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Ind. Eng. Chem. Res. 2001, 40, 4102-4110
Indirect Adaptive Backstepping Control of a pH Neutralization Process Based on Recursive Prediction Error Method for Combined State and Parameter Estimation Tae Chul Lee and Dae Ryook Yang* Department of Chemical Engineering, Korea University, 1-Anamdong, Sungbukku, Seoul 136-701, Korea
Kwang Soon Lee Department of Chemical Engineering, Sogang University, 1-Shinsoodong, Mapogu, Seoul 121-742, Korea
Tae-Woong Yoon School of Electrical Engineering, Korea University, 1-Anamdong, Sungbukku, Seoul 136-701, Korea
A novel nonlinear adaptive technique has been proposed for control of a pH neutralization process. For this, a standard nonlinear dynamic pH neutralization model was parametrized assuming that some key parameters of the buffer and feed streams, which sensitively affect the titration curve, are unknown. On this model, an indirect adaptive backstepping controller is designed by combining a backstepping controller and the recursive prediction error method for real-time parameter and state estimation. Through simulation studies, it has been shown that the estimated parameters and state variables are in good agreement with the actual values and that the proposed adaptive controller has excellent tracking and regulation performance. Introduction Neutralization of a pH process has long been taken as a representative benchmark problem of nonlinear chemical process control. It is not only due to its importance in various chemical and related industries but also due to the intricate and tricky intrinsic nonlinearities that may change sensitively to small changes in process conditions. Addressing such nonlinear characteristics, recent pH control studies are mostly directed to the development and/or application of model-based nonlinear control techniques. However, as has been indicated in Gustafsson and Waller1 and Henson and Seborg,2 performance enhancement from the employment of nonlinear control techniques may be only marginal compared to that of well-tuned linear controllers despite the computational complexity. One of the reasons for this is the limitation of the nonlinear pH models. Most models are tuned over a narrow operating region or constructed to be valid under restricted assumptions such as constant buffer compositions and/ or constant feed compositions, and so on. Because a small change in buffer may cause a large change in the titration curve, nonlinear model-based control techniques may not be successful in real situations unless some provisions are furnished to relax such restrictive assumptions. Another potential reason is the limitation of the existing nonlinear control techniques, which are still in an infant stage compared to the abundant, mature linear control techniques. A general dynamic model of the pH neutralization process had been discussed earlier by McAvoy et al.3 They derived a mathematical model from the first principles, i.e., ionic balances and chemical equilibria. * Author to whom correspondence should be addressed. Tel: +82-2-3290-3298.Fax: +82-2-926-6102.E-mail: dryang@infosys. korea.ac.kr.
Later, Gustafsson and Waller4 proposed use of the concept of reaction invariant5 in modeling and control of a pH process. Component balance with respect to reaction invariants greatly simplifies the modeling procedure and has gained wide acceptance as a standard method for pH neutralization process modeling. Recently, Gustafsson et al.6 have developed a more general pH model for control, which can account for such practical factors as precipitation, formation of chemical complexes, and so on. Instead of considering a detailed component balance, Wright and Kravaris7 proposed a simplified model using the concept of strong acid equivalent. The strong acid equivalent is a formula to account for the total contribution of the acidic ions to pH. They developed a technique to control the strong acid equivalent7 and evaluated the performance in an experimental pH neutralization apparatus.8 With the progress in the modeling technique for pH processes, many different model-based control methods have been proposed under different problem settings. Among them are inline process-model-based control,9 nonlinear inferential control,10 and nonlinear control using strong acid equivalent,7 etc. While different nonadaptive techniques have been attempted, adaptive nonlinear control has also been studied by many researchers to overcome the intrinsic uncertainties of the process model. Gustafsson and Waller1 designed a nonlinear adaptive controller. Through simulation study, they found that the proposed nonlinear adaptive controller outperforms those of the conventional proportional integral differential (PID) and linear adaptive controllers. Wright et al.11 introduced an online identification algorithm for unknown chemical species in the estimation of the strong acid equivalent and constructed a nonlinear adaptive controller. Lakshmi et al.12 proposed an adaptive internal model control technique which is based on the concepts of nonlinear internal
