State Estimation in Nonlinear Processes. Application to pH Process

Ind. Eng. Chem. Res. 2002, 41, 4777-4785

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State Estimation in Nonlinear Processes. Application to pH Process Control Silvina I. Biagiola and Jose´ L. Figueroa* Planta Piloto de Ingenierıá Quı´mica, CONICET, Camino La Carrindanga Km. 7, 8000 Bahıá Blanca, Argentina

In the field of process engineering, there are many factors that demand the knowledge of one part of the state vector, if not the full one. Among others, the implementation of nonlinear control methods as well as monitoring of some relevant process variables can be mentioned. The purpose of this paper is to introduce a nonlinear state observer of full order (i.e., the order of the observer matches the order of the nonlinear system observed) in order to estimate all of the state variables of the process. Though the estimated states can be used for many purposes, in this work we aim at using the estimations for output reference tracking. In particular, we deal with the pH neutralization process, which is widely recognized as a difficult problem for the purpose of control. To achieve the pH control goal, a simple algorithm is proposed which lies on a stable reference error dynamics model. Finally, computer simulations are developed for showing the performance of both the nonlinear observer and the control strategy based on pH measurement. For this purpose, a neutralization reaction between a strong acid and a strong base in the presence of a buffer agent is dealt with. 1. Introduction The aims of increasing product yields, improving product quality, and reducing production costs have led to great improvements connected with chemical engineering processes instrumentation, modeling, and control. It is a well-known issue that in order to monitor and control many technological processes, the problem of states estimation constitutes a strategic topic. With the goal of process control and optimization, the knowledge of some physical state variables provides useful information. This is the case of many widely diffused process control strategies. For instance, model predictive control (MPC) algorithms can be mentioned. In such cases, the state values are required to calculate the control input. Therefore, the presence of unknown states becomes a difficulty which can be solved by means of the inclusion of an appropriate state estimator.1 For this reason, many researchers have focused their attention on the development of suitable algorithms to perform the estimation. In this sense, several techniques have been introduced to estimate state variables from the available measurements, usually related to meaningful physicochemical variables. Depending on the obtainable information about the process, there exist many possible kinds of estimators to be used depending on the mathematical structure of the process model.2-4 Notwithstanding the fact that theories and applications for linear systems are well developed, the highly nonlinear essence of many processes has given rise to the development of nonlinear observers (NO). These observers are designed in such a way that they can cope with the intrinsic nonlinearities. However, the construction of NO still provides an open research field because the advance in the area of NO often faces many typical * Corresponding author. Phone: +54 291 4861700. Fax: +54 291 4861600. E-mail: [email protected]. Also with Departamento de Ingenierıa Elećtrica, Universidad Nacional del Sur, Bahıá Blanca, Argentina.

obstacles. Among others, the main barriers are the very restrictive conditions to be satisfied, uncertainty in the performance and robustness, and/or poor estimation results in the presence of noisy sensors.5 As regards the nonlinear estimation techniques developed up to now, the extended Kalman filter (EKF) is one of the most widely diffused observers among other NO based on linearization techniques.3,6 The main drawback of these techniques lies in the difficulties in determining a priori its convergence and speed of convergence. In the EKF approach, a Riccati equation must be solved to obtain the estimator gain. An important drawback is that the noise model must be known for obtaining the optimum estimated value. Because this model is often unknown, it must be assumed. In such a case, wrong noise assumptions could lead to biased estimates or even diverge.7 In addition, speed of convergence is strongly influenced by the initial value of the covariance matrix. Because this value is unknown, it must be guessed in order to start the EKF algorithm. Moreover, when linearization techniques are applied, convergence and speed of convergence are local properties; i.e., the estimation error can converge in a given time interval and diverge in another one. A method based on extended linearization has also been developed to carry out state estimation.8 The procedure is based on linearizing with respect to a fixed operating point and involves finding a function of the output in order to keep the system poles invariant in the vicinity of the mentioned point. Hence, the design procedure is subject to very tight conditions, and even when the output function is found (which is not an easy task), only local performance is ensured. Another estimation approach includes the sliding observers.9-11 The design procedure consists of determining a switching gain. One restrictive aspect is that the outputs must lie on specified sliding surfaces to achieve the estimation. Moreover, performance is rarely

