BINARY DISTILLATION COLUMN CONTROL BASED ON STATE AND INPUT OBSERVABILITY

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1 BINARY DISTILLATION COLUMN CONTROL BASED ON STATE AND INPUT OBSERVABILITY Addison Ríos-Bolívar, Ferenc Szigeti Universidad de Los Andes Facultad de Ingeniería Departamento de Control Av. Tulio Febres Cordero Mérida Venezuela fax: ilich@ula.ve, szigeti@ula.ve Keywords: Nonlinear observers. Input observability. State elimination. Composition control of binary distillation columns. System Inversion. Abstract In this paper, a method for the design of nonlinear observers and input reconstruction is proposed. The state and input observers is computed following the Diop s differential algebraic state elimination approach. In the input reconstruction case the input observability is equivalent to system inversion condition. The elimination procedure is utilized for dual composition control of binary distillation columns. The particular observer allows us to recover the unmeasured feed composition, i.e., the input reconstruction, which is used in the controller design in order to maintain desired compositions. The method is an alternative against the disturbance decoupling procedure. 1 Introduction State elimination approach, via output derivatives, has been broadly applied in order to design full order state observers. For instance, 1. In [6], D. Guillaume et al. have proposed simple observers for polymerization reactors. The nonlinear dynamics was linearized by output injection. Then linear observer was designed to the linear model. 2. In [19], a FDI filter was designed for the same binary distillation columns, which was considered by J. Levine and P. Rouchon, in [10]. The main idea to design a FDI filter, based on nonlinear observers was similar to the method applied in [6]. Linear dynamics was achieved by injection of the output and output derivatives. Now, the following question arises: How can asymptotic observer be designed for a nonlinear control system? Injection output and output derivatives are the elementary steps of the Diop s state elimination, [3]. T. Glad has considered the controller design problem in a differential algebraic setting, [5]. The construction of nonlinear observer is, traditionally, based on differential geometry. The well-known Bracket Vanishing Algorithm, see [8, 9] is due to Krener, Isidori, and Respondek. The Bracket Spanning Algorithm is due to Bestle and Zeitz, in [1], is just as hard computationally as Bracket Vanishing Algorithm due to its requirements of Lie bracket calculations. Phelps has improved those algorithms by the substitution of the bracket vanishing condition by another one, [13]. By another hand, the input reconstruction problem have been studied in [17]. The idea is to reconstruct an unknown input from system output. In [7] input observability and reconstruction, and fault detection and isolation FDI) problem have been related. In [15, 20] the input observability and system inversion problem have been related to design FDI filters.

2 In this paper a nonlinear observer and the input observability condition are used for binary distillation column control. The input observer and nonlinear observer are designed from state elimination setting, simultaneously. Thus, in the section 2 an observer for nonlinear systems based on state elimination is presented. In the section 3 the unknown input reconstruction problem is considered. Conditions for input observability are presented and input observers based on state elimination are proposed. In the section 4 the binary distillation column control problem is studied. In this case, the composition of the feed flow is an unknown input. Thus, the column control proposed is based on state and input observers. 