Nonlinear State Estimation in the Presence of Multiple Steady States

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Ind. Eng. Chem. Res. 1996, 35, 2645-2659

2645

Nonlinear State Estimation in the Presence of Multiple Steady States Sairam Valluri and Masoud Soroush* Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19104

This paper concerns the problem of state estimation in nonlinear deterministic processes. For moderately nonlinear deterministic processes, the inadequacy of full-order Luenberger observer and extended Luenberger observer with constant gain is shown analytically and through numerical simulation of a chemical reactor. The results indicate that the convergence of an extended Luenberger observer with constant gain may be poorer than that of a simple linear Luenberger observer. A nonlinear observer design method with varying gain is applied to a classical exothermic stirred-tank reactor with multiple steady states. Under significant observerinitialization errors and several observer gains, the global asymptotic convergence of the nonlinear observer is demonstrated through numerical simulations. In the presence of the multiple steady states, the nonlinear observer converges to the same steady-state operating point at which the reactor operates, irrespective of the observer initial conditions. 1. Introduction One of the key requirements for the implementation of any controller, which includes a state feedback, is the availability of accurate information on all state variables. In practice, often on-line measurements of all the state variables are not available due to high cost of measurement, lack of a reliable sensor, or both. This difficulty can be overcome by using a state observer/ estimator, which is a deterministic/stochastic dynamic system designed based on a process model. An observer/ estimator can estimate the nonmeasurable state variables from the available measurements. Observers/ estimators are also used in process monitoring, where early detection of hazardous conditions is needed to ensure a safe operation. The foundation of linear observer design was laid by Luenberger (1964), who developed a deterministic version of the well-known Kalman filter (Kalman and Bucy, 1961), known as Luenberger observer. The theoretical properties of Luenberger observer and Kalman filter are well-understood and can be found in the pertinent textbooks (Chen, 1984; Friedland, 1986). Since the introduction of Kalman filter and Luenberger observer, numerous attempts have been made to study the issues involved in nonlinear state estimation and to develop reliable observer design methods for nonlinear processes. These efforts have led to the development of several observer design methods, which, according to their underlying derivation concept, can be divided into two main classes. The first class includes those nonlinear observer design methods which are a result of a straightforward extension of Luenberger observer or Kalman filter to nonlinear processes. Examples of this class are extended Luenberger observer and extended Kalman filter (Gelb, 1974; Jazwinski, 1970; Zeitz, 1987). A full-order nonlinear observer of this type may not possess a globally asymptotically stable observer-error dynamics. Furthermore, if the observer-error dynamics is indeed globally asymptotically stable, this stability is difficult to prove analytically. A reduced-order extended Luen* To whom correspondence should be addressed. Email: [email protected]. Phone: (215) 895-1710. Fax: (215) 895-5837.

berger observer with constant gain may, however, be adequate to achieve global convergence in many physically meaningful systems (Soroush, 1996). Extended Luenberger observers and extended Kalman filters (EKF) have been applied successfully to many chemical processes (Adebekun and Schork, 1989; Ellis et al., 1988; Jo and Bankoff, 1976; Kim and Choi, 1991; Kozub and MacGregor, 1992; Ogunnaike, 1994; Quintero-Marmol et al., 1991; Robertson et al., 1995). Recently, an optimization approach to state estimation has also been proposed (Michalska and Mayne, 1995). The second class includes the observer design methods whose derivations are based on the notion of linearization through coordinate transformation, adopted from nonlinear control theory. Most of the recent new results on nonlinear observer design are of the latter class. Examples include the method of output injection of Krener and Isidori (1983) and Krener and Respondek (1985), the Luenberger-like observer design method of Gauthier et al. (1992) and Ciccarella et al. (1993), and the reduced-order Luenberger observer design method of Kazantzis and Kravaris (1995). The first contribution of the second class was made by Krener and Isidori (1983) and Krener and Respondek (1985), who derived a set of conditions under which a nonlinear state-space model is transformed into a linearin-state-variables but nonlinear-in-input-and-output model. For this linearized-in-state-variables model, a linear observer is then designed, leading to an observer with a globally asymptotically stable error dynamics, for the original nonlinear process. Kantor (1989) applied this approach to an exothermic chemical reactor with a single first-order irreversible reaction. This observer design method is applicable to a very restricted class of nonlinear processes. For instance, it is not applicable to a chemical reactor where a single reaction with an order other than 1 or 0 takes place (Kantor, 1989). Adaptive versions of the nonlinear observer design method have also been implemented or studied (Limqueco et al., 1991; Marino, 1990; Bastin and Gevers, 1988). Hammouri and Gauthier (1988, 1992) also presented a nonlinear observer design method, which is known as observer design by bilinearization up to output injection. Another contribution of this class is the high-gain Luenberger-like observer of Tornambe` (1989), Gauthier et al. (1992), and Deza et al. (1992).

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2646 Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996

In particular, Gauthier et al. (1992) derived a set of conditions for global convergence of the nonlinear Luenberger-like observer. This observer has been applied to biological and free-radical polymerization reactors (van Dootingh et al., 1992; Gibbon-Fargeot et al., 1994), as well as a chemical reactor with classical reactions (Gibon-Fargeot et al., 1994). The work of Gauthier et al. (1992) was later generalized to systems with measurable inputs by Ciccarella et al. (1993), who addressed several limitations of the aforementioned existing approaches. This paper deals with the issues involved in state estimation in nonlinear deterministic processes. It demonstrates analytically and through simulation a case for which the Luenberger observer and extended Luenberger observer are inadequate. The nonlinear observer design method of Ciccarella et al. (1993) is applied to three continuous-stirred-tank reactors which exhibit multiple steady-state operating points. This study explores the potentials of the nonlinear observer design method, to estimate the state variables of a fairly nonlinear chemical reactor. Section 2 demonstrates the inadequacy of Luenberger and extended Luenberger observers for fairly nonlinear processes. This inadequacy is also shown by numerical simulation of an exothermic continuous-stirred-tank reactor (CSTR). This section will serve as a motivation for using nonlinear observers for nonlinear processes. Section 3 briefly reviews some definitions and the nonlinear observer design method of Ciccarella et al. (1993). Section 4 describes the performance of the nonlinear observer applied to three CSTR examples. Finally, the conclusions section evaluates the advantages and disadvantages of the nonlinear observer method. 2. Motivation: Inadequacy of Luenberger and Extended Luenberger Observers for Nonlinear Processes Most of the chemical processes are nonlinear and operate over wide ranges of operating conditions. For such a process, one can simply use a linear observer (Luenberger, 1964), which is designed based on a linear approximation of the process model around a steadystate operating point, or an extended Luenberger observer with constant gain (deterministic version of EKF with constant gain). It is the purpose of this section to demonstrate the inadequacy of these observers, even for a moderately nonlinear process. Consider a single-input single-output nonlinear system of the form:

{

x˘ ) f(x) + g(x) u, y ) h(x)

x(0) ) x0

(1)

where x ∈ IRn is the vector of state variables; y ∈ IR and u ∈ IR are measurable output and input, respectively; f and g are analytic vector functions; and h is an analytic scalar function. The pair (xeq, ueq) is said to be an equilibrium (steady-state) pair of the system of (1), if and only if x˘ eq ) f(xeq) + g(xeq) ueq ≡ 0, u˘ eq ≡ 0. Property 1: A process of the form of (1) is said to possess property 1, if (i) it has multiple steady-state operating points for a given ueq; that is, the process has a set of equilibrium pairs (xeq1, ueq), ..., (xeqp˜ , ueq), p˜ g 2 such that xeqi * xeqj, ∀i, j, i * j; and (ii) two of the equilibrium pairs, let us say (xeq1, ueq) and (xeq2, ueq), are locally asymptotically stable; i.e., all the eigenvalues of

A)

∂f(x) ∂g(x) + u ∂x ∂x

evaluated at (xeq1, ueq) and (xeq2, ueq) lie in the left half of the complex plane. 2.1. Luenberger Observer. For a process of the form of (1) with property 1, a straightforward application of Luenberger observer (Luenberger, 1964) results in the following linear observer:

xˆ˘ ) Ai(xˆ - xeqi) + bi(u - ueq) + Li[y - h(xeqi) ci(xˆ - xeqi)],

xˆ (0) ) xˆ 0 (2)

where xˆ represents the vector of estimated state variables;

Ai )

|

∂f(x) ∂x

x)xeq

+ ueq i

|

∂g(x) ∂x

;

x)xeq

bi ) g(xeqi);

i

ci )

|

∂h(x) ∂x

x)xeq

i

and Li ) [Li1...Lin]T ∈ IRn is the observer gain, which is chosen such that all the eigenvalues of the matrix (Ai - Lici) lie in the left half of the complex plane. Consider the case that u(t) ≡ ueq and the observer of (2) is designed based on the assumption that the process operates only in a vicinity of the steady state xeq2, i.e., i ) 2 in (2):

xˆ˘ ) A2(xˆ - xeq2) + L2[y - h(xeq2) - c2(xˆ - xeq2)], xˆ (0) ) xˆ 0 (3) where L2 is chosen such that all the eigenvalues of the matrix (A2 - L2c2) lie in the left half of the complex plane. The performance of this linear observer is examined under the two different sets of process initial conditions: (a) when the reactor is operated such that at steady state x ) xeq1 (in mathematical terms, when x0 is in the domain of attraction of xeq1); (b) when the reactor is operated such that at steady state x ) xeq2 (i.e., when x0 is in the domain of attraction of xeq2). In case a, since x0 is in the domain of attraction of xeq1 and (xeq1, ueq) is a locally asymptotically stable equilibrium pair of the process (1), x(t) governed by (1) will go to xeq1 and thus y to h(xeq1) in the limit as t f ∞. The linear observer of (3) is also asymptotically stable, and therefore in the limit as t f ∞, for all xˆ 0, the estimated values of the state variables approach the solution for ζ of the equation

