# Modeling and Estimation of Thermal Flows Based on Transport and Balance Equations

Authors: Angelo Alessandri; Patrizia Bagnerini; Roberto Cianci; Roberto Revetria
Journal: Advances in Mathematical Physics
 (2020)

---

## Abstract

Heat transfer in counterflow heat exchangers is modeled by using transport and balance equations with the temperatures of cold fluid, hot fluid, and metal pipe as state variables distributed along the entire pipe length. Using such models, boundary value problems can be solved to estimate the temperatures over all the length by means of measurements taken only at the boundaries. Conditions for the stability of the estimation error given by the difference between the temperatures and their estimates are established by using a Lyapunov approach. Toward this end, a method to construct nonlinear Lyapunov functionals is addressed by relying on a polynomial diagonal structure. This stability analysis is extended in case of the presence of bounded modeling uncertainty. The theoretical findings are illustrated with numerical results, which show the effectiveness of the proposed approach.


---

## Body

## 1. Introduction

The mathematical modeling of heat exchangers is really important since these devices are employed in chemical processes, petroleum refineries, power systems, and heating/refrigeration of air-conditioning plants [1, 2]. In this paper, we address the modeling of a counterflow heat exchanger based on transport and balance PDEs and use such models for the purpose of monitoring the thermodynamic process by solving boundary value problems that exploit only few temperature measurements at the boundaries. A rigorous stability analysis is addressed to prove the effectiveness of the resulting temperature estimates from the theoretical and numerical points of view in line with the widespread methods based on the Lyapunov approach in a number of different applications [3–6].Heat exchangers for industrial processes allow recovering lost energy from hot fluid streams or to heat a cold fluid for the purpose of air conditioning. Finite-dimensional models of such processes are used for fouling detection by using Kalman filtering methods [1, 2]. As compared with these models, the proposed modeling framework is in an infinite dimension since it relies on hyperbolic PDEs. Based on such equations, first we will focus on a model with the dynamics of the temperatures of cold fluid, hot fluid, and metal pipe. In addition, a reduced model is considered with only the temperatures of cold and hot fluid as state variables in line with [7]. Since such temperatures cannot be determined by direct, distributed measurements, it is fundamental to estimate them by using a few numbers of measures. Toward this end, using such models, we address the solution of boundary value problems to estimate the distributed state, which provides such estimate by having at our disposal measurements of temperature only at the boundaries. We will refer to them also as boundary-output Luenberger observers or simply state observers (see [8] and the references therein). Stability conditions on the estimation error (i.e., the difference between the temperatures and their estimates) will be presented to construct such estimators in case of perfect modeling and assuming bounded uncertainty. In such a case, we will rely on the notion of quadratic boundedness (QB, for short) [9]. Finally, numerical results will be given to showcase the theoretical findings.The stability of the estimation error is analyzed in two different settings. First, we will consider perfect modeling and hence asymptotic stability results will be provided. Next, we will assume the presence of additive noises, which account for bounded modeling uncertainties and stability is studied by using QB. Generally speaking, QB allows dealing with positively invariant sets and derive upper bounds on the trajectories of the state of a dynamic system subject to bounded disturbances [9]. Thus, QB has been successfully used for both output feedback control [10] and state estimation [11]. Here, we will extend the use of QB for the purpose of estimation in the considered infinite dimensional context. It is worth noting that, in lieu of searching for exact solutions of the flow equations (see, e.g., [12, 13]), here we deal with the construction of boundary-output observers with guaranteed stability properties on the estimation error.All the stability conditions that will be presented demand the selection of nonlinear Lyapunov functionals. In this paper, we will address this task by using the SOS (sum-of-squares) approach [14, 15], which was originally developed for the analysis and control of polynomial systems described by ODEs (see [16] and the references therein) and recently applied also to systems described by PDEs [17–19]. SOS methods are quite computationally efficient since they are based on semidefinite programming (SDP) [20, 21]. In this respect, the SOS approach allows finding Lyapunov functionals just like the methods based on LMIs (linear matrix inequalities) enable computing Lyapunov functions for linear systems based on ODE models [22]. As compared with the search of classical Lyapunov functions to investigate local stability [23], we do not rely on linearized models but provide conditions of global stability.The paper is organized as follows: Section2 reports the basic definitions that will be used in the following discussions. Modeling of thermal flows in one-dimensional heat exchangers is given in Section 3. The proposed estimation approach is presented in Section 4, where stability is analyzed in a noise-free modeling framework. This analysis is extended to models affected by bounded disturbances in Section 5. A brief account on the use of the SOS method to find the Lyapunov functionals and thus ensure stability is given in Section 6. Section 7 illustrates the numerical results, while conclusions are drawn in Section 8.