10.1021/ie010974f CCC: $22.00 © 2002 American Chemical Society Published on Web 08/15/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002

guaranteed, especially when the outputs are corrupted with noise. Other procedures for observer construction make use of transformed canonical forms in order to design the estimator gain. Gauthier et al.12 proposed a simple observer for input affine systems, whose design involves solving a Riccati equation. The NO construction is subject to the input affine property of the system. Moreover, the observer performance is not guaranteed. A detailed discussion on the current available state estimation techniques applicable to a broad class of nonlinear systems is provided by Mouyon.2 Another comprehensive evaluation of various NO was presented by Wang et al.11 Taking into account the characteristics of the observers discussed above, the objective of this work is to present a nonlinear efficient state estimator for later multipurpose applications. The approach herein proposed guarantees that the estimation error exponentially converges toward zero whenever the observer gain is adequately chosen. The estimation procedure is oriented to those nonlinear control methods that require the knowledge of the internal state of the process. The observer implementation is simple, and it requires small computational effort. The design approach can be applied for any general type of nonlinear process model, because no fixed model structure is required. An advantageous feature of the proposed NO is that it shows robust performance in the presence of noisy measurements. In particular, both the state estimation methodology and a nonlinear regulation technique based on linearizing control are here orientated to pH processes. It must be highlighted that pH neutralization processes came to be widely studied for two different reasons. First of all, because of their environmental implication, especially when effluents or wastewater have to be neutralized before being discharged into the environment.13 Second, pH neutralization processes are highly nonlinear and are known to be an interesting challenge to overcome by any new control technique proposal.14,15 The work is organized as follows. The observer design procedure is developed in section 2. In section 3, the controller synthesis is dealt with. The evaluation of the observer/controller performance is presented via simulation in section 4. Finally, in section 5, the conclusions are presented. 2. Statement of a Nonlinear Full-Order Observer Design The objective of this section is to design an observer for estimating the whole state vector. To attend to the pH neutralization process, in which the reaction is typically followed up by measuring the effluent with a pH sensor and where the acid (or base) is easily neutralized by the addition of a relatively small amount of strong base (or acid), the following nonlinear single input/single output (SISO) general model is proposed for the process:

x˘ ) f(x) + g(x) u + p(x) d

(1)

y ) h(x)

(2)

where the vector x stands for the n state variables x1, ..., xn that are associated with the concentrations of the chemical species inside the reactor. The input u repre-

sents the manipulated variable to accomplish pH control, i.e., the inlet base (or acid) flow. Equation 1 considers the possibility of the existence of a measured disturbance d which influences the state dynamics. It must be highlighted that, although eq 2 implies the output is an explicit function of the states, this is not the case for pH processes. As was mentioned above, in these processes the measured variable is often the pH, which is an implicit function of the states (the so-called pH equation16). However, there is a way to overcome this problem and to formulate the model as in eqs 1 and 2. This procedure is further detailed in the application presented in section 4. To estimate the states of the system represented by eqs 1 and 2, the following observer is proposed:

xˆ˙ ) f(xˆ ) + g(xˆ ) u + p(xˆ ) d + O-1(xˆ ) G[y-h(xˆ )] (3) where the symbol ˆ stands for the estimated variables. The estimator given by eq 3 is a nonlinear observer for the states x. The difference between the measured output y and its evaluation on the estimated states [h(xˆ )] is used as a correction factor in the algorithm. The observer contains a nonlinear gain O-1(xˆ ) G, where G is a vector of constants to be designed and O is the gradient of the matrix Φ(x) defined as follows:

[ ]

h(x) Lfh(x) Φ(x) ) l h(x) Ln-1 f

(4)

The element Lfh(x) represents the Lie derivative of h(x) in the direction of f(x).17 Then, the following equations are satisfied:

∂h(x) f(x) ∂x

(5)

∂Ln-1 h(x) f f(x) ∂x

(6)

Lfh(x) ) Lnf h(x) )