0 ) (A2 - L2c2)(ζ - xeq2) + L2[h(xeq1) - h(xeq2)] Recall that L2 is chosen such that all the eigenvalues of the matrix (A2 - L2c2) lie in the left half of the complex plane. Therefore, the matrix (A2 - L2c2) is always nonsingular, and

ζ - xeq2 ) -(A2 - L2c2)-1L2[h(xeq1) - h(xeq2)] which implies that ζ * xeq1, unless

L2 ) -

1 A (x - xeq2) h(xeq1) - h(xeq2) - c2(xeq1 - xeq2) 2 eq1

Thus, the use of a linear observer designed based on an equilibrium point at which the process will not operate will lead to a permanent mismatch between the

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Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2647

On the other hand, in case b, since x0 is in the domain of attraction of xeq2, x(t) will go to xeq2 and therefore y to h(xeq2) as t f ∞. Moreover, in the limit as t f ∞ the estimated values of the state variables approach xeq2, because xeq2 is the only equilibrium point of the observer (3): xeq2 is the only solution for ζ of the equation

0 ) (A2 - L2c2)(ζ - xeq2)

Figure 1. Simulated exothermic CSTR.

Thus, in this case, the difference between the estimated and actual values of the state variables will always decay to zero asymptotically. The above analysis indicated that in the presence of multiple steady states, if a linear observer is not designed based on the same steady states at which the process will operate, there will be a permanent mismatch between the actual and estimated values of the state variables. 2.2. Extended Luenberger Observer. For a process of the form of (1) with property 1, let us use an extended Luenberger observer with constant gain of the form

xˆ˘ ) f(xˆ ) + g(xˆ ) u + Li[y - h(xˆ )],

k1

A 98 U1 k2

A 98 U2

xˆ (0) ) xˆ 0 (4)

where Li ) [Li1...Lin ∈ is the observer gain, which is chosen such that all the eigenvalues of the matrix (Ai - Lici) lie in the left half of the complex plane. In what follows, we show that the global convergence of this extended Luenberger observer (which is nonlinear and has a constant gain) is not guaranteed. Let u(t) ≡ ueq, and design the preceding nonlinear observer based on the assumption that the process operates only in a vicinity of the steady state xeq2 [i ) 2 in (4)]: ]T

observers for a moderately nonlinear process. The next subsection describes an exothermic reactor example which is used to show this inadequacy through numerical simulations. 2.3. Exothermic Reactor Example. Consider the CSTR shown in Figure 1, where the irreversible parallel reactions

kd

A 98 D

IRn

xˆ˘ ) f(xˆ ) + g(xˆ ) ueq + L2[y - h(xˆ )],

xˆ (0) ) xˆ 0

(5)

where L2 is chosen such that all the eigenvalues of the matrix (A2 - L2c2) lie in the left half of the complex plane. The performance of this nonlinear observer is examined under the two different sets of process initial conditions: (a) when the reactor is operated such that at steady state x ) xeq1 (when x0 is in the domain of attraction of xeq1); (b) when the reactor is operated such that at steady state x ) xeq2 (when x0 is in the domain of attraction of xeq2). In case a, since x0 is in the domain of attraction of xeq1 and the (xeq1, ueq) is a locally asymptotically stable equilibrium pair of (1), x will go to xeq1 and thus y to h(xeq1) as t f ∞. The observer of (5) has certainly one equilibrium point at xeq1. Additionally, it may have “spurious” equilibrium points which are the other roots of the equation

0 ) f(ζ) + g(ζ) ueq + L2[h(xeq1) - h(ζ)] Let us assume that one of the spurious equilibrium points of the observer, denoted by β (ζ ) β * xeq1), is locally asymptotically stable and that xˆ 0 is in the domain of attraction of this spurious equilibrium point of the observer. In this case, xˆ (t) goes to β * xeq1 in the limit as t f ∞, leading to a permanent mismatch between the actual and estimated values of the state variables. Thus, an extended Luenberger observer with constant gain may be inadequate for a nonlinear process. The same problem may arise in case b when x0 is in the domain of attraction of xeq2. These preceding analyses demonstrated analytically

take place. U1 and U2 are undesirable side products, and D is the desired product. The inlet stream does not contain U1, U2, or D. The dependence of the reaction rate constants k1, k2, and kd on temperature is given by ki ) Zi exp(-Eai/RT), i ) 1, 2, and kd ) Zd exp(-Ead/ RT). The parameters of the reactor are the same as in Soroush and Kravaris (1992). 2.3.1. Mathematical Model and Observer Problem. Under standard assumptions, a mole balance on the reactant A and an energy balance for the reactor yield the following dynamic model of the reactor:

{

CAi - CA dCA ) RA(CA,T) + dt τ q dT RH(CA,T) Ti - T ) + + dt Fc τ FcV

(6)

where

RA(CA,T) ) -k1CA3 - k2CA0.5 - kdCA RH(CA,T) ) (-∆H1)k1CA3 + (-∆H2)k2CA0.5 + (-∆Hd)kdCA and q is the rate of heat input to the reactor. It is assumed that the reactor temperature, T, is a measurable output, and CAi and q are measurable inputs. This CSTR example can exhibit multiple steady states. For example, for qss ) -1.030 kJ‚s-1 there are three steady-state operating points SS1 ) (CAss1, Tss1) ) (8.0 kmol‚m-3, 310.0 K), SS2 ) (CAss2, Tss2) ) (3.3 kmol‚m-3, 370.0 K), and SS3 ) (CAss3, Tss3) ) (1.3 kmol‚m-3, 400.0 K). The steady-state operating points SS1 and SS3 are stable, whereas the steady-state operating point SS2 is unstable. 2.4. State Estimation in the CSTR. In this example, the purpose of using an observer is to estimate the state variables CA and T by using the reactor dynamic model and the measurable inputs and output. It is assumed that q(t) ≡ qss ) -1.030 kJ‚s-1. 2.4.1. Luenberger Observer. For this reactor

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[ ] [

][]

C ˆ A - CAss Li Â d C i ) Ai + L 1 (T - T ˆ) T ˆ T ˆ - Tssi dt i2

Table 1. Reactor Initial Conditions

(7)

initial conditions

CA(0)

T(0)

IC1 IC2

6.3 0.3

300.0 410.0

where C ˆ A and T ˆ are the estimated values of the state variables CA and T, respectively;

[

∂RA(CA,T) 1 ∂CA τ Ai ) 1 ∂RH(CA,T) Fc ∂CA

∂RA(CA,T) ∂T ∂R 1 1 H(CA,T) Fc ∂T τ

]

(8) (CA ,Tss ) ssi

i

The process response is simulated by integrating the set of nonlinear ordinary differential equations in (6) with initial conditions IC1 or IC2, listed in Table 1. The estimated values of temperature and concentration are obtained by integrating the system (7) with initial conditions C ˆ A(0) ) 5.0 kmol‚m-3 and T ˆ (0) ) 350.0 K. A numerical integration step size of 10 s is used. The following two cases are considered: (a) when the observer (7) is designed based on the assumption that the process operates in the vicinity of the steady-state operating point SS1 [observer (7) with i ) 1]; (b) when the observer (7) is designed based on the assumption that the process operates in the vicinity of the steadystate operating point SS3 [observer (7) with i ) 3]. For both cases, the observer gains L1 and L3 are chosen to be [4.0 × 10-4 4.0 × 10-4]T, so that the eigenvalues of the matrices A1 - L1c1 and A3 - L3c3 all lie in the left half plane. Figure 2a depicts the actual and estimated values of the state variables when the observer (7) is designed based on the assumption that the process operates only in the vicinity of the steady-state operating point SS1 [observer (7) with i ) 1]. As this figure shows, (1) when the reactor initial conditions are those of IC1, both the observer and process reach the same steady-state values; that is, there is a zero steady-state observer error, and (2) when the reactor initial conditions are those of IC2, the process finally operates at the steadystate SS3, whereas the observer asymptotically approaches a value not equal to the steady-state SS3; that is, there exits a permanent observer error. Figure 2b depicts the actual and estimated values of the state variables when the observer (7) is designed based on the assumption that the process operates in the vicinity of the steady-state operating point SS3 [observer (7) with i ) 3]. As this figure shows, (1) when the reactor initial conditions are those of IC2, both the observer and process reach the same steady-state values; that is, there is a zero steady-state observer error, and (2) when the reactor initial conditions are those of IC1, the process finally operates at the steadystate SS1, whereas the observer asymptotically approaches a value not equal to the steady-state SS1; that is, there exits a permanent observer error. Figure 2c depicts the actual and estimated values of the state variables, for various observer gains listed in Table 2. Here the observer (7) is designed based on the assumption that the process operates in the vicinity of the steady-state operating point SS3 [observer (7) with i ) 3]. The observer and process initial conditions are C ˆ A(0) ) 0.2 kmol‚m-3; T ˆ (0) ) 340.0 K; CA(0) ) 6.0 kmol‚m-3; T ˆ (0) ) 360.0 K. As this figure shows, the farther from the origin the location of the dominant eigenvalue of the matrix (A3 - L3c3) , the faster but more oscillatory the profiles of the estimated state variables. These simulation results indeed confirm the theoretical insight on the poor performance of Luenberger