## 2. Notations and Definitions

The set of real numbers is denoted byℝ and hence ℝ+ will be used for the set of strictly positive real numbers. The set of the integer numbers equal to or greater than zero is denoted by ℕ.The minimum and maximum eigenvalues of a symmetric matrixP∈ℝn×n are denoted by λminP and λmaxP, respectively. Moreover, P>0 (P<0) means that it is also positive (negative) definite. Given a generic matrix M, M≔λmaxM⊤M1/2=λmaxMM⊤1/2, and hence, for a vector x=x1,x2,⋯,xn∈ℝn, ∣x∣≔x⊤x1/2 is its Euclidean norm. ℝx denotes the ring of real polynomials in x∈ℝn; ℝxd is the set of real polynomials of a degree equal or less than d∈ℕ.Given the compact setΩ⊂ℝn, L2Ω denotes the Hilbert space of square integrable functions γ:Ω×0,+∞⟶ℝm with the norm γ·,tL2=∫Ωγx,t2dx1/2<∞ for all t≥0. The solution ϕx,t∈L2Ω of a PDE with the initial condition ϕx,0=ϕ0x∈L2Ω is said to be
(i)
L2 is stable (to zero) if, for all ε>0, there exists δε>0 such that
(1)ϕ0L2<δε⇒ϕ·,tL2<ε,for all t≥0
(ii)
L2 is asymptotically stable (to zero) if it is stable and
(2)limt→+∞ϕ·,tL2=0
(iii)
L2 is exponentially stable (to zero) if there exists λ>0 such that
(3)ϕ·,tL2≤c,ϕ0L2exp−λt,for some c>0 and all t≥0


## 3. Modeling Heat Transfer in Heat Exchangers

Consider the one-dimensional model of counterflow heat exchangers with three distributed state variables, i.e.,Tcx,t, Thx,t, and Tmx,t for x∈0,L and t≥0, as depicted in Figure 1; such variables represent the temperatures of cold fluid, hot fluid, and pipe, respectively (the subscript indicating that it is made of some metal in such a way to maximize the heat transfer). Based on the balance of enthalpy for the hot fluid, we get
(4)ρhchSh∂Th∂t+ṁhch∂Th∂x=UhphTm−Th,where ρh is the density (in kg/m3), ch is the specific heat (in J/(kg K)), ṁh is the mass flow rate (in kg/s), Uh is the transfer coefficient (in W/(m2K)), Sh (in m2) is the internal surface of the pipe, and ph is its perimeter (in m). Concerning the cold fluid, the balance of enthalpy is given by
(5)ρcccSc∂Tc∂t−ṁccc∂Tc∂x=UcpcTm−Tc,where ρc is density (in kg/m3), cc is the specific heat (in J/(kg K)), and Uc is the transfer coefficient (in W/(m2)). Moreover, Sc (in m2) is the external surface of the pipe and pc is its perimeter (in m). Finally, for the metal pipe it follows that
(6)ρmcmSm∂Tm∂t=UhphTh−Tm+UcpcTc−Tm,where ρm is the density (in kg/m3) of the metal, cm is the specific heat (in J/(kg K)) of the metal, and Sm≔Sc−Sh (in m2) is the surface of the pipe section.Figure 1
Sketch of the heat exchanger model.After dividing each of the three previous equations for the first coefficient in each respective l.h.s., we get(7)∂Th∂t+a1∂Th∂x=b1Tm−Th,(8)∂Tc∂t−a2∂Tc∂x=b2Tm−Tc,(9)∂Tm∂t=b3Th−Tm+b4Tc−Tm,where
(10)a1=ṁhρhSh,a2≔ṁcρcSc,b1≔UhphρhchSh,b2≔UcpcρcccSc,b3≔UhphρmcmSm,b4≔UcpcρmcmSm,or, more concisely, as follows:
(11)∂tT+A1∂xT=B1T,where Tx,t≔colThx,t,Tcx,t,Tmx,t∈ℝ3 and
(12)A1≔a1000−a20000,B1≔−b10b10−b2b2b3b4−b3−b4.The system (11) is solved with initial conditions
(13)Thx,0=Th0x,Tcx,0=Tc0x,Tmx,0=Tm0x,x∈0,L,and Dirichlet boundary conditions
(14)Th0,t=T¯h0,t,TcL,t=T¯cL,t,for all t≥0. The boundary conditions on Tm are not necessary since equation (9) does not contain spatial derivatives of Tm.Likewise, in [7], a reduced model can be easily derived from (11) by neglecting the dynamics of the temperature in the pipe, i.e., assuming that such a temperature is fixed by the values of the temperatures of cold and hot fluids, i.e., imposing that the r.h.s. of (6) is null. Therefore, we obtain
(15)∂Th∂t+a1∂Th∂x=−b1b4b3+b4Th+b1b4b3+b4Tc,∂Tc∂t−a2∂Tc∂x=b2b3b3+b4Th−b2b3b3+b4Tc,or, more compactly,
(16)∂tT+A2∂xT=B2T,where Tx,t≔colThx,t,Tcx,t∈ℝ2 and
(17)A2≔a100−a2,B2≔1b3+b4−b1b4b1b4b2b3−b2b3,with initial conditions
(18)Thx,0=Th0x,Tcx,0=Tc0x,x∈0,L,and Dirichlet boundary conditions
(19)Th0,t=T¯h0,t,TcL,t=T¯cL,t,for all t≥0.Model (11) is of a hyperbolic type but not strictly hyperbolic since matrix A1 has a zero eigenvalue. By contrast, (16) is strictly hyperbolic since the eigenvalues of A2 have a non-null-real part. In Section 4, we will consider the problem to reconstruct all the temperatures by using a generic hyperbolic model.