The matrix Φ constitutes a nonlinear change of coordinates, and it is used to design G such that the dynamics of eq 3 is stable. Provided that O is invertible and that the input u and the disturbance d are bounded (see the Appendix), given the initial condition xˆ (0), the following property behaves for any R > 0:

||x(t) - xˆ (t)|| e δe-Rt||x(0) - xˆ (0)||

(7)

with δ ∈ R+. It must be remarked that G must be appropriately chosen to ensure stability. For this purpose, a complete demonstration based on Lyapunov arguments is presented in the Appendix where the observer convergence and its relationship with the gain selection are treated. Then we have developed a stable observer for the nonlinear system that, under certain hypotheses, provide an online estimation of the whole state variables of the disturbed process. Note that the observer can be easily implemented and it is only based on the output measurement. Moreover, the observer was built using the whole process model, and this nonlinear procedure avoids losing information about the dynamics as well as simplifications, order reduction, or the frequently used linearization methods.

Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4779

In the application considered herein, the observer provides the estimation of each chemical species inside the reactor based exclusively on the pH measurement. Once the internal state of the system can be observed, an appropriate control technique based on state knowledge can be performed to achieve a desired trajectory for the pH inside the reactor. Therefore, we now turn to the control design problem. 3. Formulation of the Control Strategy Although in many applications in the field of nonlinear processes the control problem is solved via Taylor linearization techniques, it is possible to achieve an improved control performance from an exploitation of the exact nonlinear model structure using nonlinear control design. In this work we propose to control the nonlinear process through a design technique based on exact linearizing control. In this way, we obtain a nonlinear controller which is constructed to achieve a linear stable closed loop. The objective is to control a scalar output variable which is a measured function of the state variables. Then the goal is to track a reference output signal denoted h*(t). In some cases, this signal can be a constant reference (indeed, this is the case of many wastewater treatments where the desired value pH ) 7 must be obtained), and in such cases h* is called setpoint and the control is referred to as regulation.3 The basis of the control action proposed herein is to find a control law u(x1,...,xn,d,h*) which consists of a nonlinear function of (x1, ..., xn, d, h*) such that the tracking error (h* - h) is governed by a prespecified stable linear differential equation denominated the reference model. The linearizing control law uses an input/output model which connects the manipulated variable (u) with the output variable (h) we want to track. The model consists of an rth differential equation as follows:

d rh ) f0(t) + f1u(t) dtr

(8)

where r is known as the relative degree.18 A stable linear reference model of the tracking error (h* - h) can be stated as follows: r

∑ i)0

λr-i

di dti

(h* - h) ) 0

(9)

where λ0 is set equal to 1 in order to attain a monotonic expression. The constant coefficients λr-i can be arbitrarily selected, whenever the resulting differential equation (9) is stable. The principle of this control action is to treat the reference error dynamics model as a linear stable system; consequently, the Laplace transform of eq 9 must be a Hurwitz polynomial.19 It is worth mentioning that when the relative degree (r) is less than the order of the system (n), apart from designing a stable closed-loop system (i.e., a stable I/O system), internal stability of the model used must be achieved. This issue has been dealt with in previous works for pH control processes.20,21 After combination of eqs 8 and 9, the control action u is given by the following explicit expression:

Figure 1. Scheme of the pH neutralization process.

u)

1 f1(t)

[

-f0(t) +

r-1

∑ i)0

λr-i

di dti

(h* - h) +

drh* dtr

]

(10)