2.4.2. Extended Luenberger Observer. For the reactor example of (6), the extended Luenberger observer of (4) takes the form

[

][]

Â CAi - C ss R (C ˆ ,T ˆ ) + Li1 C ˆ A A d A τ ) + ˆ) ˆ Li2 (T - T dt T RH(C ˆ A,T ˆ ) Ti - T ˆ qss + + Fc τ FcV (9)

[ ]

The process response is simulated again by integrating the set of ordinary differential equations in (6) with initial conditions IC1 or IC2 listed in Table 1. The estimated values of temperature and concentration are obtained by integrating the system (9) with initial conditions C ˆ A(0) ) 5.0 kmol‚m-3 and T ˆ (0) ) 350.0 K. A numerical integration step size of 10 s is used. To illustrate the performance of the extended Luenberger observer and to compare this performance with that of Luenberger observer, the following two cases are considered: (a) when the observer (9) is designed based on the assumption that the process operates in the vicinity of the steady-state operating point SS1 [observer (9) with i ) 1]; (b) when the observer (9) is designed based on the assumption that the process operates in the vicinity of the steady-state operating point SS3 [observer (9) with i ) 3]. We choose the observer gains L1 ) [3.9 × 10-5 -5.7 × 10-4]T and L3 ) [-3.6 × 10-4 3.2 × 10-3]T, which place all the eigenvalues of the matrices (A1 - L1c1) and (A3 - L3c3) in the left halfplane. Figure 3a depicts the simulation results corresponding to the first case. When the reactor initial conditions are those of IC1, the reactor operates at the steady-state operating point SS1, but the extended Luenberger observer converges to a value which is physically meaningless and is neither SS1 nor SS3. This performance is observed, while the observer was designed based on the assumption that the reactor operates at the low-temperature low-conversion steady-state SS1. Therefore, the performance of the extended Luenberger observer is worse than that of the Luenberger observer; surprisingly, the nonlinear observer exhibits a poorer performance, compared to the linear observer. On the other hand, when the reactor initial conditions are those of IC2, the process operates at the steady-state operating point SS3, but the extended Luenberger observer converges to another value which is neither the operating point SS1 nor SS3. In summary, in this case, the use of the nonlinear (extended Luenberger) observer led to a poorer performance when the process was operated around the desired steady state (SS1) and to a better performance when the process was operated around the undesirable steady state (SS3). Figure 3b depicts the simulation results corresponding to the second case. When the reactor initial conditions are those of IC2, the reactor operates at the steadystate operating point SS3, but the extended Luenberger observer again converges to a totally new steady state, which is neither SS1 nor SS3. Recall that, in this case, the observer was designed based on the assumption that the reactor operates at the high-temperature highconversion steady-state SS3. Thus, the performance of

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Figure 2. (a) Actual and estimated values of the state variables (estimated by using Luenberger observer of (7) designed for steady-state SS1): thick solid line, actual state variable under IC1; thin solid line, actual state variable under IC2; dashed line, estimated state variable corresponding to IC1; dotted line, estimated state variable corresponding to IC2. (b) Actual and estimated values of the state variables (estimated by using Luenberger observer of (7) designed for steady-state SS3): thick solid line, actual state variable under IC1; thin solid line, actual state variable under IC2; dashed line, estimated state variable corresponding to IC1; dotted line, estimated state variable corresponding to IC2. (c) Actual and estimated values of the state variables (estimated by using Luenberger observer of (7)), for various observer gains (see Table 2). Table 2. Luenberger Observer Gains

Table 3. Extended Luenberger Observer Gains

case

L21

L22

case

L21

L22

A1 A2 A3

1.0 × 10-4 1.0 × 10-2 3.0 × 10-2

1.0 × 10-4 1.0 × 10-3 1.0 × 10-3

B1 B2 B3

3.0 × 10-4 -3.6 × 10-4 -2.1 × 10-4

3.0 × 10-4 3.2 × 10-3 1.9 × 10-3

that of the Luenberger observer; the nonlinear observer exhibits a poorer performance, compared to the linear observer. On the other hand, when the reactor initial conditions are those of IC1, the extended Luenberger observer converges to the steady-state operating point SS1, the same steady-state operating point at which the reactor operates; the estimated values of the state variables asymptotically converge to their actual values. In summary, in this case, the use of the nonlinear observer led to a better performance around the undesirable steady state (SS1) but a poorer performance around the desired steady state (SS3). From these simulation results, it can be concluded that (1) the extended Luenberger of (9) is also inadequate for this fairly nonlinear reactor because of its inability to estimate the exact values of the state variables asymptotically and (2) the convergence performance of a linear observer may be better than that of a poorly-designed nonlinear observer. Figure 3c depicts the actual and estimated values of the state variables, for various observer gains listed in Table 3. These simulation results indicate that it is possible to improve the poor performance of the extended Luenberger observer, by “fine-tuning” the observer gain. Here the observer (9) is designed based on the assumption that the process operates in the vicinity of the steady-state operating point SS3 [observer (9) with i ) 3]. The same observer initial conditions as in the simulations of parts a and b of Figure 3 are used. The reactor initial conditions are those of IC2. Although for all the listed values of the observer gain (see Table 3) both eigenvalues of the matrix (A3 - L3c3) have negative real parts, only the estimated state variables represented by line B1 asymptotically converge to their actual values. The estimated state variables repre-

values, resulting in significant permanent errors. Recall that having negative real parts for the eigenvalues of the matrix (Ai - Lici) is a necessary condition but not sufficient to guarantee global observer convergence. These simulation results confirm the theoretical analysis presented in subsection 2.2 that this nonlinear observer may lack a global convergence. 3. Nonlinear Observer Design This section briefly reviews a few definitions and the nonlinear observer design method of Ciccarella et al. (1993). This observer design method will be used later to reconstruct the state variables of three chemical reactor examples. The global convergence of the nonlinear observer will be shown through numerical simulations. 3.1. Preliminaries. Lie derivative, directional derivative, of the scalar function h(x) in the direction of the vector function f(x) is denoted by Lfh(x) and is defined by: n

Lfh(x) )

∑ i)1

∂h(x) ∂xi

fi(x)

Higher order Lie derivatives can be obtained recursively:

Lfkh(x) ) Lf(Lfk-1h(x)); LgLfk-1h(x) ) Lg(Lfk-1h(x)),

k ) 1, 2, ...

where Lf0h(x) ) h(x). Relative order of the system of (1), denoted by r, is

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Figure 3. (a) Actual and estimated values of the state variables (estimated by using extended Luenberger observer of (9) designed for steady-state SS1): thick solid line, actual state variable under IC1; thin solid line, actual state variable under IC2; dashed line, estimated state variable corresponding to IC1; dotted line, estimated state variable corresponding to IC2. (b) Actual and estimated values of the state variables (estimated by using extended Luenberger observer of (9) designed for steady-state SS3): thick solid line, actual state variable under IC1; thin solid line, actual state variable under IC2; dashed line, estimated state variable corresponding to IC1; dotted line, estimated state variable corresponding to IC2. (c) Actual and estimated values of the state variables (estimated by using extended Luenberger observer of (9)), for various observer gains (see Table 3).

y which depends explicitly on the input u: r is the smallest integer for which LgLfr-1h(x) * 0. Definition 1: For a process of the form of (1), the n × n matrix

Q)

∂ ∂x

{[ ]}

(10)

is called the observability matrix. Remark 1: By using the definition of the relative order r, the observability matrix Q is given by

[ ][ y dy dt · · ·

∂Φ(x,U) ∂x

h(x) Lfh(x) · · · Lfr-1h(x)

dr-1y dtr-1 r Φ(x,U) ) d y ) Lfrh(x) + LgLfr-1h(x) u dtr ψr+1(x,U) dr+1y · r+1 · dt · · ψn-1(x,U) · · n-1 d y n-1

[

ψl(x,U) ) Lf+guψl-1(x,U) +

dn-1y dtn-1

where

u(1)(t)...u(n-r-1)(t)]T ) du(t) dun-r-1(t) ... u(t) dt dtn-r-1 m(l)

y dy dt · · ·

Q(x,U) )

U(t) ) [u(0)(t)

(11)

]

]

T

∂ψl-1(x,U) , ∂u(k) l ) r + 1, ..., n - 1

u(k+1) ∑ k)0

with ψr(x,U) ) Lfrh(x) + LgLfr-1h(x) u and m(l) ) min (l - 3, r - 1). Remark 2: In the special case of a linear system, i.e., h(x) ) cx and f(x) ) Ax, the observability matrix Q(x,U) ) [c cA2...cAn-1]T, which is the well-known observability matrix and is independent of x and u. This implies that, in linear systems, observability is a global property; if a linear system is observable, it is observable everywhere. Remark 3: State variables of a process can be reconstructed by using a state observer, if the process is observable. As mentioned earlier, in the case of a linear system, observability is a global property and can be checked simply by calculating the observability matrix. For a nonlinear system, however, observability may not be a global property and the observability concept is more complex (Kantor, 1989). Here we will say that a nonlinear process is locally observable on a set, if the observability matrix of the process is nonsingular everywhere in the set. Definition 2: A function f(x) is said to be Lipschitz on X ⊂ IRn if and only if, for every x1, x2 ∈ X, there exists a finite constant γ, which is known as the Lipschitz constant, such that

|f(x1) - f(x2)| e γ|x1 - x2| Definition 3: A function f(x) is said to be globally Lipschitz if and only if it is Lipschitz on X ) IRn. 3.2. Luenberger-Like Nonlinear Observer Design Method. A key step in the derivation of the