10.1021/ie0002632 CCC: $20.00 © 2001 American Chemical Society Published on Web 08/24/2001
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model control, strong acid equivalent, and a simplified adaptive mechanism.12 Although the techniques showed a satisfactory performance in the exemplary pH process model, it may not work properly for more complex neutralization processes when there is a change in the buffering solution due to the use of strong acid equivalent and simple adaptation mechanism. Among different nonlinear control techniques, feedback linearization and backstepping design are the representative up-to-date ones and have recently attracted researchers’ attention more than others. Feedback linearization is for linearizing a nonlinear system by canceling its nonlinearities in order to apply abundant linear control theories to the linearized system.13 Meanwhile, backstepping design provides a systematic stabilizing procedure avoiding unwanted cancellation of favorable nonlinearities.14 Adaptive nonlinear control schemes can be designed by combining the nonlinear controllers with parameter estimation algorithms in a variety of ways. For adaptive feedback linearization, see, e.g., refs 15-17. Adaptive backstepping design methods can be found in refs 14, 18, and 19. Henson and Seborg20 developed an adaptive inputoutput linearizing control strategy for a pH neutralization process to account for the unmeasured changes in the buffer solution; however, some theoretical justifications such as internal stability were not given. Yoon et al.21 presented an adaptive backstepping state feedback controller for a pH process with a proof on internal stability. Henson and Seborg2 compared several adaptive control strategies for pH processes and concluded that indirect adaptive controllers are superior to the direct schemes. Especially, they advocated indirect adaptive input-output linearizing control as a highperformance pH control strategy. However, they introduced a rather unrealistic assumption that the reaction invariants are available for feedback. This problem may be resolved by estimating the reaction invariants from the output measurements using an appropriate state estimator. On the basis of the above general background, in this study, we have designed an indirect adaptive backstepping control scheme for a pH neutralization process. The process is assumed to have an arbitrary acidic feed stream and a buffer stream whose key reaction invariants are time-varying and unknown. To design backstepping control under this rather general problem setting, the unknown stream conditions as well as state information should be available. The theoretical backups behind the proposed controller are the proofs of nonlinear state observability and internal stability by Yoon et al.20 for the concerned pH process. These proofs allow us to design an adaptive backstepping control on the basis of the pH measurement. The simultaneous estimation of state and unknown parameters, which are unknown stream conditions, was conducted using an recursive prediction error method (RPEM). Because the backstepping design in the discrete-time domain has not been developed appropriately yet, the feedback control runs in the continuous-time domain while the identification is conducted on a discrete-time base. For this, a continuous-discrete RPEM algorithm is used for identification.22 To enhance the performance of identification, variable forgetting and covariance resetting were employed for efficient tracking of the chosen time-
Figure 1. pH neutralization process.
varying parameters. The performance of the proposed technique was demonstrated through numerical simulation. Model of pH Neutralization Process We consider a pH neutralization process as considered in ref 2. A schematic diagram is shown in Figure 1. The process consists of an acid stream (q1), a buffer stream (q2), a base stream (q3), and an effluent stream (q4). The flow rate of the base stream (q3) is manipulated to regulate the pH of the effluent stream, which is denoted by pH4. A dynamic model of the process is obtained from the component material balance and the equilibrium relationship under the assumptions of perfect mixing, constant vessel volume V, and so on. The chemical reactions occurring in the system are
H2CO3 T HCO3- + H+
(1a)
HCO3- T CO32- + H+
(1b)
H2O T OH- + H+
(1c)
The equilibrium constants for the reactions are
Ka1 )
Ka2 )
[HCO3-][H+] [H2CO3] [CO32-][H+] [HCO3-]
Kw ) [H+][OH-]
(2a)
(2b) (2c)
The chemical equilibria are modeled using the concept of a reaction invariant.4,5 For this system, two reaction invariants are concerned for each stream (i ) 1-4)
Wai ) [H+]i - [OH-]i - [HCO3-]i - 2[CO32-]i Wbi ) [H2CO3]i + [HCO3-]i + [CO32-]i
(3a) (3b)
The invariant Wa is a charge-related quantity, while Wb represents the concentration of the carbonate ion. These invariants are independent of the extent of the reaction.