Hence, we see that, for control action achievement, the reference h* and its derivatives must be evaluated for calculating u. However, this is not an obstacle because the reference is a known function and the expressions for its time derivatives can be easily calculated. Moreover, eq 10 requires the rth time derivatives of the signal h. However, this does not imply that the measured signal has to be differentiated; instead, the differential equations of the model as well as the output algebraic equation have to be used in order to obtain the expressions of the output derivatives as functions of the state variables. It must be remarked that the control law in eq 10 can be straightforwardly attained when the output h is an explicit function of the state variables. On the other hand, when h is an implicit function of the states, the time derivatives (from the first to the rth order) turn out to depend not only on the states but also on the output h. This fact can complicate the expression of the control law u, which can turn out to be a strongly nonlinear expression of the measured output h (apart from the states x1, ..., xn, the measured disturbance d, the reference h*, and its rth time derivatives, which also appeared in the explicit case). 4. Evaluation of the Observer/Controller Performance via Simulation The efficacy of the proposed algorithm will be illustrated through an example. The application case deals with the neutralization reaction between a strong acid (HA) and a strong base (BOH) in the presence of a buffer agent (BX) as described by Galań.22 The reaction takes place in a continuous stirred tank reactor with a constant volume V. An acidic solution with a time-varying volumetric flow qA(t) of a composition x1i(t) is neutralized with an alkaline solution with volumetric flow qB(t) of known composition made up of base x2i and buffer agent x3i. Figure 1 shows a scheme of this continuous pH neutralization process. Because of the high reaction rates of the acid-base neutralization, chemical equilibrium conditions are instantaneously achieved. Additionally, under the assumptions that the acid, the base, and the buffer are strong enough, then the total dissociation of the three compounds takes place. The dynamic behavior of the process can be obtained considering the electroneutrality condition (which is always preserved) and through mass balances of equivalent chemical species that were introduced by Gustafs-

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Table 1. Model Parameters parameter

value

x1i x2i x3i Kx Kw V

0.0012 mol of HCl/L 0.002 mol of NaOH/L 0.0025 mol of NaHCO3/L 10-7 mol/L 10-14 mol2/L2 2.5 L

son and Waller.23 These species are known as chemical invariants. For this specific case, under the assumptions introduced above, the dynamic behavior of the process can be described considering the following state variables:

x1 ) [A-]

(11)

x2 ) [B+]

(12)

x3 ) [X-]

(13)

Figure 2. Measured output (pH).

Afterward, the process model is given by the following equations:22

1 1 x˘ 1 ) (x1i - x1) - x1qB θ V

(14)

1 1 x˘ 2 ) - x2 + (x2i - x2)qB θ V

(15)

1 1 x˘ 3 ) - x3 + (x3i - x3)qB θ V

(16)

F(x,ξ) ≡ ξ + x2 + x3 - x1 -

Kw ξ

x3 )0 Kxξ 1+ Kw

(17)

where ξ ) 10-pH and θ ) V/qA. Kx and Kw are the dissociation constants of the buffer and the water, respectively. The parameters of the system represented by eqs 14-17 are addressed in Table 1. Equation 17 was deduced by McAvoy et al.,16 and it takes the standard form of the widely used implicit expression that connects pH with the states of the process. To continue with the state estimation and the pH control, by simple inspection of eqs 2 and 10, it is clear that we need an expression of ξ ) h(x). Moreover, eq 10 imposes h(x) to be r times differentiable with respect to x. We will show that both requirements are satisfied in the current application. Note that eq 17 can be rewritten as a third-order polynomial:

ξ3 +

[

]

Kw Kw + x3 + x2 - x1 ξ2 + (x2 - x1 - Kx) ξ Kx Kx Kw2 ) 0 (18) Kx

It is now clear that this polynomial has only one root ξ(x) with physical meaning. It is the real root of eq 18, and the solution ξ ) h(x) is easily obtained. The second condition, i.e., the differentiability of h(x), is satisfied for every operation point. This can be straightforwardly deduced through the application of the implicit function theorem17 to F(x,ξ). To evaluate the state observer’s performance, the system was simulated assuming x1i as constant (see

Figure 3. Concentration of species x1: actual (s) and estimated (-‚-) values.

Table 1) and qA ) 1 L/min. The process was excited through a bounded input signal qB (real control valves have a limited rangeability), and that gave rise to the output (i.e., the measured pH) shown in Figure 2. The state initial conditions were set to x1(0) ) 8.8 × 10-4 mol/L, x2(0) ) 5.4 × 10-4 mol/L, x3(0) ) 6.8 × 10-4 mol/L, xˆ 1(0) ) 0.80x1(0), xˆ 2(0) ) 0.80x2(0), and xˆ 3(0) ) 1.20x3(0). The estimation results obtained for G ) [0.05, 0.05, 0.05]′ are depicted in Figures 3-5. To evaluate the performance of the whole observer/ controller structure, further simulations were developed. In this case, the input qA was considered as a measured disturbance (see Figure 6). A variable step perturbation was assumed, which is typically found in many industrial facilities because of construction features.24 The inlet acid flow composition was assumed to exhibit a sinusoidal behavior {x1i(t) ) 0.0012[1 + 0.1 sin(0.5t)] mol/L}, which is the common disturbance type due to typical homogenization and dilution effects.21 The states initial conditions were x1(0) ) 8.8 × 10-4 mol/L, x2(0) ) 5.4 × 10-4 mol/L, and x3(0) ) 6.8 × 10-4 mol/L, and the estimated value was xˆ (0) ) 1.2x(0). The design parameters G and λ were set equal to [0.05, 0.05,