2 The nonlinear observer In this paper an algebraic observer design method is proposed, based on the Diop s state elimination algorithm. The idea is the systematization of the naive state elimination steps, mentioned in the introduction, into the celebrated state elimination in control. Let us consider an algebraic control system ẋ = fx, u), y = gx); 1) where x R n is the state vector, u R p is the control input, y R m is the output vector. The functions f, g are of appropriate dimensions. f, g are algebraic analytic) functions, respectively. The proposed design scheme is the following: The construction of the nonlinear observer starts from the equations ˆx = fx, u) + Dx ˆx), ŷ = gx) + dx ˆx), 2) for the observer states ˆx, where the gains D, d are algebraic functions, such that D0) = 0, d0) = 0, and ė = De), is an asymptotically stable algebraic differential equation. System 2) is considered as an observer, coupled with the plant states x. Now, let us eliminate x from the coupled system ẋ = fx, u), ˆx = fx, u) + Dx ˆx) y = gx), ŷ = gx) + dx ˆx). Hence, an implicit observer equation will be obtained F ˆx, ˆx, u, u, ü,..., y, ẏ,..., ŷ, ŷ, )... = 0, with a non equality G ˆx, ˆx, u, u, ü,..., y, ẏ,..., ŷ, ŷ, ) Theorem 2.1 Let us suppose that 1) is algebraically observable. Then, the plant state x can be eliminated from 2) by the Diop s algorithm. Considering the differential algebraic setting of the definition of observability, that is, all elements of the dynamics is non differentially) algebraic over the differential field R < u, y >, see [4]. Theorem 1 can be extended by factorization with respect to the zero dynamics of the unobservable system. The suggested design method, based on state elimination, gives the same result for linear systems. For simplicity, let us consider the linear system in canonical form ẋ 1 = x 2,. ẋ n1 = x n, n ẋ n = a i x i + u, i=1 y = x 1. Then, the proposed observer can be computed from the system ˆx x 1. ˆx n1 = x n1 +Dxˆx) ˆx n a 1 a 2 a n x n

3 where the linear system ė = Dx ˆx) is asymptotically stable. The states in terms of the output derivatives are x 1 = y, x 2 = ẏ,..., x n = y n1). Let us denote the structure matrix of the system by A, the vector y, ẏ,..., y n1)) T by Y, and 0,..., 0, 1) T by e n. Then, the observer is ˆx = AY + DY ˆx) + e n u = Aˆx + e n u + A + D)Y ˆx). The induced error equation for e = x ˆx, is ė = De, which is asymptotically stable. 2.1 Example Let us consider the bilinear system in order to illustrate the importance of the persistent excitation and the universality of an input, for arbitrary universal input which distinguish all pair of initial states, see [4]. The system ẋ 1 ẋ 2 ẋ 3 = 0 u 1 u 2 u 1 0 u 3 u 2 u 3 0 x 1 x 2 x 3, y = x 2 is observable. The state elimination can be achieved by computing output derivatives in order to use, for example, Gauss elimination. Hence, x 1 = u 3ÿ + u 1 u 2 u 3 )ẏ + u 3 u u 2 3), 3) u 1 u 3 u 1 u 3 u 2 u u 2 3) x 2 = y, 4) x 3 = u 1ÿ u 1 + u 2 u 3 )ẏ + u 1 u u 2 3)y.5) u 1 u 3 u 1 u 3 u 2 u u 2 3) Considering the observer in the form ˆẋ 1 ˆẋ 2 ˆẋ 3 = 0 u 1 u 2 u 1 0 u 3 u 2 u 3 0 +D x 1 ˆx 1 x 2 ˆx 2 x 3 ˆx 3 x 1 x 2 x 3 + 6) the resulting error equation is ė 1 ė 2 ė 3 = D e 1 e 2 e 3. Hence, if the linear system ė = De is asymptotically stable, observer 6) is convergent. The computable observer can be obtained by state elimination. Indeed, using the expressions 3), 4), 5), for x 1, x 2, x 3, the corresponding observer s dynamics are ˆẋ 1 ˆẋ 2 ˆẋ 3 = u 1 u 2 ÿu 2 u 1 +u 2 u 3 )ẏ+u 1 u 1 u 3 u 1 u 3 )y u 1 u 3 u 1 u 3 u 2 u 2 1 +u2 3 ) ẏ u 2 u 3 ÿ+u 2 u 3 u 1 u 2 )ẏu 3 u 1 u 3 u 1 u 3 )y u 1 u 3 u 1 u 3 u 2 u 2 1 +u2 3 ) +D u 3 ÿ+u 1 u 2 u 3 )ẏ+u 3 u 2 1 +u2 3 ) u 1 u 3 u 1 u 3 u 2 u 2 1 +u2 3 ) ˆx 1 y ˆx 2 u 1 ÿ u 1 +u 2 u 3 )ẏ+u 1 u 2 1 +u2 3 )y u 1 u 3 u 1 u 3 u 2 u 2 1 +u2 3 ) ˆx 3 Next, simulations will be shown with u 1 = cos t), u 2 = 1, u 3 = sin t). The figure 1 shows the system output and the estimated output. The figure 2 shows the output error. It is clear the potentiality of the method proposed. Outputs Time t Figure 1: Output and estimated output. Now, let us consider the non-universal controls u 1 = u 3 = 1, u 2 = 0. The obtained system ẋ 1 = x 2, ẋ 2 = x 1 + x 3, ẋ 3 = x 2, y = x 2 is not observable. After the state transformation x 4 = x 3 x 1, x 5 = x 3 + x 1, the unobservable state can be easily computed from ẋ 2 = x 4, ẋ 4 = 2x 2, ẋ 5 = 0.