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As in ref 2, it follows from eqs 2 and 3 that
Ka1/[H+] + 2Ka1Ka2/[H+]2
Kw Wbi + Wai + + + + 2 1 + Ka1/[H ] + Ka1Ka2/[H ] [H ] [H+] ) 0, i ) 1-4 (4) The concentration of hydrogen ion can be determined from the above equations for given Wai and Wbi, and the pH is calculated by the following relationship:
pH ) -log([H+])
(5)
A dynamic model for the pH neutralization process can be obtained from the component balance for the reaction invariants around the reactor vessel.
V
V
dWa4 ) q1(Wa1 - Wa4) + q2(Wa2 - Wa4) + q3(Wa3 dt Wa4) (6a) dWb4 ) q1(Wb1 - Wb4) + q2(Wb2 - Wb4) + q3(Wb3 dt Wb4) (6b)
The output equation is given in eq 4 for i ) 4. In the above model, we can reasonably assume that all of the flow rates and the conditions of the base stream are known. Then the remaining unknowns are the conditions for the buffer and feed streams. Among them, we assumed that Wa2 and Wb1 are available and chose Wa1 and Wb2 as unknown parameters. Then, the concerned pH process model can be written in the following nonlinear state space model:
x3 ) f(x,t) + g(x,t) u + Fp c(x,y) ) 0
(7)
a neutralization process. Unless a neutralization process involves only strong acid and base, a buffer stream must be added to avoid an erratic titration curve. Although the buffer stream is a prepared one, Wb2 is treated as an unknown parameter because a change in Wb2, which is possible by dissolution of CO2 in the air, can result in a significant change in the characteristics of the neutralization process. Adaptive Backstepping Controller The model of the pH neutralization process given in the previous section reveals nonlinearity and contains unknown time-varying parameters. To effectively deal with the posed process situation, various nonlinear control strategies are proposed and adaptive control of nonlinear systems has drawn much attention recently. Among existing nonlinear control techniques, backstepping control is regarded as a new breakthrough.23 The backstepping-based control offers a systematic stabilizing procedure, while unwanted cancellation of favorable nonlinearities can be avoided.14 In addition, a wide class of uncertain nonlinear systems can be handled by backstepping. The detailed design procedure of a general backstepping controller can be found in Krstic´ et al.14 Yoon et al.21 designed an direct adaptive backstepping controller for the pH process considered in this study and proved the internal stability of the closed loop under the assumption that the state variables are measured. To estimate the state together with the unknown parameters, in this study, we introduced an RPEM technique and designed an indirect adaptive backstepping scheme. First, the key procedures of the backstepping design for the pH neutralization process are briefly reviewed as given in eq 7. The standard backstepping procedure is slightly modified to achieve a desired rise time. For integral control action, define an error variable z1 as
where
z1 )
[ ]