0.05]′ and 10, respectively. The states estimation results are shown in Figures 7-9, and these values were used in the control law to track the desired pH trajectory. In this case, the control law qB calculated through eq 10 is

qB(t) )

-VR(t) S(t) - qAM(t)

(19)

N(t)

with

R(t) ) 1 +

Kw 2

h

+

x3KwKx

(20)

(Kw + Kxh)2

S(t) ) λ[h* - h] + h*

[

M(t) ) -x2 - 1 -

]

(21)

Kw x - (x1i - x1) (22) Kw + Kxh 3

[

N(t) ) x1 + x2i - x2 + (x3i - x3) 1 -

Kw Kw + Kxh

]

Figure 6. Input acid flow (qA).

(23)

Figure 10 shows the measured pH, which was obtained by adding to the real deterministic system output a white noise signal previously filtered by a first-order transfer function. Although many authors have considered the measured pH only corrupted with white noise,25 herein we contemplate the more realistic situation where the sensor dynamics is considered. This is an important assumption in order to test the proposed observer/controller structure under more disadvantageous operation conditions. In Figure 11 the manipulated signal qB is depicted, and Figure 12 shows the pH reference as well as the pH value inside the reactor (free from noise). To evaluate the performance achieved with the proposed observer/controller structure, other control strategies are evaluated. For this purpose, two different nonlinear control methodologies were selected. First, the trajectory tracking problem is solved using a gain scheduling PI controller. The method includes the traditional PI control and adds a variant gain to cope with the intrinsic nonlinear nature of the pH neutral-

ization process. Second, a nonlinear MPC (NMPC) strategy is considered to accomplish the same control goal. Although the PI controller is usually a suitable regulator for linear systems, it faces an important obstacle when implemented for pH process control. The slope of a chemical system’s titration curve can vary several orders of magnitude over a modest range of pH values, causing the overall process gain to vary accordingly. The regions of high and low slope on the titration curve correspond to conditions of high and low gain for a pH control loop. Shinskey26 thoroughly documented the process gain characteristics, indicated the control difficulties, and discussed several PID controller applications. Despite the nonlinear features mentioned above, PI controllers are frequently adapted for nonlinear control purposes. For instance, in their work Hall and Seborg14 evaluated the use of a digital PI controller to cope with the series of load changes in acid and buffer flow-rate manipulation. PI control is chosen rather than PID control because preliminary studies have reported the

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counterproductive/unfavorable effect of the derivative action in the presence of noisy measurements.21 Another attempt in using PI control was made by Proudfoot et al.27 They selected a self-tuning PI for an industrial waste treatment process. One of the first attempts in considering variant gain controllers for pH neutralization processes was by Shinskey,28 who applied a piecewise proportional controller to an industrial process. Wright et al.25 tuned a PI controller following the closed-loop poles assignment approach. They used it to control an online-identified pH process, and they also evaluated the effect of measurement noise. To cope with the variant gain nature of the pH process, a PI controller with gain scheduling is considered. Figure 13 shows that the PI tuning allows noise amplitude reduction. It also points out the quite fast response achieved in following the desired pH trajectory. However, visible overshoots are obtained. An overshoot appearance under PI regulation has already been detected.25 Additionaly, it can be noticed that, although


Figure 10. Noise-corrupted measured output (pH).

the output tracks closer to the desired trajectory, the PI controller with gain scheduling shows a deficient capacity to reject the sinusoidal disturbance in the inlet concentration x1i as well as in the step changes in qA. A variant proportional gain (Kp) was included to achieve a gain scheduling control strategy together with an integral action (Ti ) 3). The variant gain was selected in accordance with the titration curve depicted in Figure 14. The second control alternative herein considered for comparison aims is NMPC.29-31 MPC is a control strategy which seeks to minimize a performance function with the aid of a process model. The model is used to predict the system response and, consequently, to optimize it subject to constraints on input, output, and state variables. The underlying idea is to minimize the error between the desired output and the predicted output over a number of time intervals, called the “prediction horizon”. MPC has been welcome in the chemical process industries because it appears as a suitable choice because of its setpoint tracking capabilities in the presence of constraints such as physical restrictions related to valve sizes and actuator dynam-


Figure 11. Manipulated variable.