4 Output error Time t Figure 2: Output error. The elimination procedure computes the states x 2 = y, x 4 = ẏ, however the input-output relation does not allows us to compute the state variable x 5 ; ÿ = ẋ 4 = 2y. The unobservable state is not asymptotically stable, hence our system is not detectable. By using the general procedure by state elimination must fail. Indeed, from the observer equation ˆx 2 ˆx 4 ˆx 5 = x 4 2x D x 2 ˆx 2 x 4 ˆx 4 x 5 ˆx 5, with asymptotically stable gain matrix-d, the state x 5 can not be eliminated. Remark Rigorous proof of theorem requires the consideration of the bad, the nonuniversal inputs, which constitutes the real obstacle to the existence of observers, see the second part of Example. 2. Let us also distinguish the initialized and noninitialized observers. The non initialized observers work for any initial state of the observer, while the initialized observers only work for some initial state of the observer. The dependence of the observer on its initial states originates another difficulty, see [12]. 3 Input observability In the problem of input reconstruction, the first task consists in evaluating the input observability, distinguishing whether the changes of the input of a dynamic system are reflected as changes at the output, [17]. If a system is input observable, the input reconstruction problem consists in the synthesis of a device or a mechanism which has as input the measured outputs, and it should take place as output a signal that should converge to the observable input. One can easily notice that the input reconstruction problem is closely linked to the problem of system inversion. Definition 3.1 Consider the system 1). The input ut) is said observable if that input can be distinguished from zero by the output yt), i.e., if yt) = 0 for t 0, implies ut) = 0 for t > 0. The input observability of LTI systems is almost equivalent to their left invertibility, see [7]. Indeed, the input observability of ẋt) = Axt) + But) yt) = Cxt) + Dut), is equivalent to the following property: The outputs y 1,y 2, corresponding to the initial states x 1, x 2 and inputs u 1, u 2, respectively, are equal, if and only if x 1 and x 2 are undistinguishable and u 1 = u 2, [15]. More direct relationship can be established between these concepts, if the input set will be restricted to the smooth inputs U = { u; u0) = 0,..., u n1) 0) = 0 }. Indeed in this case input observability and system invertibility are equivalent. Thus, if the input is observable, we can reconstruct it. This properties have been used for fault detection and isolation, [14, 15, 21]. For nonlinear systems, the input observability and system invertibility have been related, [20]. The system inversion implies the input observability. System inverse can be obtain by state elimination. Let us consider a dynamic system ẋt) = fx, u) + gx, u)νt), x0) = x 0 yt) = hx); 7) where the functions f, g, h are of appropriate dimensions. f, g, h are algebraic analytic) functions, respectively. ν is an unknown input. In this context, the detectability observability) of ν can be defined by:

5 Definition 3.2 The input ν is said to be non detectable if for ν 0 the relation yx 0, x, u, 0) = yx 0, x, u, ν) is satisfied; otherwise the input ν is said detectable. Consequently, if the input ν is detectable, then it is possible to construct an input observer. Input observer, which can be based on state elimination, is an other dynamic system, with inputs u, y, ζt) = F ζ, u, u,..., y, ẏ,...) ηt) = Hζ, u, u,..., y, ẏ,...) 