W x ) Wa4 , u ) q3, y ) pH4, pK1 ) log Ka1, b4 pK2 ) log Ka2 f)
[
]
[
]
1 q2(Wa2 - x1) - q1x1 1 W -x , g ) Wa3 - x1 , V q1(Wb1 - x2) - q2x2 V b3 2 q W 0 1 1 F) , p ) Wa1 q V 0 3 b2
[ ]
[ ]
y-pK2
1 + 2 × 10 c(x,y) ) x1 + 10y-14 - 10-y + x2 1 + 10pK1-y + 10y-pK2 In the above, p contains the unknown parameters we chose. The reasoning behind this choice is as follows: Wa1 represents a charge-related quantity and gives the hydrogen ion related information of the feed stream. In practical situations, major ionic species contained in the feed stream are usually fixed, but the composition of each species may vary significantly. In our case, the feed consists of a large part of nitric acid and a very small part of carbonic acid. Hence, we set Wb1 at zero and left the effects of nonzero Wb1 on the resulting pH to be compensated by Wb2. Wb2 is the key condition of the buffer stream. As is well-known, the buffer stream plays a very important role in deciding the characteristics of
∫0t(y - yd) dt
(8)
or, equivalently
z˘ 1 ) y - yd
(9)
where yd is the set point for y. Considering y - yd as the so-called virtual control for eq 9, define another error variable z2 as follows:
1 z2 ) (y - yd - R1) η
(10)
where the stabilizing function R1 is given by
R1 ) -c1z1
(11)
with c1 and η being positive constants. Note that the design constant η introduced as the standard backstepping design may fail to satisfy the desired closed-loop specifications and the function R1 stabilizes eq 9 when y - yd is equal to R1. Using eqs 10 and 11, z˘ 1 is written as
z˘ 1 ) -c1z1 + ηz2
(12)
Here, the first Lyapunov-like function is defined as
Ind. Eng. Chem. Res., Vol. 40, No. 19, 2001 4105
V1 ) (1/2)z12
(13)
Then the derivative of V1 is
V˙ 1 ) -c1z12 + ηz1z2
(14)
Clearly, the boundedness of V1 does not follow from this equation because of the term ηz1z2 in V˙ 1. Now we define the second Lyapunov-like function
V2 ) V1 + (1/2)z22
(15)
To consider V˙ 2, we differentiate z2 and obtain
(
)
(16)
where y˘ is -cy-1cxgu - cy-1cxf - cy-1cxFp and thus we have
{
}
(17)
{
1 + 2 × 10y-pK2 1 + 10pK1-y + 10y-pK2
]
cy ) (ln 10) 10y-14 + 10-y +
}
10pK1-y + 10y-pK2 + 4 × 10pK1-pK2 x2 (1 + 10pK1-y + 10y-pK2)2
(18)
Further, by defining the nonlinear functions
1 l1 ) -cxg ) - (10y-14 - 10-y + Wa3 + cx2Wb3) V 1 q2 l2 ) -cxf ) - (q1 + q2)(10y-14 - 10-y) - (Wa2 + V V cx2Wb1) (19) cx2 )
1 + 2 × 10y-pK2 1 + 10pK1-y + 10y-pK2
and the regressor φ(x) as
φ(x) ) (cxF)T
(20)
we finally express z˘ 2 in the following compact form:
z˘ 2 )
{
l2 φTp ∂R1 1 l1 u+ + - y˘ d z˘ η cy cy cy ∂z1 1
}
(21)
For stabilization of the second Lyapunov-like function in eq 15, the backstepping controller is now derived as follows:
u)
with k1 and k2 being positive constants and l3 defined by
l3 ) -
cy2 {-η2z1 + y˘ d + c12z1 - c1ηz1} η
V˙ 2 ) -c1z12 + ηz1z2 + z2z˘ 2 2
2
(24)
e -c1z1 - cj2z2
(25)
2 1 |x˜ 2| 1 ||p˜ ||2 + + 4k1 |c |2 4k2 |c |2 y
e -2c0V2 +
y
1 |x˜ 2| 1 ||p˜ ||2 + 2 4k1 |c | 4k2 |c |2 y
where cx and cy are the partial derivatives of c(x,y) with respect to x and y, i.e.