## 4. Estimation for Heat Exchangers with Stable Estimation Error

Consider the hyperbolic equation(20)∂tT+A∂xT=BT,where Tx,t∈ℝn, A∈ℝn×n diagonal, and B∈ℝn×n. Consider also the observer for (20) given by
(21)∂tT^+A∂xT^=BT^,where T^x,t∈ℝn is the estimate of Tx,t. Such PDEs need some suitable boundary conditions, which will be given later. The stability of the estimation error can be guaranteed under suitable conditions as follows.Theorem 1.
The estimation errorex,t≔Tx,t−T^x,t∈ℝn is L2 asymptotically stable if there exists a diagonal matrix Px∈ℝn×n with Px>0 for all x∈0,L such that
(22)e0,t⊤AP0e0,t−eL,t⊤APLeL,t≤0,and
(23)A∂xPx+B⊤Px+PxB<0,for all x∈0,L.Proof.
Consider the Lyapunov functional(24)Vt≔∫0Lex,t⊤Pxex,tdx,and note that the asymptotic stability of the estimation error is ensured if
(25)V̇t<0,for all ex,t∈ℝn since Px>0 for all x∈0,L. Since P does not depend on time, we get
(26)V̇t=∫0L−∂xe⊤AΤPe−e⊤PA∂xe+e⊤B⊤P+PBedx,and, owing to the diagonal structure of A and P,
(27)−∂xe⊤A⊤Pe−e⊤PA∂xe=−∂xe⊤APe+e⊤A∂xPe.
Thus, it follows(28)V̇t=−eL,t⊤APLeL,t+e0,t⊤AP0e0,t+∫0Le⊤A∂xP+B⊤P+PBedx,and hence (25) holds owing to (22) and (23).Note that if, instead of (23), we consider
(29)A∂xPx+B⊤Px+PxB+αPx<0,with α>0 for all x∈0,L, the estimation error is L2 exponentially stable with a rate of decrease equal to α. Thus, a design goal may consist in maximizing α, which can be regarded as a sort of generalized eigenvalue problem for Lyapunov functionals instead of Lyapunov functions [22].In case of the reduced model (16) (i.e., with A=A2 and B=B2), the stability conditions of Theorem 1 can be greatly simplified according to [7] by using
(30)Px≔diagexpμ1x,expμ2x,and boundary conditions
(31)T^h0,t=Th0,t+ℓ1Tc0,t−T^c0,t,(32)T^cL,t=TcL,t+ℓ2ThL,t−T^hL,t,for all t≥0 with ℓ1,ℓ2,μ1,μ2∈ℝ to be suitably chosen. More specifically, in [7] it is proposed to satisfy (23) by using an LMI-based discretized approach to get μ1,μ2 and then select ℓ1,ℓ2 such that
(33)a2ℓ22expμ2L−a1expμ1L≤0,a1ℓ12−a2≤0,by choosing, for example,
(34)ℓ1=a2a1,ℓ2=a1a2expμ1−μ2L2.The construction of Lyapunov functionals is not an easy task to solve in general and will be addressed later in Section6.