8) where in the limits t ), η ν. The observer can be an inverse system, [15, 20]. The reconstruction of ν may take place in different information granulation, for example, ν: 1) is zero or not binary granulation); 2) is reconstructed as functions. Nevertheless, each of these tasks can be solved by system inversion. Thus, the detectability notion of ν can be formulated based on the invertibility condition for the system 7) considering the state elimination: Definition 3.3 ν is said detectable if there exists a differential polynomial P u, u,..., y, ẏ,..., ν, ν,...) such that system 7) with input ν and output y, is left invertible. Then, there exists a dynamic system 8), the inverse system from u, y) ν, which allows to reconstruct ν, supposing that u, y, ν) satisfies the non equality P u, u,..., y, ẏ,..., ν, ν,...) 0. Remark 3.1 The role of the differential polynomial P u, u,..., y, ẏ,..., ν, ν,...) is the invertibility condition: certain determinant of a functional matrix is not zero. The inversion algorithm have been showed in [20]. Thus, the existence of an input observer in order to estimate an unknown input is based on a necessary and sufficient condition for the invertibility of systems,[15]. We can notice that the observer can depend on the derivatives of the output. These derivatives are also required to system invertibility, [14]. 4 Binary distillation column control case A distillation column is one of the most important example in process control theory and practice. Columns handle multicomponent feeds, however, many can be approximated by binary mixtures. A single feed stream is fed as saturated liquid at its bubblepoint onto the m-th) feed tray. But it is very difficult to obtain accurate, continuous measurements of the composition of nearly pure streams. The large holdup of liquid in the column, reboiler and reflux drum tend to make distillation control systems sluggish. Besides, there is also an effective time delay of up several minutes before a change in the feed or reflux stream is noticed at the intermediate trays. Feed flow rate mol/min) is measured, and its composition is an unmeasured disturbance. Many efforts have been made to robustly control distillation process, against the unmeasured disturbance. Unfortunately, the decoupling matrix of the standard model of an idealized binary distillation column, is ill-posed, see [11]. Its decoupling matrix is almost singular, becaused the compositions of successive trays are closed to each other. Dual composition control was proposed in [2]. Singular perturbation technics lead to aggregated models, see [10]. Takamatsu et al. have solved the disturbance attenuation problem, using differential geometric methods, [22]. The qualitative description of the stability of the binary distillation columns can be found in [16]. Our purpose is to design nonlinear observers, in order to recover the unmeasured disturbance for the dual composition control as it has been proposed in [18]. In our case, a binary system two components) is assumed with constant relative volatility throughout the column, that is, the vapour leaving the tray is in equilibrium with the liquid on the tray. The relative volatility is a measure of the ease of separation. This is the ratio between the tendency to vaporize two components. The greater value means easier separation see Figure 3). A simple vapour-liquid equilibrium relationship can be used kx) = αx 1 + α 1)x. On the other hand, the relative volatility is effected