[
(23)
2
cx cx ∂R1 1 cx - gu - f + Fp - y˘ d z˘ η cy cy cy ∂z1 1
cx ) [cx1 cx2 ]) 1
l3 ||η-1φ||2 χu ) -k1η - k2η z2 cˆ y cˆ y
As a result of this backstepping controller, we have
∂R1 1 z˘ z˘ 2 ) y˘ - y˘ d η ∂z1 1
z˘ 2 )
where c2 is a positive constant, cˆ y is the estimate for cy calculated with the actual states replaced by their estimates, and χu is defined by
{
l2 φTpˆ cˆ y -c2ηz2 - η2z1 - + + y˘ d + c12z1 l1 cˆ y cˆ y c1ηz1 - c1ηz2 + χu
}
(22)
y
where cj2 is c2cˆ y/cy, c0 ) min {c1, cj2}, and x˜ and p˜ denote the state and parameter estimation errors, respectively. The above inequality does not say that V˙ 2 may be negative; hence, V2 is not a Lyapunov function. However, in the case that both x˜ and p˜ are bounded, V2 cannot grow indefinitely and remains bounded because a positive V˙ 2 results in an increase of V2, which again leads V˙ 2 back to negative. Consequently, V2 values are bounded as far as the state and parameter estimation errors are bounded. From this fact, it can be derived that the error variables of the system and controller remain bounded. More details on the internal stability and the controller design can be found in ref 24. Note that the presence of z1 in eq 22 implies an integral action in the controller. Also the parameter estimate p˜ is used to evaluate φTp˜ in eq 22. Recursive Prediction Error Method (RPEM) If we scrutinize each term in eq 22, it can be seen that both the state and parameter information is needed to compute u. In this study, we employed an RPEM25,26 for simultaneous estimation of x and p. Typically, the RPEM has been developed for discrete-time systems with sampled measurements. The backstepping controller was, however, designed in the continuous-time domain because the design procedure for discrete-time backstepping controllers has not been established yet. To accommodate the two different time frames, we employed a continuous-discrete version of the RPEM applied to a nonlinear system.22 We briefly describe the algorithm. Consider the following general nonlinear continuoustime state-space model:
x3 (t) ) f(x(t),u,p) + v(t) y(t) ) h(x(t),p) + w(t)
(26)
where v and w are white noises and p is the unknown parameter vector. Here, it is assumed that the output is sampled at discrete times, ti (i ) 0, 1, 2, ...), with a fixed sampling interval, i.e., ti+1 - ti ) h for all i. For this continuous-time, sampled-data process, the follow-
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ing predictor is assumed:
i
xˆ (t|ti) ) f(xˆ (t|ti),u(t),θˆ ), t∈ [ti, ti+1]
Λ(ti) )
yˆ(ti+1) ) h(xˆ (ti+1|ti+1),θˆ )
(27)
where xˆ (ti+1|ti) and xˆ (ti+1|ti+1) represent the time and measurement updates, respectively; (ti+1,θˆ ) is the prediction error given by y(ti+1) - h(xˆ (ti+1|ti),θˆ ); K denotes the parametrized steady-state Kalman gain matrix; and θ is the parameter vector to estimate including process parameter, p and the diagonal elements of Kalman gain. In RPEM, parameter estimation is performed by solving the following minimization problem:
{
θˆ ) arg min V(θ) ) θ
1
∑T(ti,θ) Λ-1(ti,θ)
2Ni)1
}
(29a)
yˆ (ti+1|ti) ) h(x(ti+1|ti),θˆ )
(29b)
(ti+1) ) y(ti+1) - yˆ (ti+1|ti)
(29c)
(
)
T
ˆ (ti) + γ(ti+1) [(ti+1) T(ti+1) - Λ ˆ (ti)] Λ ˆ (ti+1) ) Λ
(29d) (29e)
R(ti+1) ) R(ti) + γ(ti+1) [Ψ(ti+1) Λ ˆ -1(ti+1) ΨT(ti+1) + δI - R(ti)] (29f) ˆ -1(ti+1) θˆ (ti+1) ) θˆ (ti) + γ(ti+1) R-1(ti+1) Ψ(ti+1) Λ (ti+1) (29g) xˆ (ti+1|ti+1) ) xˆ (ti+1|ti) + K(θˆ ) (ti+1,θˆ )
i
β(i,j) )
λ(tj), ∏ k)j
0 < λ(tj) e 1
(32)
One of the useful forms of λ(ti) is the variable forgetting factor28 given by
λ˜ (ti) ) 1 -
xˆ (t|ti) ) f(xˆ (t|ti),u(t),θˆ ), t∈ [ti, ti+1]
dyˆ (ti+1|ti) dθˆ
where β(i,j) represents the discount factor of the past information. Because the information in the distant past needs to be discounted more, it is conveniently represented by the product of the forgetting factors at each in the past
(28)
Following Zhou and Blanke’s procedure,22 a recursive formula is obtained as
Ψ(ti+1) )
(31)
i
β(i,j) ∑ j)0
xˆ (ti+1|ti+1) ) xˆ (ti+1|ti) + K(θˆ ) (ti+1,θˆ )
N
β(i,j) (tj) (tj)T ∑ j)0
2(ti) 1 σ 1 + Ψ(t ) R-1(t ) ΨT(t ) i
i
i
λ(ti) ) max {λ˜ (ti), λmin}, 0 < λmin e 1
(33)