Figure 13. pH reference and process output (PI + gain scheduling control).

Figure 12. pH reference and process output (observer + linearizing control).

Figure 14. Gain selection based on the titration curve.

ics. The model predictive controller is typically tuned by means of the control and prediction horizons (H and M, respectively) and the weighting factors in the performance function

f ) ETQEE + UTQUU

(24)

where

E ) [e1, ..., eH]′

and

U ) [∆u0, ..., ∆uM-1]′ (25)

The symbols ei stand for the difference between the desired and predicted outputs i intervals into the future, and ∆ui is the change in input i intervals into the future. The matrices QE and QU are the weighting factors. In the simulation shown in Figure 15, the values QE ) 1 and QU ) 20 were considered. The parameters H and M were set to 20 and 10, respectively. The figure shows a sluggish phenomenon, which can also be observed in the results reported by Gerksˇicˇ et al.31 To compensate for this control feature, the design parameter QU could be increased. However, this increment would lead to a more vigorous control action, and it could lead to

Figure 15. pH reference and process output (NMPC).

instability. Moreover, a higher value for QU would increase the observed effect of measurement noise on the controlled variable.

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Therefore, Figure 12 shows an improved observer/ controller performance in comparison with the gain scheduling PI controller and NMPC strategy. From inspection of this figure, the effect of the sinusoidal exogenous input x1i on the pH can hardly be recognized. This is because the disturbance has been canceled by the control action. Additionally, the simulation results show that the whole structure (observer/controller) exhibits a robust performance in the presence of known disturbances and a measured output corrupted with noise.

z˘ ) Az + b Lnf h[Φ-1(z)] + LgΦ[Φ-1(z)] u + LpΦ[Φ-1(z)] d y ) cz with

5. Conclusions In the present work the problem of state variables estimation has been tackled. In particular, the analysis has been focused on the estimation of the concentrations of chemical reaction invariants in pH neutralization processes. To perform the estimation, we proposed a high-gain full-order observer that robustly estimates the whole state vector based only on the output measurement. Moreover, the observer design was used in a control strategy to track a desired pH reference. The controller has been developed following the principle of linearizing control, which, provided internal stability, guarantees a stable closed-loop dynamics. When this regulation strategy was applied to the pH control process, it gave rise to a highly nonlinear control action demanding the knowledge of the internal state of the system. Because these variables cannot be measured, this fact was overcome by incorporating the full-order observer to the control structure. Finally, computer simulations have been developed to illustrate the performance of both the nonlinear observer and the control strategy based on pH measurement. The results showed a good agreement between the actual and estimated states, as well as a successful behavior of the controller. Both the observer and the controller showed a robust performance when exposed to disturbances influencing the states dynamics and noise-corrupted measured output.

[

0 1 0 ... 0 0 1 ... A) l 0 0 ... 0 and

0 0 l 0

] [] 0 l b) 0 1

[ ]

Lgh(x) LgLfh(x) u; LgΦ(.) u ) l h(x) Lg Ln-1 f

(28)

c ) [1 0 ... 0 ] (29)

[ ]

Lph(x) LpLfh(x) LpΦ(.) d ) d l h(x) Lp Ln-1 f

(30)

At this time, the following observer in the z domain is proposed:

zˆ˙ ) (A - Gc)zˆ + Gy + b Lnf h[Φ-1(zˆ )] + LgΦ[Φ-1(zˆ )]u + LpΦ[Φ-1(zˆ )]d (31) Therefore, the time derivative of the estimation error (z - zˆ ) can be written as follows:

e˘ z ) z˘ - zˆ˙ ) (A + Gc)ez + b{ Lnf h[Φ-1(z)] Lnf h[Φ-1(zˆ )]} + {LgΦ[Φ-1(z)] LgΦ[Φ-1(zˆ )]}u + {LpΦ[Φ-1(z)] -