+

+


(1993) is to rewrite a process model in the form of (1) in the state coordinates

z ) Φ(x,U)

(12)

In these new coordinates, the process of (1) takes the Brunowsky canonical form

{

z˘ ) A ˜ z + b˜ ψn[Φ-1(z,U),U,u(n-r)] y ) c˜ z

[

0

1

0

0 A ˜ ) ·· · 0 0

0 · · · 0 0

1

· · ·

· · · ·· 0 · · · · 0 0 · · ·

]

[]

u(k+1) ∑ k)0

dψn-1[x,U] du(k)

0 · · · n×n 0 ∈ IR ; 1 0

K[y - h(xˆ )], xˆ (0) ) xˆ 0 (17)

∀xˆ (0) ∈ IRn

Implementation of the state observer of (17), which can have a globally asymptotically decaying observer error, requires the first (n - r - 1)th time derivatives of the measurable input to be available. In practice, these derivatives are not available but can be calculated approximately. 3.2.1. Computation of the Time Derivatives of u(t). One way to calculate the derivatives of the measurable input approximately is to assume that the measurable input can be represented by piecewise continuous polynomial functions of order p:

∀l g p + 1

(18)

Equivalently, these functions can be represented by the following state-space dynamic system:

c˜ ) [1

{

0...0] ∈ IR1×n

For a moment, let us assume that the first (n - r 1) time derivatives of the measurable input u are available. Then, for the above transformed system, use a Luenberger-like observer of the form

zˆ˘ ) A ˜ zˆ + b˜ ψn[Φ-1(zˆ ,U),U,u(n-r)] + K(y - c˜ zˆ ), zˆ (0) ) Φ(xˆ 0,U0) (14) where zˆ is the estimated value of the vector z; K ∈ IRn is the observer gain. In this case, the estimation error e˜ ) zˆ - z is governed by

ψn[Φ-1(z,U),U,u(n-r)]},

{

e˜ (t) ) e(A˜ -Kc˜ )te˜ 0 +

{[ ]

[In-r | 0], I , C′ ) n-r Ip+1 , 0

Ciccarella et al. (1993) have shown that the observer (14) will have a globally asymptotically decaying observer error; that is,

∀zˆ (0) ∈ IRn

if (i) the observer gain K is chosen such that all the eigenvalues of the matrix (A ˜ - Kc˜ ) lie strictly in the left half of the complex plane; (ii) the observability matrix Q(x,U) has full rank for every (x,U) ∈ IRn × IRn-r; and (iii) ψn(Φ-1[z,U],U,ξ) is globally Lipschitz in (z, U, ξ). The observer of (14) in the original coordinates takes

(19)

n-rp+1

Here Il represents the identity matrix of dimension l. 3.2.2. Observer Structure. Consider the process model of (1) together with the input model of (19):

{

x˘ ) f(x) + g(x) D1 D˙ ) A′D

∫0te(A˜ -Kc˜)(t-τ)b˜ {ψn[Φ-1(zˆ (τ),U(τ)),U(τ),u(n-r)(τ)] ψn[Φ-1(z(τ),U(τ)),U(τ),u(n-r)(τ)]} dτ (16)

D˙ ) A′D U ) C′D

where D ) [D1...Dp+1]T; A′ is a Brunowsky canonical matrix of dimension p + 1; C′ is an [(n - r) × (p + 1)] matrix, defined by

e˜ (0) ) e˜ 0 (15)

The solution for e˜ (t) is given by

D˙ 1 ) D2 · · · D˙ p ) Dp+1 D˙ p+1 ) 0

where Dl(t) ) u(l-1)(t), l ) 1, ..., p + 1, which, in a compact form, can be written as

e˜˘ ) (A ˜ - Kc˜ )e˜ + b˜ {ψn[Φ-1(zˆ ,U),U,u(n-r)] -

tf∞

]

lim ||xˆ (t) - x(t)|| ) 0,

u(l)(t) ≡ 0,

0 0 · b˜ ) · ∈ IRn×1; · 0 1

lim ||e˜ (t)|| ) 0,

∂xˆ

-1

tf∞

ψn[x,U,u(n-r)] ) m(n)

[

∂Φ(xˆ ,U)

where xˆ is the estimated value of the state vector x, which under the three aforementioned assumptions has the desirable property that

(13)

where

Lf+guψn-1[x,U] +

xˆ˘ ) f(xˆ ) + g(xˆ ) u +

(20)

while measurable outputs are h(x) ) y and D1 ) u. For this augmented system, design an observer of the form of (14):

[] [

] ][ ]

zˆ˘ A ˜ zˆ + b˜ ψn[Φ-1(zˆ ,C′D ˆ ),C′D ˆ ,uˆ (n-r)] ) + A′D ˆ D ˆ˙

[

K′ 0

y - zˆ 1 0 ˆ1 K′′ u - D

(21)

where K′ ) [K′1...K′n]T and K′′ ) [K′1′...K′p′+1]T are the observer gains, which are chosen such that all the roots of the two equations

λn + K′ λn-1 + ... + K′

λ + K′ ) 0

(22)

+

+


λp+1 + K′1′λp + ... + K′p′λ + K′p′+1 ) 0

(23)

lie strictly in the left half of the complex plane. The observer of (21) in the original coordinates takes the form described in the following theorem. Theorem 1 (Ciccarella et al., 1993): For a process of the form of (1), the nonlinear state observer

{[

xˆ˘ ) f(xˆ ) + g(xˆ ) D ˆ1 + ∂Φ(xˆ ,C′D ˆ) ∂Φ(xˆ ,C′D ˆ ) -1 K′′[u - D ˆ 1] K′[y - h(xˆ )] ∂xˆ ∂(C′D ˆ) D ˆ˙ ) A′D ˆ + K′′[u - D ˆ 1] (24)

]{

}

with a proper choice of the observer gains K′ and K′′, will have a globally asymptotically decaying observer error:

lim ||xˆ (t) - x(t)|| ) 0, tf∞

∀xˆ (0) ∈ IRn

if (i) Q(x,U) has full rank for every (x,U) ∈ IRn × IRn-r and (ii) ψn[Φ-1(ζ,U),U,ξ] is globally Lipschitz in (ζ, U, ξ). A physically meaningful system of the form of (1) normally operates (is defined) only within a physically feasible region; that is, for such a process there exists a bounded, connected set Ω ⊂ IRn × IRn-r such that (x(t), U(t)) ∈ Ω for every t g 0. In such a case, the conditions for global asymptotic decay of the observer error are greatly weakened, as stated in the following corollary. Corollary 1 (Ciccarella et al., 1993): The observer error converges exponentially to zero, if the following conditions are satisfied: (i) there is a bounded, connected set Ω such that (x(t), U(t)) ∈ Ω for every t g 0; (ii) Q(x,U) has full rank for every (x, U) ∈ Ω; (iii) ||xˆ (0) - x(0)|| e β, for a suitable β > 0; (iv) roots of the equations (22) and (23) all lie strictly in the left half of the complex plane; (v) ψn[φ-1(ζ,U),U,ξ] is locally Lipschitz in (ζ, U, ξ). The observer of (24) can be simplified by considering special cases such as (I) systems with a constant measurable input or no measurable input and (II) systems whose relative orders are equal to its number of state equations. 3.2.3. Special Case I: Processes with Constant Input. In the case of processes with constant input, the observability matrix Q(x,U) ) Q(x), and therefore there is no need for the calculation of a time derivative of the measurable input. For these processes, Theorem 1 and Corollary 1 simplify into the following theorem and corollary. Theorem 2 (Ciccarella et al., 1993): For a process of the form of (1) with u(t) ≡ uo, u˘ o ) 0, the nonlinear state observer

xˆ˘ ) f(xˆ ) + g(xˆ ) uo + [Q(xˆ )]-1K′[y - h(xˆ )], xˆ (0) ) xˆ 0 (25) with a proper choice of the observer gain K′ ∈ IRn will have a globally asymptotically decaying observer error:

lim ||xˆ (t) - x(t)|| ) 0, tf∞

∀xˆ (0) ∈ IRn

if (i) Q(x) has full rank for every x ∈ IRn and (ii) Lfnh(φ-1(z)) is globally Lipschitz. Corollary 2 (Ciccarella et al., 1993): The observer error converges exponentially to zero, if the following

nected set Ωx such that x(t) ∈ Ωx for every t g 0; (ii) Q(x) has full rank for every x ∈ Ωx; (iii) ||xˆ (0) - x(0)|| e β, for a suitable β > 0; (iv) roots of the equation (22) all lie strictly in the left half of the complex plane; (v) Lf+guonh[Φ-1(ξ)] is Lipschitz on Ωx. 3.2.4. Special Case II: Processes with a Varying Input and r ) n. In the case of processes with a varying input and r ) n, the observability matrix Q(x,U) ) Q(x), and therefore there is no need for the calculation of a time derivative of the measurable input. For these processes, Theorem 1 and Corollary 1 simplify into the following theorem and corollary. Theorem 3 (Ciccarella et al., 1993): For a process of the form of (1) with r ) n, the nonlinear state observer