## 5. Estimation under Modeling Uncertainties

Let us focus on estimation in the presence of bounded disturbances. More specifically, instead of (20) we consider
(35)∂tT+A∂xT=BT+Dw,where D∈ℝn×q and wx,t∈ℝq such that wix,t≤1, i=1,⋯,q for all x∈0,L and t≥0 without loss of generality (different upper bounds can be taken into account by suitably scaling the coefficients of D), where q is an integer number no larger than n. Estimation will be performed by using the same observer adopted in the noise-free case, i.e., (21).The presence of the system noises prevents from ensuring asymptotic stability, but since such disturbances are bounded, an invariant set for the estimation error exists and can be studied by using QB [9].Toward this end, first of all we need to define QB in theL2 sense. More specifically, the estimation error is said to be L2 quadratically bounded with the Lyapunov functional Vt as defined in (24) if
(36)Vt>L⇒V̇t<0,∀w∈−1,1q.As a matter of fact, any positive constant can be chosen instead ofL, but with such a choice we reduce the notational burden. This does not entail loss of generality since the conditions ensuring QB are homogeneous in the design parameters, which scales with L and thus allows for this simplification.Owing to (36), the set
(37)E≔e∈L20,L:∫0Lex⊤Pxexdx≤L,turns out to be positively invariant and it is attractive (i.e., if the error is out of E, it approaches E asymptotically).Theorem 2.
The estimation error is quadratically bounded if there exists a diagonal matrixPx∈ℝn×n with Px>0 for all x∈0,L and α∈ℝ>0q, β>0 such that (22) and
(38)A∂xPx+B⊤Px+PxB+βPxPxDD⊤Px−diagα<0,(39)∑i=lqαi−β≤0,hold for all x∈0,L.Proof.
Likewise in the proof of Theorem1, we easily compute the time derivative of the Lyapunov functional (24) and get that V̇t<0 subject to (22) is equivalent to
(40)e⊤A∂xP+B⊤P+PBe+w⊤D⊤Pe+e⊤PDw<0.
Since−e⊤Pe+1>0 implies Vt>L and −wi2+1>0 holds, using the well-known S-procedure (see [22], p. 23) we obtain that QB holds if
(41)e⊤A∂xP+B⊤P+PB+βPe+w⊤D⊤Pe+e⊤PDw−∑i=1qαiwi2+∑i=1qαi−β<0,for some α∈ℝ>0q and β>0 or equivalently if (38) and (39) hold.Clearly, (38) implies
(42)A∂xPx+B⊤Px+PxB+βPx<0,for all x∈0,L, which ensures that the estimation error in the absence of noises is L2 exponentially stable with a rate of decrease equal to β. Thus, a convenient design goal may consist in maximizing β, as pointed out in [7]. Another, quite popular objective of design is related to the L2 gain between disturbance and estimation error.Theorem2 provides conditions ensuring that the set E is attractive. Thus, one can design the estimator in such a way to keep E as small as possible. Toward this end, note that
(43)λminPxe2≤e⊤Pxe,x∈0,L,for all e∈E and hence, as a design objective, we may maximize the minimum eigenvalue of Px since
(44)ex,t≤1λminPx,for all t≥0, x∈0,L. The inequality above allows proving the boundedness in the L2 sense as
(45)e·,tL2≤Lminx∈0,LλminPx,for all t≥0.To compare the effectiveness of a given observer in terms of rejection of the noise, it is quite popular to rely on the notion of theL2 gain by assuming finite-energy noises, i.e., w·,t∈L2Ω. More specifically, we say that the proposed observer admits and L2 gain between disturbance and estimation error is equal to γ>0 if
(46)∫0Te·,tL2dt≤γ2∫0Tw·,tL2dt,for T>0. It is straightforward to show that the bound of the L2 gain holds if
(47)A∂xPx+B⊤Px+PxB+1γ2IPxDD⊤Px−I<0.Note also that this condition is quite similar to (38). Moreover, minimizing γ is equivalent to maximizing 1/γ2 in accordance with the goal of maximizing β.