Acknowledgment This work was financially supported by the National Council of Scientific and Technological Research (CONICET). Appendix As was previously mentioned, the selection of the design parameter G is a relevant issue in regards to the estimation algorithm convergence. The construction of the nonlinear observer proposed herein makes use of a change of coordinates. This approach has been frequently followed in the field of NO design.12,32 The change of coordinates selected in this work is the one given by eq 4. It allows us to transform the original system by defining the transform variable z

z ) Φ(x)

(26)

LpΦ[Φ-1(zˆ )]}d (32) To carry out the selection of the constant vector gain G, the following Lyapunov candidate function is introduced:

V ) ezTPez

(33)

with P a positive definite matrix. Then

V˙ ) e˘ zTPez + ezTPe˘ z

(34)

) ezT[(A + Gc)TP + P(A + Gc)]ez + ˆ )TPezu + 2(γ - γˆ )TPez + 2(ω - ω 2(F - Fˆ )TPezd (35)

and

x ) Φ-1(z)

(27)

which constitutes a change of coordinates in Rn. Therefore, the original system given by eqs 1 and 2 can be rewritten in the new coordinates as follows:

where γ(.), ω(.), and F(.) stand for Lnf h[Φ-1(.)], LgΦ[Φ-1(.)], and LpΦ[Φ-1(.)], respectively. Now, P and a positive definite matrix Q should satisfy the equation

(A + Gc)TP + P(A + Gc) ) -Q

(36)


with qm and pM the minimum and maximum eigenvalues of Q and P, respectively. Under the assumptions that

||u|| e U ||d|| e D ||γ - γˆ || e Lγ||z - zˆ || ||ω - ω ˆ || e Lω||z - zˆ || ||F - Fˆ || e LF||z - zˆ ||

(37)

where Lγ, Lω, and LF are the Lipschitz constants of the respective functions, and provided the previous conditions behave, the following inequality can be obtained:

V˙ e [-qm + 2pM(Lγ + LωU + LFD)]||ez||2

(38)

If the gain G is selected such that pM and qm satisfy

-qm + 2pM(Lγ + LωU + LFD) < 0

(39)

then V˙ turns out to be negative and the norm of the estimation error goes to zero as t f ∞. In other words, the convergence of the algorithm is guaranteed. At this time, if the transform Φ(x) is nonsingular and Φ-1 is uniformly Lipschitz, then when eqs 26 and 27 are revisited, the condition given by eq 7 is obtained. Note that eq 38 establishes a sufficient condition to guarantee stability. However, in some cases it may be rather conservative. As a consequence of this fact, in many applications a good estimation performance could be achieved even when the gain G does not satisfy eq 38. Literature Cited (1) Gattu, G.; Zafiriou, E. Nonlinear quadratic matrix control with state estimation. Ind. Eng. Chem. Res. 1992, 31, 1096. (2) Mouyon, Ph. Tools for nonlinear observer design. IEEE International Symposium on Diagnostics and Drivers (SDEMPED′97), Carry-Le-Rouet, France, Sept 1-3, 1997. (3) Bastin, G.; Dochain, D. On-line estimation and adaptive control of bioreactors; Elsevier Science Publishers: New York, 1990. (4) Soroush, M. Nonlinear state-observer design with application to reactors. Chem. Eng. Sci. 1997, 52 (3), 387. (5) Garcıá, R. A.; Troparevsky, M. I.; Mancilla Aguilar, J. L. An observer for nonlinear noisy systems. Lat. Am. Appl. Res. 2000, 30 (2), 87. (6) Stephanopoulos, G.; San, K. Y. Studies on On-line Bioreactor Identification. Biotechnol. Bioeng. 1984, 26, 1176. (7) Ljung, L. Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems. IEEE Trans. Autom. Control 1979, AC-24, 36. (8) Baumann, W. T.; Rugh, W. J. Feedback control of nonlinear systems by extended linearization. IEEE Trans. Autom. Control 1986, AC-31, 40.