where σ is a scalar used to speed up adaptation for sudden changes in the set point or load disturbances, λmin is the predefined minimum value of the forgetting factor. The above formula discounts the past information more in proportion to the information content in the new measurement. In addition to the variable forgetting factor, the covariance resetting was employed together to effectively track the parameters that may change abruptly but infrequently. The covariance is reset on the basis of the residual change between the measured and estimated outputs. When applying the above recursive formula, Ψ(t) was computed numerically because the output is given as an implicit function. The above RPEM algorithm is combined with the backstepping controller for adaptive control. Note that the pH neutralization process is found to be unobservable when the observability is tested for a linearized model.20 However, recent investigation of nonlinear observability has revealed that the concerned process is observable.24
(29h)
where ˆ represents the estimated values; -Ψ(t) denotes the parameter sensitivity of the prediction error; Λ(t) is the covariance matrix of the prediction error; and R(t) is the Gauss-Newton direction to update the parameter estimate. In the original RPEM algorithm, the updating formula of R(t) is given by
ˆ -1(ti+1) ΨT(ti+1) R(ti+1) ) R(ti) + γ(ti+1) [Ψ(ti+1) Λ R(ti)] (30) However, R(t) in the above formula is apt to be singular when the input signal is not rich enough and may cause a numerical problem when the inversion is computed. To remedy this problem, we adopted the LevenbergMarquardt regularization algorithm as given in eq 29f.27 Finally, γ(ti) represents the contribution of the information at ti to the parameter update in relation to that of the weighted integral of past information. For example, Λ(t) is evaluated such that
Simulation Results and Discussion Performance of the proposed adaptive nonlinear controller has been investigated through simulation study. The nominal operating conditions are given in Table 1. The states to be estimated are the reaction invariants in the effluent stream. The parameters to be estimated are the reaction invariants Wa1 and Wb2 in the input streams and the two elements of the Kalman gain matrix K. The Wa1 is a charge-related quantity of the stream q1, and Wb2 represents the concentration of carbonate ion in the buffer stream q2. Before adaptive control was initiated, an identification experiment was conducted for 150 samples by superimposing a series of PRBS signals of magnitude (2 [mL/ s] on the nominal flow rate of the base stream, q3. Then adaptive backstepping control was turned on and lasted for 474 sample steps without any external excitation of the input. The initial values of the state and parameter estimates were assumed to be deviated from their true values by (10%. Also the pH measurement was made
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Figure 2. Performance of the RPEM: true values (solid) and estimates (dashed).
Figure 3. Performance of the RPEM when there is a change in Wa1: true values (solid) and estimates (dashed).
to be corrupted by a zero-mean Gaussian noise of variance 0.035. The sampling time for identification was chosen to be 5 s, while the control action was exerted continuously. Figure 2 presents the performance of the RPEM during the identification experiment. We can see that the state as well as the parameter estimates very closely follow their actual values after some initial transients. Also the value of the pH estimate is observed to be almost exact over the whole duration. Between the two ˆ b2 parameters, W ˆ a1 is found to converge faster than W to the true values. This result indicates that the pH is
less sensitive to a change in the carbonate ion in the input stream. The same phenomenon is duplicated in other simulation experiments shown in the next two figures. In Figures 3 and 4, we applied abrupt changes in Wa1 and Wb2, respectively. It is shown that the RPEM works satisfactorily, providing quite accurate estimates of both states and parameters as in the previous case. Figure 5 shows the tracking performance of the adaptive backstepping controller when the set point is changed from 7 to 6 at about 25 min (300th sample) and back to 8 at about 37.5 min (450th sample) while the parameters are kept constant. The control input is
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Figure 4. Performance of the RPEM when there is a change in Wb2: true values (solid) and estimates (dashed).