## 6. Search of Lyapunov Functionals Using the SOS Approach

Stability conditions such as (23), (29), (38), or (47) (each together with Px>0) demand the satisfaction of LMIs in Px for all x∈0,L, i.e., the solution of semi-infinite programming problems. Such problems arise in a variety of applications and are usually approximately solved in discretized form on a sufficiently fine mesh of points (see, e.g., [24]). The main difficulty to address semi-infinite programming problems concerns both the choice of a sufficiently large number of points and especially the local minima trapping, by which nonlinear programming solvers may be affected in trying to find the solution. A more appealing way to solve such problems consists in resorting to the SOS paradigm, which enables turning the construction of a Lyapunov functional into a convex problem that can be efficiently solved by using SDP without encountering such issues due to local minima.The idea behind such an approach is the SOS decomposition of a candidate Lyapunov functional as well as of the opposite of its time derivative by using a positivity certification, which does not depend on the characteristics of the chosen polynomial [21, 25]. More specifically, the following result holds.Theorem 3.
A polynomialpx∈ℝx2d in x=x1,⋯,xn∈ℝn has sum-of-squares decomposition (or is said to be SOS) if and only if there exists a real symmetric and positive semidefinite matrix Q∈ℝsd×sd such that px=vdxΤQvdx, where vdx is the vector of all the monomials in the components of x∈ℝn of degree equal to or less than d∈ℕ, i.e.,
(48)vdx≔1,x1,⋯,xn,x1x2,⋯,xn−1xn,xn2,⋯,x1d,⋯,xnd,of dimension
(49)sd≔n+dd.Proof.
See Proposition 2.1 in [26] (p. 17).Consider, for example, the estimation of the state variables of (7), (8), and (9) by using the results of Theorem 1. Based on Theorem 3, a procedure can be adopted to find suitable SOS polynomials as diagonal elements of Px=diagP11x,P22x,P33x, all to be taken in ℝx2d with x∈ℝ for increasing values of d. Toward this end, we say that px∈ℝx2d is ε-SOS polynomial if px−ε∑i=1nxi2∈ℝx2d with d∈ℕ, d≥1 is SOS for some “small” tolerance ε>0 [27]. In addition, a polynomial, square matrix Mx with Mijx∈ℝx2d for i,j=1,⋯,m is said to be an ε-SOS matrix if yΤMxy is ε-SOS in ℝx,y with x∈ℝ and y∈ℝm.Givend∈ℕ with d≥1 and ε>0, the problem to solve when dealing, for example, with the stability conditions of Theorem 1 is the following:Problem 4.
FindP11x,P22x,P33x∈ℝx2d such that
(50)P11x,P22x,P33xareε‐SOS(51)−AdiagP11′x,P22′x,P33′x−BTdiagP11x,P22x,P33x,−diagP11x,P22x,P33xBisaε‐SOSmatrix,for all x∈0,L.If Problem4 admits a solution, we can follow the same reasoning that leads to (34) to ensure the satisfaction of (22) by choosing
(52)ℓ1=a2P220a1P110,ℓ2=a1P11La2P22L.Similar problems may be formulated by replacing the SOS description of (52) with those corresponding to (29), (38), and (47).