xˆ˘ ) f(xˆ ) + g(xˆ ) u + [Q(xˆ )]-1K′[y - h(xˆ )], xˆ (0) ) xˆ 0 (26) with a proper choice of the observer gain K′ ∈ IRn will have a globally asymptotically decaying observer error:

lim ||xˆ (t) - x(t)|| ) 0, tf∞

∀xˆ (0) ∈ IRn

if (i) Q(x) has full rank for every x ∈ IRn and (ii) there is a γ > 0 such that

sup ||Lfnh(Φ-1(ζ1)) + uLgLfn-1h(Φ-1(ζ1)) -

||u||eM

Lfnh(Φ-1(ζ2)) - uLgLfn-1h(Φ-1(ζ2))|| e γ||ζ1 - ζ2|| for every ζ1, ζ2 ∈ IRn. Corollary 3 (Ciccarella et al., 1993): The observer error converges exponentially to zero, if the following conditions are satisfied: (i) there is a bounded, connected set Ω′ such that (x(t), u(t)) ∈Ω′ for every t g 0; (ii) Q(x) has full rank for every x ∈Ωx; (iii) ||xˆ (0) - x(0)|| < β, for a suitable β > 0; (iv) roots of the equation (22) all lie strictly in the left half of the complex plane; (v) Lfnh(Φ-1(ξ)) and LgLfn-1h(Φ-1(ξ)) are locally Lipschitz in ξ. The work of Ciccarella et al. (1993) states the conditions for global asymptotic convergence of the nonlinear observer, which is applicable to a wide class of nonlinear processes. Apart from the need for differentiation of measurable inputs to a process, a major limitation of this observer design method is the need for checking a few functions to be globally Lipschitz. Each of these functions is normally in an implicit form, which makes checking the global Lipschitz property almost impossible. This limitation will be explained further when this observer design method is applied to three fairly nonlinear chemical reactors in the next section. 4. Application to Reactor Examples This section demonstrates the application of the nonlinear observer method, described in the previous section, to three chemical reactor examples. The first reactor has a constant measurable input; the second reactor has a time-varying measurable input and its relative order is equal to the number of its state equations (r ) n); and the third reactor has a timevarying measurable input and its relative order is less than the number of its state equations (r < n). Indeed, the three reactors will be used to illustrate the application of the observers of Theorems 1, 2, and 3, respectively. 4.1. Example 1: A CSTR with Constant Input.

+

+

[]

measurable input q(t) ≡ qss:

{

dCA ss ) RA(CA,T) + dt τ (C ,T) T R T qss H A i dT ) + + dt Fc τ FcV

x)

[ ]

CA ; T

f(x) )

[

RA(CA,T) +

CAi - CA ss

(27)

]

τ ; RH(CA,T) Ti - T + Fc τ 0 g(x) ) 1 ; FcV

[ ]

{

]

(28)

This observability matrix, Q(x), is nonsingular almost everywhere, and therefore the observer of Theorem 2 can be used for this example. The observer takes the form:

{

[

RH(CA,z1) ) Fc z2 -

y)T

∂RH(CA,T) ) ∂CA 3(-∆H1)k1CA2 + 0.5(-∆H2)k2CA-0.5 + (-∆Hd)kd ∂RH(CA,T) ) ∂T (-∆H1)k1E1CA2 + (-∆H2)k2E2CA0.5 + (-∆Hd)kdEdCA

ˆ A) (CAi - C dC Â ss ) RA(C ˆ A,T ˆ) + dt τ 1 1 ∂RH (C ˆ A,T ˆ) - K′2 K′1 Fc ∂T τ (T - T ˆ) 1 ∂RH (C ˆ ,T ˆ) Fc ∂CA A ˆ A,T ˆ ) Ti - T ˆ qss dT ˆ RH(C ) + + + K′1(T - T ˆ) dt Fc τ FcV (29)

(

]

(30)

]

(31)

where P(z1,z2) represents the implicit function corresponding to the solution for CA of

where

RT2

[

CA P(z1,z2) ) Φ-1(z1,z2) ) z1 T

T ∂ R (C ,T) T - T qss ) Q(x) ) i ∂x H A + + Fc τ FcV 0 1 1 ∂RH(CA,T) 1 ∂RH(CA,T) 1 Fc ∂CA Fc ∂T τ

[

[

T qss RH(CA,T) Ti - T + + Fc τ FcV

[ ]

)

The observer gain is selected such that the roots of λ2 + K′1λ+ K′2 ) 0 have negative real parts. To prove the local convergence of the observer, it is essential to verify that the function Lf+guo2h(Φ-1(z)) is locally Lipschitz. Here, z ) Φ(CA,T) can be calculated

]

qss Ti - z1 τ FcV

Lf)guo2h(Φ-1(z1,z2)) is calculated by substituting (31) in Lf+guo2h(x), where:

Lf+guo2h(x) )

]

]

However, an explicit expression for Φ-1(z1,z2) cannot be found:

4.1.1. Observer Design. The observability matrix of this reactor is (using the definition (10)):

[

[

h(x) z1 z2 ) Φ(CA,T) ) Lf+guoh(x) )

CAi - CA

In terms of the general form of (1),


(

( )(

)

CAi - CA ss 1 ∂RH(CA,T) RA(CA,T) + + Fc ∂CA τ

)

qss 1 ∂RH(CA,T) 1 RH(CA,T) Ti - T + + Fc ∂T τ Fc τ FcV

By using the chain rule, it can be verified that Lf+guo2h(Φ-1(z1,z2)) is continuously differentiable at least locally. This implies that Lf+guo2h(Φ-1(z1,z2)) is locally Lipschitz (Khalil, 1992). However, it is impossible to prove analytically that Lf+guo2h(Φ-1(z1,z2)) is globally Lipschitz. With this limitation to prove the global Lipschitz property, we have to rely on numerical simulations to show the global convergence of the observer. 4.1.2. Simulation Results. Numerical simulations are performed to verify the theoretical properties of the nonlinear observer. The actual process response is obtained by integrating the set of differential equations in (6). The estimated values of the concentration and temperature are obtained by integrating the observer system of (29). A numerical-integration step size of 10 s is used. Simulation results are organized as follows. The effect of observer gain on the rate of convergence is first studied. Observer tolerance to various initialization errors is then examined. Finally, the global convergence of the observer is demonstrated in the presence of multiple steady states. Figure 4a depicts the actual and estimated values of the state variables for various observer gains, listed in Table 4. The initial conditions of the actual process and observer are CA(0) ) 6.0 kmol‚m-3, T(0) ) 360 K, C ˆ A(0) ) 0.2 kmol‚m-3, and T ˆ (0) ) 340 K. Note that there is a significant mismatch between the initial conditions of the actual process and those of the observer. Since the process initial conditions lie in the domain of attraction of the high-conversion high-temperature steady-state operating point, the reactor operates at this operating point asymptotically. As this figure shows, the smaller max [Real(λ1),Real(λ2)], where λ1 and λ2 are the roots of λ2 + λK′1 + K′2 ) 0, the faster the rate of convergence of the observer. This is in complete agreement with the theoretical results presented in the previous section. Figure 4b depicts the actual and estimated values of the state variables for various observer initial conditions, listed in Table 5. The initial conditions of the actual process are CA(0) ) 6.0 kmol‚m-3 and T(0) ) 360

+

+


Figure 4. (a) Actual and estimated values of the state variables (estimated by using nonlinear observer of (29)), for various observer gains (see Table 4). (b) Actual and estimated values of the state variables (estimated by using nonlinear observer of (29)), for various observer initial conditions (see Table 5). (c) Actual and estimated values of the state variables (estimated by using nonlinear observer of (29)): thick solid line, actual state variable under IC1; thin solid line, actual state variable under IC2; dashed line, estimated state variable corresponding to IC1; dotted line, estimated state variable corresponding to IC2. Table 4. Nonlinear Observer Gains case

K′1

K′2

C1 C2 C3

1.0 × 10-2 1.0 × 10-1 1.0 × 100

1.0 × 10-5 1.0 × 10-4 1.0 × 10-2

Table 5. Initial Conditions for Nonlinear Observer case

C ˆ A(0)

T ˆ (0)