6 Thus, the nonlinear state equations are kx), mole fraction in vapor α=10 α=5 α=2 α= x, mole fraction in liquid Figure 3: Relative Volatility. by the pressure. The relative volatility reduces as the pressure is raised. Besides, the relative volatility approaches unity as the pseudo-critical point of the mixture is reached. Any time delay will be neglected. We assume that the vapour rate vt) through all trays of the column is the same, both at transient time and steady state as well. The liquid flow rate leaving the trays is lt). The measured feed flow rate is ft), and the unmeasured composition is zt) at time t, which is considered as an unknown input. Figure 4 shows a schematic representation of the nomenclature used. Tray2 Traym-1 Traym Trayn-2 Trayn-1 Bottoms vt) vt) vt) vt) lt) lt) x n lt) lt) Reflux Reboiler Trayn Condenser Tray1 Reflux Drum Figure 4: Distillation column. x 1 H 1 ẋ 1 = vt)x 1 + vt)kx 2 ), H i ẋ i = lt) x i1 x i ) + +vt) kx i+1 ) kx i )), i = 2,..., m 1, H m ẋ m = lt) x m1 x m ) ft)x m + +vt) kx m+1 ) kx m )) + ft)z, H i ẋ i = lt) x i1 x i ) + ft)x i1 x i ) + +vt) kx i+1 ) kx i )), i = m + 1,..., n 1, H n ẋ n = lt) x n1 x n ) + ft)x n1 x n ) + +vt)x n vt)kx n ), where y 1 = x 1, z 1 = x n are the top and the bottom components, respectively. Our purpose is to design a nonlinear observer and input observer to estimate the unmeasured states and recover the unmeasured disturbance zt), in order to control robustly, the top and bottom compositions y 1 and z Observer design The observer is based on the elementary state elimination procedure. In this case, due to the distillation column nonlinear model, is not necessary to select the gain D as we will show next. We notice that from the last equation lt) + ft)) x n1 = H n ż 1 + lt) + ft)) z 1 vt)z 1 + vt)kz 1 ), 9) hence x n1 can be expressed by z 1. Let us define new, computed outputs z 2, z 3,..., z nm+2 and y 2, y 3,..., y m+1 in order to express the unmeasured states x 2, x 3,..., x n1, similarly to expression 9): z I = n i=ni+2 H i ẋ i + +l + f v)x n, 2 < I n m + 2, y 2 = H 1 ẋ 1 + vx 1 ), y I = lx 1 x I1 ) vx 1 I1 H i ẋ i, 2 < I m + 1. i=1

7 Then, recursively x I = 1 l + f z ni+1 + vkx I+1 )), I = n 1, n 2,..., m, x m1 = 1 l z nm+2 + vkx m ) fz). On the other hand, y I x I =, I = 2,..., m. αv + α 1)y I Let us consider the sum of all equations of the dynamics using the expressions from y m+1 and z nm+1 : z nm+1 y m+1 lx m = fz 10) The equation 10) allows to reconstruct the unknown input z input observer). H 1 ė 1 = v + 1)e 1, Then, the observer is H i ė i = l + 1)e i, i = 2,..., n, H 1 ˆx1 = v + 1)ˆx 1 + ) H i ė i = l + f + 1)e i, i = m, m + 1,..., n. y 2 vk + y 1 αv + α 1)y 2 ) In this particular case, it is easy to recognize that the y i+1 H i ˆxi = l + 1)ˆx i + v k observer gains are as follow αv + α 1)y )) i+1 y i d 1 = v + 1), k αv + α 1)y i d i = l + 1), i = 2,..., n, ly i1 y i d, i = l + f + 1), i = m, m + 1,..., n. αv + α 1)y i1 αv + α 1)y i i = 2,..., m 1, The vapour rate vt), liquid flow rate lt), feed flow rate ft), and the physical constants H H m ˆxm = l + f + 1)ˆx m + v kx m+1 ) i, i = 1,..., n are positive and slowly variants in the time, )) y m hence the error equation is asymptotically stable. k αv + α 1)y m ly m1 4.2 Controller design αv + α 1)y m1 l 1)y m + z αv + α 1)y m H m ˆxm = l + f + 1)ˆx m + v kx m+1 ) )) y m k αv + α 1)y m ly m1 αv + α 1)y m1 l 1)y m + z nm+1. αv + α 1)y m Here, formula 10) is expressed in terms of the outputs, and x m+1 is also computed recursively from z 1,..., z nm. Analogously, for i = m+1,..., n1, the corresponding x i, x i+1 can be computed from outputs z 1, z 2,...,: H i ˆxi = l + f + 1)ˆx i + vkx i+1 ) + z ni l + f z ni+1 + vkx i+1 )), i = m + 1,..., n 1, H n ˆxn = v l f)ˆx n + z 2 + z 1 ˆx n. The observer must be stable, i.e. the error et) = xt) ˆxt) must tend to zero as t goes infinity. The error equation is The dual composition problem supposes the measurement modelling) of the quality of the top or bottom product. The quality is measured by analyzers, for example, by using a final point analyzer. The final point temperature may depend on the states, and the disturbance terms. This dependence can be identified, and it is supposed to be functions of all variables y 1 t) = g Top xt), vt), lt), ft), zt)), z 1 t) = g Bottom xt), vt), lt), ft), zt)).