## 7. Numerical Results

For the sake of brevity, in the following discussion, we will describe only the numerical solution of the boundary value problem that results from the discretization of the model (11) since this includes that of the reduced model (16) as a special case. We have discretized the domain 0,L with M equally spaced points xj for j=1,⋯,M, and both (16) and (21) are treated by using a finite difference method. Equations (7) and (8) composed of an advection term (a1∂xTh and −a2∂xTc) and a reaction one (the right side of the equalities) are semidiscretized in space by an upwind finite difference scheme [28–30]. The choice of an upwind scheme is more appropriate for the convective term in which there is a propagation direction. The boundary conditions (19) for (16) (and (31) and (32) for (21)) are given for Th and Tc only in the left (x=0) and right (x=L) parts of the domain, respectively. The term a1∂xTh in (7) is therefore discretized by using backward finite difference since a1 is positive, whereas the term −a2∂xTc in (8) is discretized by a forward one since −a2 is negative.Setting byThjt the solution at the discretization point xj at time t (and analogously for Tc and Tm), we obtain the following system of ODEs:
(53)Thj′t=−a1Thjt−Thj−1t+b1Tmjt−Thjt,j=2,⋯,M,Tcj′t=a2Tcj+1t−Tcjt+b2Tmjt−Tcjt,j=1,⋯,M−1,Tmj′t=b3Thjt−Tmjt+b4Tcjt−Tmjt,j=1,⋯,M,Th1′t=0,TcM′t=0,with initial conditions
(54)Thj0=Th0xj,Tcj0=Tc0xj.The boundary conditions in (19) are translated in the system by setting
(55)Th1t=T¯h0,t,TcMt=T¯cL,t.The temperaturesTh and Tc at the initial time are chosen constant and equal to the values at the boundary points where their measures are available, i.e., equals to T¯h in x=0 and T¯c in x=L such as
(56)Th0xj=T¯h0,0,Tc0xj=T¯cL,0,j=1,⋯,M,with a constant value belonging to the interval Th,Tc taken as initial conditions for Tm, i.e.,
(57)Tm0xj=T¯m,j=1,⋯,M.The scheme for the observer (21) is the same by replacing Thjt with T^hjt, Tcjt with T^cjt and so on, except for the boundary conditions, which are the following:
(58)T^h1t=T¯ht+l1Tc1t−T^c1t,T^cMt=T¯ct+l2ThMt−T^hMt.For the observer, the initial conditions in the simulation are chosen as:(59)T^h0x=T¯h0,0+3sin1.2πx,T^c0x=6sin1.2πx−9.539x+0.535x2+18,T^m0x=T¯m.The system of ODEs (52) is solved by using a fourth-order Runge-Kutta method (implemented in theode45 function of MATLAB) with variable time step. In the first instants of the simulation, the fast dynamics of the convective part prevails, while later the reaction term predominates. To accurately simulate the model, it is therefore necessary to select a very small time step in the first time instants, while it may be chosen larger afterwards, but of course, always respecting the stability condition of the numerical scheme. Implicit methods (e.g., the scheme implemented in the routine ode15s of MATLAB) are tested, since they generally allow using a larger time step. This resulted in a longer running time, due to the higher complexity to complete each iteration. Then, explicit methods seem to be the most appropriate choice for this problem.In the numerical case study, we have fixedL=5 m, a1=1 m/s, a2=1 m/s, b1=0.01 1/s, b2=0.01 1/s, b3=0.01 1/s, and b4=0.01 1/s. Concerning the proposed estimation approach with (11) and (16), we have solved Problem 4 by using the SOS toolbox [27] and the numerical solver Yalmip [31] with ε=10−6. In the first case (i.e., A=A1 and B=B1), we have got a solution to such a problem with d=1 given by
(60)Px=diag1.096·10−4x+1.0172+0.0017x−1.096·10−42,2.8477·10−4x+0.98642+0.0017x+2.8477·10−42,1.4496·10−4x+1.00932+0.0017x+1.4496·10−42,and select ℓ1,ℓ2 according to (52). The same has been done for the reduced model with A=A2 and B=B2 by obtaining
(61)Px=diag1.9955·10−4x+0.95462+0.001x+1.9955·10−42,4.1525·10−4x+0.91242+0.001x+4.1525·10−42,and taking ℓ1,ℓ2 again according to (52). We have chosen M=500, T¯h0,0=55, T¯cL,0=−15, and T¯m=50.Figure2 shows the behavior of the temperatures Th, Tc, and Tm and their estimates given by T^h, T^c, and T^m at different time instants, all based on (11). Figure 3 shows the behavior of the temperatures Th, Tc and the corresponding estimates T^h, T^c, based on the reduced model (16).Figure 2
Simulation run in a noise-free case at different time steps for the model (11).Figure 3
Simulation run in a noise-free case at different time steps for the reduced model (16).


## 8. Conclusions

Two models have been presented to account for the temperature dynamics in heat exchangers. Such models are based on hyperbolic PDEs with the most complete having the temperatures of cold fluid, hot fluid, and pipe as state variables, whereas the reduced one relies only on the temperatures of cold and hot fluids. Estimators resulting from the solution of boundary value problems based on such models have been proposed and provided with a rigorous stability analysis with the Lyapunov approach. To this end, the existence of nonlinear polynomial Lyapunov functionals has been addressed by using the SOS approach.Future work will concern the investigation on the stability of systems described by nonlinear hyperbolic equations (see, e.g., [32]). Another direction of research is the study of stability of cascaded hyperbolic and other nonlinear PDEs such as the normal flow equation to deal with multiphase problems, which turn out to be increasingly difficult but with a wide range of potential applications [33].



---

*Source: 9621308-2020-02-01.xml*