturbed. The fol-lowing conclusions are reached:(i) The sidCSTR systems may be formulated into asidSPHS structure if the opposite entropy function is T − T * ( K ) Fig. 2. Temperature response of the sidCSTR system. de v i a t i on m o l a r m a ss ( m o l ) N A −N *A N B −N *B Fig. 3. Molar mass responses of the sidCSTR system. −4 time(s) q ( m / s ) Fig. 4. Input/output volume flow evolution. T w ( K ) Fig. 5. Jacket temperature evolution. taken as the Hamiltonian. Moreover, the degenerateddeterministic version can represent the first law and thesecond law of thermodynamics simultaneously.(ii) The sidCSTR systems have the property ofstochastic passivity with respect to the Hamiltonianwhen the Hamiltonian is transformed from the oppositeentropy function to the availability function.(iii) The sidCSTR systems can be locally asymptoti-cally stabilized if a stochastic passivity based controlleris connected in negative feedback.
Acknowledgements
This work was supported by the National Natural Sci-ence Foundation of China under Grant No. 11671418,11271326 and 61611130124, and the Research Fund forthe Doctoral Program of Higher Education of China un-der Grant No. 20130101110040.
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