Figure 5. Tracking performance: true values (solid) and estimates (dashed).
seen to be reduced or increased according to the change of set point. The control and estimation performance is also very satisfactory, and the set-point changes are found not to affect parameter identification. Also, it seems that the output follows the constant set point quite well after the control loop is closed. Figures 6 and 7 show the combined tracking and regulation performance of the controller when there are changes in parameter values. The set-point change occurs during the period between 33 min (400th sample) and 41 min (500th sample) after Wa1 is changed by 15% at about 25 min (300th sample), which means that the
hydrogen ion in the feed stream is abruptly increased. In Figure 7, Wb2 is increased by 15% at about 25 min (300th sample), which changes the characteristics of the neutralization process by changing the buffer concentration. For these two cases, the control performance is quite satisfactory. However, the estimation of the quantities related to the carbonate ion, x2 and Wb2, revealed some difficulties. Part of the cause is thought to be the low sensitivity of the pH to a change in carbonate ion as mentioned earlier. To remedy the trouble, the covariance resetting is conducted whenever the prediction error of the pH tends to settle below some
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Figure 6. Combined tracking and regulation performance when there is a change in Wa1: true values (solid) and estimates (dashed).
Figure 7. Combined tracking and regulation performance when there is a change in Wb2: true values (solid) and estimates (dashed).
threshold value. This made the RPEM algorithm immediately detect the change in Wa1 or Wb2 and update the estimates such that the true values of the parameters are tracked very rapidly. Through the simulation study, it is found that the combination of the continuous-time backstepping controller and the discrete-time RPEM can provide a plausible control framework for nonlinear time-varying processes. However, there are some limitations in this technique yet. Because the current backstepping design cannot handle the time delay while the real process may reveal a significant time-delay effect, one of the future
research topics should be focused on the time-delay issue: either enhancing the backstepping design to directly handle the time delay or combining the Smith predictor with the present current backstepping design or others. Conclusions In this paper, a new indirect adaptive nonlinear control method has been proposed for a pH neutralization process by combining a RPEM and a backstepping controller. The full state information required by the
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Table 1. Nominal Process Parameters symbol
value
symbol
value
V Ka1 Ka2 [q1]
2900 mL 4.47 × 10-7 5.62 × 10-11 0.003 M HNO3 5.0 × 10-5 M H2CO3 0.03 M NaHCO3 0.003 M NaOH 5.0 × 10-5 M NaHCO3 16.6 mL/s 0.55 mL/s
q3 pH4 Wa1 Wb1 Wa2 Wb2 Wa3 Wb3 Wa4 Wb4
15.8 mL/s 7.00 0.003 M 5.00 × 10-5 M -0.03 M 0.03 M -3.05 × 10-3 M 5.00 × 10-5 M -4.50 × 10-4 M 5.50 × 10-4 M
[q2] [q3] q1 q2
backstepping controller is provided by the RPEM on the discrete-time basis together with the parameter estimates. Numerical simulations demonstrated that the parameter as well as the state estimates are in good agreement with the actual values and that the proposed adaptive backstepping controller has excellent tracking and regulation performance under various changes in influent streams of the pH process. The proposed nonlinear adaptive control technique can be applied to a wide class of nonlinear chemical processes. Although the backstepping design is still in the early stage, as it is developed more, the proposed technique is believed to provide a new framework that improves the current techniques such as input-output linearization or others. Acknowledgment The authors are grateful for financial support by the Korea Science and Engineering Foundation (KOSEF). Literature Cited (1) Gustafsson, T. K.; Waller, K. V. Nonlinear and Adaptive Control of pH. Ind. Eng. Chem. Res. 1992, 31, 2681. (2) Henson, M. A.; Seborg, D. E. Adaptive Input-Output Linearization of a pH Neutralization Process. Int. J. Adapt. Control Signal Process 1997, 11, 171. (3) McAvoy, T. J.; Hsu, E.; Lowenthal, S. Dynamics of pH in a Controlled Stirred Tank Reactor. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 68. (4) Gustafsson, T. K.; Waller, K. V. Dynamic Modeling and Reaction Invariant Control of pH. Chem. Eng. Sci. 1983, 38, 389. (5) Waller, K. V.; Makila, P. M. Chemical Reaction Invariants and Variants and Their Use in Reactor Modeling, Simulation, and Control. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 1. (6) Gustafsson, T. K.; Skrifvars, B. O.; Sandstro¨m, K. V.; Waller, K. V. Modeling of pH for Control. Ind. Eng. Chem. Res. 1995, 34, 820. (7) Wright, R. A.; Kravaris, C. Nonlinear Control of pH Processes Using the Strong Acid Equivalent. Ind. Eng. Chem. Res. 1991, 30, 1561. (8) Wright, R. A.; Soroush, M.; Kravaris, C. Strong Acid Equilvalent Control of pH Processes: An Experimental Study. Ind. Eng. Chem. Res. 1991, 30, 2437.