8 The purpose is to control the top and bottom composition. We suppose that top and bottom desired composition are y 1 t) = C 1 and z 1 t) = C 2, respectively. Let us show the case n = 5, as an example. Thus, the dynamics are written by H 1 ẋ 1 = vx 1 + vkx 2 ), H 2 ẋ 2 = lx 1 x 2 ) + vkx 3 ) kx 2 )), H 3 ẋ 3 = lx 2 x 3 ) + vkx 4 ) kx 3 )) fx 3 + fz, H 4 ẋ 4 = l + f)x 3 x 4 ) + vkx 5 ) kx 4 )), H 5 ẋ 5 = l + f)x 4 x 5 ) + vx 5 vkx 5 )), y 1 = x 1 z 1 = x 5. Hence the computed outputs are y 2 = H 1 ẋ 1 + vx 1 ) = H 1 ẏ 1 + vy 1 ), y 2 x 2 =, αv + α 1)y 2 y 3 = lx 1 x 2 ) vx 1 H 1 ẋ 1 H 2 ẋ 2, y 4 = lx 1 x 3 ) fx 3 vx 1 H 1 ẋ 1 H 2 ẋ 2 H 3 ẋ 3, z 2 = H 5 ẋ 5 + l + f v)x 5, z 3 = H 4 ẋ 4 + H 5 ẋ 5 + l + f v)x 5, z 4 = H 3 ẋ 3 + H 4 ẋ 4 + H 5 ẋ 5 + l + f v)x 5. Hence, fz = z 3 y 4 lx 3. Now, let us suppose that the desired output compositions y 1 = C 1, z 1 = C 2 are constant. Then, ẋ 1 = 0, ẋ 5 = 0, and from v 0 and 0 = C H 1 0 = vc 1 + vkx 2 ), x 2 = 1 αvα1)c 1, obviously C 1 α is supposed. x α1 2 is also a constant. Therefore, the system dynamics are reduced to ) C 1 0 = l C 1 + vkx 3 ) C 1 ), α α 1)C 1 11) 0 = l + f)x 4 C 2 ) + vc 2 vkc 2 )) 12) By substituting the unmeasured term fz = H 3 ẋ 3 + H 4 ẋ 4 + l + f v)c 2 + v l)c 1 is obtained. Considering the coupled system 11), 12), 13), x 3, and x 4 can be expressed algebraically in terms of l, v, f. For example, and x 4 = 1 v ) C 2 + v l + f l + f kc 2). 14) ẋ 4 = d dt Similarly for x 3, kx 3 ), and kx 4 ). ) v kc 2 ) C 2 ). 15) l + f The substitution of all terms is elementary but rather tedious computation. The obtained equation is the implicit form of the control law with one free control term. For example choosing lt), vt) is computed. For this example, let us suppose that C 1 = 0.98, C 2 = Thus, in the steady state, if H i = 100, F = 100, z = 0.04, α = 5, then, v = and l = Let us choose v = and lt) is computed. Figures 5, 6, and 7 show the simulation results: into the third equation H 4 ẋ 4 = l + f)x 3 l + l v)c 1 α α 1)C 1 vkx 4 ) kx 3 )) + v l f)c 2 13) C Figure 5: x 1 -vs- t.