D1 D2

8.0 0.2

340.0 380.0

K′ ) [0.1 5 × 10-5]T. Since the process initial conditions lie in the domain of attraction of the highconversion high-temperature steady-state operating point, the reactor will operate at this operating point asymptotically. As this figure shows, over a wide range of mismatch between the initial conditions of the observer and those of the actual process, the estimated values of the state variables converge to the actual values. So far simulation results have concerned the convergence of the nonlinear observer when the CSTR operates in the vicinity of the high-temperature high-conversion steady-state operating point. This reactor has two stable steady states (SS1 and SS3), and hence the reactor can be operated at either steady state. The purpose of this simulation is to demonstrate the estimation capability of the nonlinear observer in the presence of multiple steady states. The observer gain is chosen to be K′ ) [1.0 × 10-2 1.0 × 10-6]T. The following observer initial conditions are used: C ˆ A(0) ) 5.0 kmol‚m-3; T ˆ (0) ) 350.0 K. The corresponding actual and estimated values of the state variables are shown in Figure 4c. Whether the reactor initial conditions are those of IC1 or IC2 (listed in Table 1), both the observer and process reach the same steady-state values; that is, there is a zero steady-state observer error. Recall that the initial conditions IC1 are in the domain of attraction of the low-conversion low-temperature steadystate SS1, whereas the initial conditions IC2 are in the domain of attraction of the high-conversion high-temperature steady-state SS3. This simulation demonstrates that the observer always converges to the same

operates, irrespective of the initial mismatch; it proves the global convergence capability of the nonlinear observer. 4.1.3. Comparison of the Performances of the Luenberger, Extended Luenberger, and Nonlinear Observers. In subsubsections 2.4.1 and 2.4.2, the state variables of this reactor were estimated under the same initial conditions, by using Luenberger observer and extended Luenberger observer, respectively. A comparison of these simulation results explains the advantages of the nonlinear observer over the Luenberger and extended Luenberger observers. The convergence of estimated values of the state variables to the actual values in the case of nonlinear observer (Figure 4a) was significantly faster as well as free from oscillations and overshoot when compared with Luenberger observer (Figures 2c) and extended Luenberger observer (Figure 3c, line A1). In the case of nonlinear observer, the rate of decay of observer error as well as the shape of the observer error profile can be adjusted easily by properly placing the roots of the equation (22) in the complex plane, whereas in the case of Luenberger and extended Luenberger observers, the relation between the shape of observer error profile and the locations of the eigenvalues of the matrix (Ai - Lici) is not that clear. While the nonlinear observer is tolerant to huge initialization errors (Figure 4c), the extended Luenberger observer (Figure 3c) resulted in spurious values for estimated state variables. The Luenberger observer is tolerant to initialization errors (Figure 2a-c), as long as the observer is designed based on the same steadystate operating point at which the process will operate asymptotically. In the case of multiple steady states, the nonlinear observer (Figure 4c) exhibits global convergence capability. While application of Luenberger observer (Figure 2a,b) is restricted by the initial mismatch between the estimated and actual values of the state variables, application of extended Luenberger observer (Figure 3a,b) is restricted by many factors including the initial mismatch between the estimated and actual values of the state variables. 4.2. Example 2: A CSTR with r ) n. The reactor

+

+


measurable input has a relative order of 2 and also a dimension of 2 (r ) n ) 2). In this case, the CSTR model is in the form of (1), where

x)

[ ]

CA ; T

[

]

CA RA(CA,T) τ f(x) ) ; RH(CA,T) Ti - T qss + + Fc τ FcV 1 y)T g(x) ) τ ; 0

[]

4.2.1. Observer Design. The observability matrix of this reactor is given by:

[

][

Q(x) )

0 T ∂ R Ti - T q ) 1 ∂RH H ∂x + + Fc ∂CA Fc τ FcV

1 1 ∂RH 1 Fc ∂T τ

]

where ∂RH/∂CA and ∂RH/∂T are given by (28). Because the observability matrix Q(x) is nonsingular almost everywhere, the observer of Theorem 3 can be used for this example. It takes the form:

{

dC Â dt

) RA(C ˆ A,T ˆ) +

ˆ A) (CAi - C

τ 1 1 ∂RH (C ˆ ,T K′1 ˆ) - K′2 Fc ∂T A τ (T - T ˆ) 1 ∂RH (C ˆ A,T ˆ) Fc ∂CA ˆ A,T ˆ ) Ti - T ˆ dT ˆ RH(C qss ) + + + K′1(T - T ˆ) dt Fc τ FcV (32)

(

)

The observer gain is selected such that the roots of λ2 + K′1λ + K′2 ) 0 have negative real parts. To prove the local asymptotic convergence of the observer, we have to verify that the functions Lf2h(Φ-1(z)) and LgLfh(Φ-1(z)) are locally Lipschitz. For this example, z ) Φ(CA,T) can be calculated easily:

[]

[

]

[

]

T z1 h(x) RH T i - T qss ) Φ(C ,T) ) ) A z2 Lfh(x) + + Fc τ FcV (33)

However, an explicit expression for Φ-1(z1,z2) cannot be found:

[ ]

[

CA P(z1,z2) ) Φ-1(z1,z2) ) z1 T

]

(34)

where P(z1,z2) represents the implicit function corresponding to the solution for CA of

[

RH(CA,z1) ) Fc z2 -

]

qss Ti - z1 τ FcV

The functions Lf2h(Φ-1(z1,z2)) and LgLfh(Φ-1(z1,z2)) can be calculated by substituting (34) for Lf2h(x) and LgLf-

Lf2h(x) )

(

) )(

CAi - CA 1 ∂RH RA + + Fc ∂CA τ

(

)

qss 1 ∂RH 1 RH Ti - T + + Fc ∂T τ Fc τ FcV

LgLfh(x) )

1 ∂RH Fcτ ∂CA

By using the chain rule, it can be verified that the functions Lf2h(Φ-1(z1,z2)) and LgLfh(Φ-1(z1,z2)) are continuously differentiable at least locally. This implies that these functions are locally Lipschitz (Khalil, 1992). However, it is impossible to prove analytically that Lf2h(Φ-1(z1,z2)) and LgLfh(Φ-1(z1,z2)) are globally Lipschitz. With this limitation, to prove the global Lipschitz property of the two functions, we have to rely again on numerical simulations to show the global convergence of the observer. 4.2.2. Simulation Results. Numerical simulations are performed to verify the theoretical properties of the nonlinear observer of (32). The actual process response again is obtained by integrating the set of differential equations in (6). A numerical-integration step size of 10 s is used. Simulation results are presented for CAi(t) ) 10 + sin(0.01t) kmol‚m-3 and organized as follows. The effect of observer gain on the rate of convergence is first studied. Observer tolerance to various initialization errors is then ascertained. Finally, the global convergence of the observer is demonstrated in the presence of multiple steady states. Figure 5a depicts the actual and estimated values of the state variables for various observer gains listed in Table 6. The initial conditions of the actual process and observer are CA(0) ) 1.3 kmol‚m-3, T(0) ) 400 K, C ˆ A(0) ) 0.2 kmol‚m-3, and T ˆ (0) ) 380 K. Note that there is a significant mismatch between the initial conditions of the actual process and those of the observer. Since the process initial conditions lie in the domain of attraction of the high-conversion high-temperature steady-state operating point, the reactor will operate at this operating point asymptotically. Figure 5a shows that the smaller max [Real(λ1),Real(λ2)], where λ1 and λ2 are the roots of λ2 + λK′1 + K′2 ) 0, the faster the rate of convergence of the observer. This is also in complete agreement with the theoretical results presented in the previous section. Figure 5b depicts the actual and estimated values of the state variables for various observer initial conditions listed in Table 7. The initial conditions of the actual process are CA(0) ) 1.3 kmol‚m-3 and T(0) ) 400.0 K. The observer gain K′ ) [0.1 1.0 × 10-6]T. Since the process initial conditions lie in the domain of attraction of the high-conversion high-temperature steady-state operating point, the reactor will operate at this operating point asymptotically. As this figure shows, over a wide range of mismatch between the initial conditions of the observer and those of the actual process, the estimated values of the state variables converged to the actual values. An observer should possess this property as it is difficult to determine initial conditions for the observer in practical cases. So far simulation results have concerned the convergence of the nonlinear observer when the CSTR operates in the vicinity of the high-temperature high-conversion steady-state operating point. However, this reactor has two stable steady states (SS1 and SS3), and hence the reactor can be operated at either steady state. The

+

+


Figure 5. (a) Actual and estimated values of the state variables (estimated by using nonlinear observer of (32)), for various observer gains (see Table 6). (b) Actual and estimated values of the state variables (estimated by using nonlinear observer of (32)), for various observer initial conditions (see Table 7). (c) Actual and estimated values of the state variables (estimated by using nonlinear observer of (32)): thick solid line, actual state variable under IC1; thin solid line, actual state variable under IC2; dashed line, estimated state variable corresponding to IC1; dotted line, estimated state variable corresponding to IC2. Table 6. Nonlinear Observer Gains case

K′1

K′2

E1 E2 E3

1.0 × 10-2 1.0 × 10-1 1.0 × 100

1.0 × 10-6 1.0 × 10-4 1.0 × 10-2

Table 7. Initial Conditions for Nonlinear Observer case

C ˆ A(0)

T ˆ (0)

F1 F2

0.2 2.0

380.0 420.0

Q(x,U) )

capability of the nonlinear observer in the presence of multiple steady states. The observer gain is chosen to be K′ ) [1.0 × 10-2 1.0 × 10-5]T. The following observer initial conditions are used: C ˆ A(0) ) 5.0 kmol‚m-3; T ˆ (0) ) 350.0 K. Actual and estimated values of the state variables are shown in Figure 5c. Whether the reactor initial conditions are those of IC1 or those of IC2 (listed in Table 1), both the observer and process reach the same steady-state values; that is, there is a zero steady-state observer error. Recall that the initial conditions IC1 are in the domain of attraction of the lowconversion low-temperature steady-state SS1, whereas the initial conditions IC2 are in the domain of attraction of the high-conversion high-temperature steady-state SS3. This simulation also demonstrates that the observer always converges to the same steady-state operating point at which the actual process operates, irrespective of the initial mismatch; it proves the global convergence capability of the nonlinear observer. 4.3. Example 3: A CSTR with r < n. Consider the reactor example of (6) with q as a varying measurable input and CAi(t) ≡ CAi . In this case, r ) 1 < n ) ss 2. The reactor model is in the general form of (1):

[ ]

C x) A ; T

f(x) )