(9) Williams, G. L.; Rhinehert, R. R.; Riggs, J. B. In-Line Process-Model-Based Control of Wastewater pH Using Dual Base Injection. Ind. Eng. Chem. Res. 1990, 29, 1254. (10) Parrish, J. R.; Brosilow, C. B. Nonlinear Inferential Control. AIChE J. 1988, 34, 633. (11) Wright, R. A.; Smith, B. E.; Kravaris C. On-Line Identification and Nonlinear Control of pH Processes. Ind. Eng. Chem. Res. 1998, 37, 2446. (12) Lakshmi Narayanan, N. R.; Krishnaswamy, P. R.; Rangaiah, G. P. An Adaptive Internal Model Control Strategy for pH Neutralization. Chem. Eng. Sci. 1997, 18, 3067. (13) Slotine, J.-J. E.; Li, W. Applied Nonlinear Control; Prentice Hall Inc.: Upper Saddle River, NJ, 1991. (14) Krstic´, M.; Kaneelankopoilos, I.; Kokotovic, P. Nonlinear and Adaptive Control Design; John Wiley & Sons: New York, 1995. (15) Campion, G.; Bastin, G. Indirect adaptive state feedback control of linearly parametrized nonlinear systems. Int. J. Adapt. Control Signal Process 1990, 11, 345. (16) Sastry, S. S.; Isidori, A. Adaptive control of linearizable systems. IEEE Trans. Autom. Control 1989, 34, 1123. (17) Teel, A.; Kadiyala, R.; Kokotovic´, P. V.; Sastry, S. S. Indirect techniques for adaptive input-output linearization of nonlinear systems. Int. J. Control 1991, 52, 193. (18) Kanellakopoulos, I.; Kokotovic´, P. V.; Morse, A. S. Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Autom. Control 1991, 36, 1241. (19) Marino, R.; Tomei, P. Nonlinear Control Design; PrenticeHall: Upper Saddle River, NJ, 1995. (20) Henson, M. A.; Seborg, D. E. Adaptive Nonlinear Control of a pH Neutralization Process. IEEE Trans. Control Syst. Technol. 1994, 2, 169. (21) Yoon, S. S.; Yoon, T. W.; Yang, D. R.; Kang, T. S. Indirect Adaptive Nonlinear Control of a pH Process. IFAC 99 (Beijing, China) 1999, N, 139. (22) Zhou, W.-W.; Blanke, M. Indetification of a Class of Nonlinear State-Space Models Using RPE Techniques. IEEE Trans. Autom. Control 1989, 34, 312. (23) Kokotovic´, P.; Arcak, M. Constructive nonlinear control: a historical perspective. Automatica 2001, 37, 637. (24) Yoon, S. S.; Yoon, T. W.; Yang, D. R.; Kang, T. S. Adaptive Output Feedback Nonlinear Control of a pH Process with an Input Constraint. 39th IEEE Conference on Decision and Control, Sidney, Australia, 2000. (25) Ljung, L.; So¨derstro¨m, T. Theory and Practice of Recursive Identification; MIT Press: Cambridge, MA, 1983. (26) Ljung, L. Asymptotic Behavior of the Extended Kalman Filter as a Parameter Estimator for Linear Systems. IEEE Trans. Autom. Control 1979, AC-24, 36 (27) Astrom, K. J.; Wittenmark, B. Adaptive Control; AddisonWesley Publishing: Reading, MA, 1995. (28) Fortescue, T. R.; Kershenbaum, L. S.; Ydstie, B. E. Implementation of Self-tuning Regulators with Variable Forgetting Factors. Automatica 1981, 17, 831.
Received for review February 22, 2000 Revised manuscript received May 19, 2001 Accepted June 27, 2001 IE0002632