9 Conclusions Figure 6: x 5 -vs- t. The reduction to the state elimination of the observer design for nonlinear systems with prescribed error dynamics and error outputs is an easy to use new method. The difficulties are really propres ones of the nonlinear systems: the excitation and the universality of the inputs, the dependence of the observer convergence on its initial states. The computational complexity is also hard, according to the hard complexity of the elimination algorithm. However, as in several example is shown, if the elimination can be reduced into, for example Gauss elimination output injection), then the method is effective. For linear systems, our method gives the same extended Luenberger s observers. Input observability for nonlinear systems based on state elimination have been defined. The state elimination procedure allows to establish an invertibility condition, which is lied to input observability condition. As application s example, the dual composition control of a binary distillation columns based on the un- Figure 7: l -vs- t. measured feed composition estimation has been proposed. Unmeasured composition unknown input) is estimated by using the state elimination procedure. This procedure is simpler than the disturbance decoupling. The difficulty of the method is due to the presence of measured composition derivatives. However, low order dynamics allows us to use low order derivatives. References [1] D. Bestle and M. Zeitz. Canonical form observer design for non-linear time-variable systems. Int. J. Control, 382): , [2] Y. Creff, J. Levine, and P. Rouchon. Qualitative behaviour of distillation columns and their control. In 1th European Control Conference, pages , Paris, [3] S. Diop. Elimination in control theory. Math. Control, Signals and Systems, 41):17 32, [4] S. Diop and M. Fliess. Nonlinear observabilitty, identifiability, and persistent trajecto-

10 ries. In 30th IEEE Conf. on Decision and Control, pages , Brighton, [5] S.T. Glad. Inonlinear regulators and ritt s remainder algorithm. In Proc. Colloque International sur l Analyse des Systémes Dynamiques Controlés, Lyon, [6] D. Guillaume, P. Rouchon, and J. Rudolph. Two simple observers for a class of polymerization reactors. In 4th European Control Conference, pages TU EF5, Brussels, [7] M. Hou and R.J. Patton. Input observability and input reconstruction. Automatica, 346): , [8] A.J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems & Control Letters, 3:47 55, [9] A.J. Krener and W. Respondek. Nonlinear observers with linearizable error dynamics. SIAM J. Control & Optimization, 23: , [10] J. Levine and P. Rouchon. Quality control of binary distillation columns based on nonlinear aggregated models. Automatica, 271): , [11] W. L. Luyben. Process Modeling, Simulation and Control for Chemical Engineers. McGraw-Hill Publishing, New York, [12] F. Olivier. Some theoretical problems in effective differential algebra and their relation to control theory. In Proc. NOLCOS 92, volume 1, pages , Barcelona, to 15th IFAC World Congress IFAC 2002), Barcelona - Spain, [16] H.H. Rosenbrock. A lyapunov function with applications to some nonlinear physical systems. Automatica, 11):31 53, [17] H.J. Sussmann. Single input observability of continuous time systems. Math Systems Theory, 1211): , [18] F. Szigeti, A. Ríos-Bolívar, and L. Rennola. Dual composition control of a binary distillation columns based on nonlinear state observer. In IEEE Conference on Decision and Control 2000, pages , [19] F. Szigeti, A. Ríos-Bolívar, and L. Rennola. Fault detection and isolation in the presence of unmeasured disturbance: Application to binary distillation columns. In 4th IFAC SAFE- PROCESS Conference, pages , Budapest - Hungary, [20] F. Szigeti and A. Rívar-Bolívar. System inversion and fault detection: The failure affine nonlinear case. In Proposed to MED 02 Control Conference, [21] F. Szigeti, C. Vera, J. Bokor, and A. Edelmayer. Inversion based fault detection and isolation. In 40th IEEE Conference on Decision and Control, pages , [22] T. Takamatsu, I. Hashimoto, and Y. Nakai. A geometric approach to multivariable system design of a distillation column. Automatica, 152): , [13] A.R. Phelps. On constructing nonlinear observers. SIAM J. Control & Optimization, 29: , [14] A. Ríos-Bolívar. Sur la Synthèse de Filtres de Détection et Diagnostic de Défauts dans les Systèmes Dynamiques. PhD thesis, Université Paul Sabatier, Toulouse, France, [15] A. Ríos-Bolívar, F. Szigeti, and G. Garcia. On fdi filters and system invertibility. In Accepted