[

RA(CA,T) +

CAi - CA ss

τ RH(CA,T) Ti - T + Fc τ 0 g(x) ) 1

]

;

[ ]

;

It is assumed that q(t) can be represented by piecewise continuous polynomial functions of order 1 (p ) 1). 4.3.1. Observer Design. For this reactor example, the observability matrix is

y)T

[ [

]

T ∂ R (C ,T) T - T u ) i ∂x H A + + Fc τ FcV 0 1 1 ∂RH(CA,T) 1 ∂RH(CA,T) 1 Fc ∂CA Fc ∂T τ

]

(35)

where ∂RH(CA,T)/∂CA and ∂RH(CA,T)/∂T are given by (28). Because the observability matrix Q(x,U) is nonsingular almost everywhere, we can use the observer of Theorem 1 for this reactor. The observer has the form:

{

dC Â dt

) RA(C ˆ A,T ˆ) +

(

Â CAi - C

)

τ

-

1 1 ∂RH (C ˆ ,T ˆ) - K′2 ˆ 1) K′′1(u - D Fc ∂T A τ (T - T ˆ) + 1 ∂RH 1 ∂RH (C ˆ A,T ˆ) (C ˆ ,T ˆ) Fc ∂CA Fc ∂CA A ˆ A,T ˆ ) Ti - T ˆ D ˆ1 dT ˆ RH(C ) + + + K′1(y - T ˆ) dt Fc τ FcV dD ˆ1 )D ˆ 2 + K′1′(u - D ˆ 1) dt dD ˆ2 ) K′2′(u - D ˆ 1) dt (36) K′1

where D ˆ 1 denotes the estimated (filtered) value of u. The observer gains are selected such that the roots of the equations λ2 + K′1λ + K′2 ) 0 and λ2 + K′1′λ + K′2′ ) 0 all have negative real parts. Remark 4: In this example, calculation of the time derivatives of the input is not required, because the observability matrix Q(x,U) ) Q(x,u) does not depend

+

+


Figure 6. (a) Actual and estimated values of the state variables (estimated by using nonlinear observer of (36)), for various observer gains (see Table 8). (b) Actual and estimated values of the state variables (estimated by using nonlinear observer of (36)), for various observer initial conditions (see Table 9). (c) Actual and estimated values of the state variables (estimated by using nonlinear observer of (36)): thick solid line, actual state variable under IC1; thin solid line, actual state variable under IC2; dashed line, estimated state variable corresponding to IC1; dotted line, estimated state variable corresponding to IC2.

is the filtered input u; D ˆ 1 is the output of a low-pass filter whose input is the input u(t). To prove the local asymptotic convergence of the observer, we have to verify that the function ψ3(Φ-1((z,U),U,u(1))) is locally Lipschitz. For this example, z ) Φ(CA,T,U) can be calculated easily:

Table 8. Nonlinear Observer Gains

[]

Table 9. Initial Conditions for Nonlinear Observer

[

]

z1 h(x) z2 ) Φ(CA,T,U) ) Lf+guh(x) ) T RH Ti - T u + + Fc τ FcV

[

]

(37)

However, an explicit expression for Φ-1(z1,z2,U) cannot be calculated:

[ ]

[

]

CA P(z1,z2,U) ) Φ-1(z1,z2,U) ) z1 T

(38)

where P(z1,z2,U) represents the implicit function corresponding to the solution for CA of

[

RH(CA,z1) ) Fc z2 -

]

Ti - z1 u τ FcV

ψ2(Φ-1((z,U),U,u(1))) can be calculated by substituting (38) for ψ2(x,U,u(1)) which is given below:

ψ2(x,U,u(1)) )

(

1 ∂RH R + Fc ∂CA A

(

)

CAi - CA ss

τ

)(

+

q(1) + FcV

)

q 1 ∂RH 1 RH Ti - T + + Fc ∂T τ Fc τ FcV

By using the chain rule, it can be verified that the function ψ2(Φ-1((z,U),U,u(1))) is continuously differentiable at least locally. This implies that this function is locally Lipschitz (Khalil, 1992). However, it is impossible to prove analytically that ψ2(Φ-1((z,U),U,u(1))) is globally Lipschitz. With this limitation to prove the global Lipschitz property of the two functions, we have to rely again on numerical simulations to show the

case

K′1

K′2

K′1′

K′2′

G1 G2 G3

1.0 × 10-2 1.0 × 10-1 1.0 × 100

1.0 × 10-6 1.0 × 10-4 1.0 × 10-2

1.0 × 10-3 1.0 × 10-2 1.0 × 10-1

1.0 × 10-7 1.0 × 10-5 1.0 × 10-4

case

C ˆ A(0)

T ˆ (0)

D ˆ 1(0)

D ˆ 2(0)

H1 H2

0.2 2.0

380.0 420.0

-0.1 -0.5

0.0 0.0

4.3.2. Simulation Results. Numerical simulations are performed to verify the theoretical properties of the nonlinear observer of (36). In all the simulations, the actual process response is obtained by integrating the set of differential equations in (6). A numerical-integration step size of 10 s is used. Simulation results are presented for q ) -1.03 - 0.4 sin(0.01t) kJ‚s-1 and organized as follows. The effect of observer gain on the rate of convergence is first studied. Observer tolerance to various initialization errors is then ascertained. Finally, the global convergence of the observer is demonstrated in the presence of multiple steady states. Figure 6a depicts the actual and estimated values of the state variables for various observer gains listed in Table 8. The initial conditions of the actual process and observer are CA(0) ) 1.3 kmol‚m-3, T(0) ) 400 K, C ˆ A(0) ) 0.2 kmol‚m-3, T ˆ (0) ) 380 K, D ˆ 1(0) ) -0.5 kJ‚s-1, and D ˆ 2(0) ) 0.0 kJ‚s-2. Since the process initial conditions lie in the domain of attraction of the high-conversion high-temperature steady-state operating point, the reactor will operate at this operating point asymptotically. Figure 6a shows that the smaller max [Real(λ1),Real(λ2)], where λ1 and λ2 are the roots of λ2 + λK′1 + K′2 ) 0, the faster the rate of convergence of the observer. This is also in complete agreement with the theoretical results presented in the previous section. Figure 6b depicts the actual and estimated values of the state variables for various observer initial conditions listed in Table 9. The initial conditions of the actual process are CA(0) ) 1.3 kmol‚m-3 and T(0) ) 400.0 K. The observer gains are chosen to be K′ ) [1.0 × 10-2 1.0 × 10-6]T and K′′ ) [1.0 × 10-3 1.0 × -7 T

+

2658

+

Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996

domain of attraction of the high-conversion high-temperature steady-state operating point, the reactor will operate at this operating point asymptotically. As this figure shows, over a wide range of mismatch between the initial conditions of the observer and those of the actual process, the estimated values of the state variables converged to the actual values. An observer should possess this property as it is difficult to determine initial conditions for an observer in many reallife situations. So far we have studied the convergence of the nonlinear observer when the CSTR operates in the vicinity of the high-temperature high-conversion steadystate operating point. Recall that this reactor has two stable steady states (SS1 and SS3), and hence the reactor can be operated at either steady state. The purpose of this simulation is to study the estimation capability of the nonlinear observer in the presence of multiple steady states. The observer gains are chosen to be K′ ) [1.0 × 10-2 1.0 × 10-5]T and K′′ ) [1.1 × 10-2 1.0 × 10-5]T. The following observer initial conditions are used: C ˆ A(0) ) 5.0 kmol‚m-3; T ˆ (0) ) 350.0 K; D ˆ 1(0) ) -0.5 kJ‚s-1; D ˆ 2(0) ) 0.0 kJ‚s-2. Actual and estimated values of the state variables are shown in Figure 6c. Whether the reactor initial conditions are those of IC1 or those of IC2 (listed in Table 1), both the observer and process reach the same steady-state values; that is, there is a zero steady-state observer error. Recall that the initial conditions IC1 are in the domain of attraction of the low-conversion low-temperature steady-state SS1, whereas the initial conditions IC2 are in the domain of attraction of the high-conversion high-temperature steady-state SS3. This simulation also demonstrates that the observer always converges to the same steadystate operating point at which the actual process operates, irrespective of the initial mismatch; it proves the global convergence capability of the nonlinear observer. Remark 5: The nonlinear observer may asymptotically provide zero error between the estimated and actual values of state variables, if all the disturbances are measurable. However, in practice all the disturbances (inputs) are often not measurable, and in such a case, the nonlinear observer, like any other state estimation method, may exhibit a permanent error. To handle this problem, the best that can be done is to develop a mathematical model for the unmeasurable disturbance and to estimate this disturbance on-line. This approach will provide the observer with an “integral action”. More details on this important practical issue can be found in (Soroush, 1996). 5. Conclusions The problem of state estimation in nonlinear deterministic processes was studied. For moderately nonlinear deterministic processes, the inadequacy of Luenberger observer and extended Luenberger observer with constant gain was shown analytically and through numerical simulation of a chemical reactor. The results indicated that the convergence of an extended Luenberger observer with constant gain may be poorer than that of a simple linear Luenberger observer. This implies that, in the design of a nonlinear observer, enough care should be taken to ensure an acceptable convergence of the observer; the use of a more complex nonlinear observer may not lead to a more accurate estimate of the state variables. A nonlinear observer design method was applied to a classical exothermic stirred-tank reactor with multiple steady states. Under