(9) Slotine, J.; Hedrick, J. K.; Misawa, E. A. On sliding observers for nonlinear systems. ASME J. Dyn. Syst. Measure. Control 1987, 109, 245. (10) Canudas de Wit, C.; Slotine, J. Sliding observers for robot manipulators. Automatica 1991, 27, 859. (11) Wang, G.; Peng, S.; Huang, H. A sliding observer for nonlinear process control. Chem. Eng. Sci. 1997, 52 (5), 787. (12) Gauthier, J.; Hammouri, H.; Othman, S. A simple observer for nonlinear systems. Applications to bioreactors. IEEE Trans. Autom. Control 1992, 37 (6), 875. (13) Williams, G.; Rhinehart, R.; Riggs, J. In-line process-modelbased control of wastewater pH using dual base injection. Ind. Eng. Chem. Res. 1990, 29, 1254. (14) Hall, R.; Seborg, D. Modeling and self-tuning control of a multivariable pH neutralization process. Part I. Modeling and multiloop control. Proc. Am. Control Conf. ACC 1989 1989, TP13 (3), 1822. (15) Longhi, L.; Lima, E.; Barrera, P.; Secchi, A. Nonlinear H∞ control of an experimental pH neutralization system. Proc. ENPROMER 2001 2001, I, 301. (16) McAvoy, T.; Hsu, E.; Lowenthal, S. Dynamics of pH in controlled stirred tank reactor. Ind. Eng. Chem. Process Des. Dev. 1972, 11 (1), 68. (17) Isidori, A. Nonlinear Control Systems, 3rd ed.; SpringerVerlag: London, 1995. (18) Khalil, H. Nonlinear Systems, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1996. (19) Kailath, T. Linear Systems; Prentice-Hall: Englewood Cliffs, NJ, 1980. (20) Yoon, S.; Yoon, T.; Yang, D.; Kang, T. Indirect adaptive nonlinear control of a pH process. Proc. 14th Triennial IFAC World Congr. 1999, N-7a, 139. (21) Alvarez, H.; Londonõ, C.; di Sciacio, F.; Carelli, R. pH neutralization process as a benchmark for testing nonlinear controllers. Ind. Eng. Chem. Res. 2001, 40, 2467. (22) Galań, O. Robust multi-linear model-based control for nonlinear plants. Ph.D. Thesis, University of Sydney, Sydney, Australia, 2000. (23) Gustafsson, T.; Waller, K. Dynamic modelling and reaction invariant control of pH. Chem. Eng. Sci. 1983, 38 (3), 389. (24) Shinskey, F. G. Control of pH. The control handbook; CRC Press: Boca Raton, FL, 1996. (25) Wright, R.; Smith, B.; Kravaris, C. On-line identification and nonlinear control of pH processes. Ind. Eng. Chem. Res. 1998, 37, 2446. (26) Shinskey, F. G. pH and pIon control in process waste streams; Wiley-Interscience: New York, 1973. (27) Proudfoot, C. G.; Gawthrop, P. J.; Jacobs, O. L. R. Selftuning PI control of a pH neutralization process. Proc. IEE D 1983, 130, 267. (28) Shinskey, F. G. Adaptive pH controller monitors nonlinear process. Control Eng. 1974, 57. (29) Norqway, S. J.; Palazoglu, A.; Romagnoli, J. A. Nonlinear model predictive control of pH neutralization using Wiener models. Proc. 13th IFAC World Congr. 1996, M, 31-36.. (30) Fruzzeti, K. P.; Palazoglu, A.; McDonald, K. A. Nonlinear model predictive control using Hammerstein models. J. Process Control, 1997, 7, 31. (31) Gerksˇicˇ, S.; Juricˇic´, D.; Strmcˇnik, S.; Matko, D. Wiener model based nonlinear predictive control. Int. J. Syst. Sci. 2000, 31 (2), 189. (32) Ciccarella, G.; Dalla Mora, M.; Germani, A. A Luenbergerlike observer for nonlinear systems. Int. J. Control 1993, 57, 537.

Received for review December 3, 2001 Revised manuscript received May 6, 2002 Accepted July 8, 2002 IE010974F

State Estimation in Nonlinear Processes. Application to pH Process

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