observer gains, the global, asymptotic convergence of the nonlinear observer was demonstrated through numerical simulations. It was shown that, in the presence of the multiple steady states, the nonlinear observer converges to the same steady-state operating point at which the reactor operates, irrespective of the observer initial conditions. Apart from that the observability matrix should be nonsingular everywhere, the major limitation of the nonlinear observer design method of Ciccarella et al. (1993) is the need for checking the global Lipschitzness of a few functions in the form of Lie derivatives. These functions are usually implicit, and therefore analytical proof of the global Lipschitzness is impossible except in very simple cases. Indeed, we were not able to prove analytically the global Lipschitzness property for these fairly complex reactors with two state variables and showed the global convergence of the observer through numerical simulations. Therefore, it is fair to state that, in almost all practical cases, one cannot check the global convergence of the observer analytically and has to rely only on numerical simulations. Another disadvantage of this observer design method is that its implementation may need the time derivatives of the measurable input(s), which have to be calculated approximately by a numerical method. Advantages of the observer design method are that (a) its design and implementation do not need a coordinate transformation and (b) the effect of observer gain on the rate of convergence is clear. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the ACS, for support of this research. Nomenclature A ) reactant, n × n matrix Ai ) n × n Jacobian matrix of f(x) + g(x) u evaluated at equilibrium point i ) 1, 2 A ˜ ) n × n Brunowsky canonical matrix A′ ) (p + 1) × (p + 1) Brunowsky canonical matrix bi ) n × 1 matrix of g(x) evaluated at equilibrium point SSi, i ) 1, 2 b˜ ) n × 1 matrix c ) heat capacity of reacting mixture, kJ‚kg-1‚K-1 ci ) 1 × n Jacobian matrix of h(x) evaluated at equilibrium point i ) 1, 2 c˜ ) 1 × n matrix CA ) concentration of reactant, kmol‚m-3 CA(0) ) actual concentration of reactant at time t ) 0, kmol‚m-3 C ˆ A ) estimated concentration of reactant, kmol‚m-3 C ˆ A(0) ) estimated concentration of reactant at time t ) 0, kmol‚m-3 CAi ) inlet concentration of reactant, kmol‚m-3 D ) desired product, vector of the input time derivatives e˜ ) vector of the errors between actual and estimated state variables e˜ 0 ) vector of the errors between actual and estimated state variables at t ) 0 Ead ) activation energy for desired reaction, kJ‚kmol-1 Eal ) activation energy for undesired reaction l ) 1, 2, kJ‚kmol-1 f, g ) vector functions h ) scalar function Il ) l × l identity matrix ICi ) initial conditions for the actual process, i ) 1, 2 kd ) reaction rate constant for desired reaction, s-1 k1 ) reaction rate constants for reaction 1, m6‚kmol-2‚s-1 k2 ) reaction rate constants for reaction 2, kmol0.5‚m-1.5‚s-1 K′, K′′, L ) observer gains

+

+

Ind. Eng. Chem. Res., Vol. 35, No. 8, 1996 2659 R ) universal gas constant, kJ‚kmol-1‚K-1 RA ) rate of consumption of A, kmol‚m-3‚s-1 RH ) overall rate of heat production by chemical reactions, kJ‚m-3‚s-1 SSi ) steady state, i q ) rate of heat input to reactor, kJ‚s-1 Q ) observability matrix t ) time, s T ) reactor temperature, K T(0) ) reactor temperature at t ) 0, K T ˆ ) estimated reactor temperature, K T ˆ (0) ) estimated temperature at t ) 0, K Ti ) temperature of inlet stream to reactor, K u ) measurable input u(i) ) ith time derivative of the measurable input U ) vector of measurable input time derivatives U1, U2 ) undesired products V ) reactor volume, m3 x ) vector of state variables x0 ) vector of state variables at t ) 0 xˆ ) vector of estimated state variables xˆ 0 ) vector of estimated state variables at t ) 0 y ) measurable output z ) vector of state variables in transformed coordinates zˆ ) vector of estimated state variables in transformed coordinates Zd ) frequency factor for desired reaction, s-1 Z1 ) frequency factor for reaction 1, m6‚kmol-2‚s-1 Z2 ) frequency factor for reaction 2, kmol0.5‚m-1.5‚s-1 Greek Letters β ) positive quantity -∆Hd ) heat of desired reaction, kJ‚kmol-1 -∆Hl ) heat of undesired reaction l ) 1, 2, kJ‚kmol-1 γ ) Lipschitz constant F ) density of reacting mixture, kg‚m-3 τ ) CSTR residence time Math Symbols ≡ ) identically equal to ∀ ) for all ∈ ) belongs to |‚| ) Eucledian norm IR ) real line sup ) supremum Subscripts eq ) evaluated at equilibrium point eqi ) evaluated at equilibrium point, i ) 1, 2 ss ) evaluated at steady state ssi ) evaluated at steady state, i ) 1, 3 Superscripts T ) transpose

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of the Molecular Weight Distribution in Batch Polymerization. AIChE J. 1988, 34, 1341. Friedland, B. Control System DesignsAn Introduction to State Space Methods; McGraw-Hill: New York, 1986. Gauthier, J. P.; Hammouri, H.; Othman, S. A Simple Observer for Nonlinear SystemssApplications to Bioreactors. IEEE Trans. Autom. Control 1992, 37, 875. Gelb, A. Applied Optimal Estimation; MIT Press: Cambridge, MA, 1974. Gibon-Fargeot, A. M.; Hammouri, H.; Celle, F. Nonlinear Observers for Chemical Reactors. Chem. Eng. Sci. 1994, 49, 2287. Hammouri, H.; Gauthier, J. P. Bilinearization up to Output Injection. Syst. Control Lett. 1988, 11, 139. Hammouri, H.; Gauthier, J. P. Global Time-Varying Linearization up to Output Injection. SIAM J. Control Optim. 1992, 30, 1295. Jazwinski, A. H. Stochastic Processes and Filtering Theory; Academic Press: New York, 1970. Jo, J. H.; Bankoff, S. G. Digital Monitoring and Estimation of Polymerization Reactors. AIChE J. 1976, 22, 361. Kalman, R. E.; Bucy, R. S. New Results in Linear Filtering and Prediction Theory. Trans. ASME J. Basic Eng. 1961, 83, 95. Kantor, J. C. A Finite Dimensional Nonlinear Observer for an Exothermic Stirred-Tank Reactor. Chem. Eng. Sci. 1989, 44, 1503. Kazantzis, N.; Kravaris, C. A Nonlinear Luenberger-Type Observer with Application to Catalytic Activity Estimation. Proc. ACC 1995, 1756. Khalil, H. K. Nonlinear Systems; Macmillan: New York, 1992. Kim, K. J.; Choi, K. Y. On-line Estimation and Control of a Continuous Stirred Tank Polymerization Reactor. J. Process Control 1991, 1, 96. Kozub, D.; MacGregor, J. F. State Estimation for Semi-Batch Polymerization Reactors. Chem. Eng. Sci. 1992, 47, 1047. Krener, A. J.; Isidori, A. Linearization by Output Injection and Nonlinear Observers. Syst. Control Lett. 1983, 3, 47. Krener, A. J.; Respondek, W. Nonlinear Observers with Linearizable Error Dynamics. SIAM J. Control Optim. 1985, 23, 197. Limqueco, L.; Kantor, J. C.; Harvey, S. Nonlinear Adaptive Observation of an Exothermic Stirred-Tank Reactor. Chem. Eng. Sci. 1991, 46, 797. Luenberger, D. G. Observing the State of a Linear System. IEEE Trans. Mil. Electron. 1964, MIL-8, 74. Marino, R. Adaptive Observer for Single Output Nonlinear Systems. IEEE Trans. Autom. Control 1990, 35, 1054. Michalska, H.; Mayne, D. Q. Moving Horizon Observers and Observer-Based Control. IEEE Trans. Autom. Control 1995, 40, 995. Ogunnaike, B. On-line Modeling and Predictive Control of an Industrial Terpolymerization Reactor. Int. J. Control 1994, 59, 711. Quintero-Marmol, E.; Luyben, W. L.; Georgakis, C. Application of an Extended Luenberger Observer to the Control of Multicomponent Batch Distillation. Ind. Eng. Chem. Res. 1991, 30, 1870. Robertson, D. G.; Kesavan, P.; Lee, J. H. Detection and Estimation of Randomly Occurring Deterministic Disturbances. Proc. ACC 1995, 4453. Soroush, M. Nonlinear Observer Design with Application to Chemical Reactors. Chem. Eng. Sci. 1996, in press. Soroush, M.; Kravaris, C. Discrete-Time Nonlinear Controller Synthesis by Input/Output Linearization. AIChE J. 1992, 38, 1923. Tornambe´, A. Use of Asymptotic Observers Having High Gains in the State and Parameter Estimation. Proceedings of the 28th Conference on Decision and Control, Tampa, FL, 1989; p 1791. van Dootingh, M.; Rakotopara, V. D.; Gauthier, J. P.; Hobbes, P. Nonlinear Deterministic Observer for State Estimation: Application to a Continuous Free Radical Polymerization Reactor. Comput. Chem. Eng. 1992, 16, 777. Zeitz, M. The Extended Luenberger Observer for Nonlinear Systems. Syst. Control Lett. 1987, 9, 149.

Received for review July 11, 1995 Accepted May 20, 1996X IE9504258 X Abstract published in Advance ACS Abstracts, July 15, 1996.

Nonlinear State Estimation in the Presence of Multiple Steady States

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