Strategies for Enhancing Geometric Nonlinear ... - ACS Publications

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Ind. Eng. Chem. Res. 2001, 40, 656-667

Strategies for Enhancing Geometric Nonlinear Control of an Industrial Evaporator System Kiew M. Kam,† Moses O. Tade´ ,*,† Gade P. Rangaiah,‡ and Yu C. Tian† School of Chemical Engineering, Curtin University of Technology, GPO Box U 1987, Perth, Western Australia 6845, and Department of Chemical and Environmental Engineering, The National University of Singapore, Singapore 119260

Two strategies, adaptive input-output internal model control (AdIOIMC) and augmented inputoutput internal model control (AuIOIMC), for enhancing the nonlinear control of an industrial multistage evaporator system are proposed. Each strategy consists of a nonlinear state feedback control law, designed by the input-output linearization technique, to linearize feedback and stabilize the open-loop unstable evaporator and the internal model control (IMC) structure for the feedback-linearized system. The combination of the nonlinear state feedback control law and the IMC structure is referred to as input-output internal model control (IOIMC). The AuIOIMC is composed of an IOIMC structure and an additional loop through which the model error is fed back and added to the input of the IMC. For AdIOIMC, the model error is fed to an adaptation loop that adjusts the model parameter within the IMC structure of IOIMC. The advantages of the proposed AdIOIMC and AuIOIMC strategies on the industrial evaporator system were evaluated through simulation studies. The results show that both strategies provide improved regulatory control performance when compared to the MIMO GLC. 1. Introduction Two main approaches for control of nonlinear systems are internal model control (IMC) and input-output (I/ O) linearization. IMC1 is regarded as the principal methodology for robust analysis and synthesis. It provides a unified approach for the analysis and synthesis of control system performance, especially robust properties. Moreover, most of the existing advanced controllers can equivalently be put into the general IMC form.2,3 Significant research has been carried out on the extension of IMC to nonlinear systems. Henson and Seborg4 provided the general extension of linear IMC to openloop stable, minimum-phase single-input single-output (SISO) nonlinear systems. In nonlinear IMC (NIMC), the controller is designed as the right inverse of the nonlinear model using the method of Hirschorn.5 Research has shown that the proposed NIMC provides the same closed-loop stability, perfect control, and zero offset properties as the linear IMC for linear systems. Hu and Rangaiah6 presented the augmented NIMC (AuIMC) and adaptive (AdIMC) strategies for enhancing the performance of NIMC in the presence of plant/model mismatch. Input-output (I/O) linearization5 is one of the most widely used techniques for nonlinear control system design. The central idea is to design a nonlinear state feedback control law such that the outputs of the nonlinear system are globally linear with respect to a set of reference signals. To this end, the nonlinear system is said to be partially feedback-linearized. Linear control theory can then be applied to the feedbacklinearized nonlinear system to ensure offset-free per* Author to whom all correspondence should be addressed. Tel.: +61 8 92667704. Fax: +61 8 92663554. E-mail: tadem@che. curtin.edu.au. † Curtin University of Technology. ‡ The National University of Singapore.

formance in the regulation and tracking of the outputs. Kravaris and Chung7 proposed the globally linearizing control (GLC) structure for minimum-phase SISO nonlinear system where an SISO proportional integral (PI) controller was used as the external controller. Kravaris and Soroush8 extended the GLC design to multi-input multi-output (MIMO) nonlinear systems. Klatt and Engell9 proposed gain-scheduling trajectory control (GSTC) for SISO nonlinnear systems. In the GSTC design, a nonlinear state feedback control law is used as a pure nonlinear feed-forward controller, and a nonlinear gain-scheduled reference controller is used for rejecting the unmeasured disturbances in the output and the plant/model mismatch. Kurtz and Henson10 combined linear model predictive control and I/O linearization to solve constrained control of SISO nonlinear systems. Krothapally et al.11 combined the concepts of I/O linearization and sliding mode control for robust control of SISO nonlinear systems in the presence of parametric uncertainty. These are just some of the examples of combining the I/O linearization concept and the well-developed linear control theory for nonlinear control system design. The industrial evaporation system, as described in Kam and Tade´,12 is an open-loop unstable, minimumphase process with high nonlinearities and interactions. The evaporator system is open-loop unstable in the sense that, when an input is perturbed, the process outputs do not reach their new equilibrium points. This can be shown by making a step change in any one of the inputs to the evaporator system. A reduced-order observer was necessary to implement the nonlinear state feedback control law because of incomplete on-line information on the required state variables. The observerbased implementation of the nonlinear state feedback control law led to degraded control of the outputs of the simulated evaporator. In control system design, plant/ model mismatch can be represented by the error between the process and the model outputs, which is also

10.1021/ie000205g CCC: $20.00 © 2001 American Chemical Society Published on Web 12/15/2000

Ind. Eng. Chem. Res., Vol. 40, No. 2, 2001 657

Figure 1. Input-output internal model control (IOIMC) of a nonlinear process.18

Figure 2. Adaptive IOIMC (AdIOIMC) structure.

referred to as the model error. Nonlinear control strategies with enhanced performance that are based on information about either the model error or the output error (i.e., the difference between the output and the corresponding set point) have been reported in the literature.6,13-17 The common approach in these strategies is on-line open-loop simulation of the nonlinear process model, which is then compared with the actual output to generate the model error as the secondary reference. However, on-line open-loop simulation of the evaporator model is not desirable as it is open-loop unstable. In this paper, I/O internal model control (IOIMC)18 is used as the underlying structure for nonlinear control system design with improved model uncertainty rejections for the industrial evaporator system. Two strategies, namely, adaptive IOIMC (AdIOIMC) and augmented IOIMC (AuIOIMC), which are simple and computationally efficient, are proposed and tested for enhanced nonlinear control of the evaporator. Section 2 presents the IOIMC structure. The IOIMC consists of a nonlinear state feedback control law and an external linear IMC structure. By using the IMC approach, tuning rules for the PI controllers of a MIMO GLC structure are presented for comparative studies. The AdIOIMC, which is composed of the IOIMC structure with adaptation loops for updating the parameters within the IMC structure, is presented in section 3. Section 4 presents the AuIOIMC, which consists of the IOIMC structure with an additional feedback loop based on the model error to improve the robustness. The application of AdIOIMC and AuIOIMC and a comparison of their performance with that of MIMO GLC (with IMC-tuned PI controllers) are presented in section 5. The effects of the tuning parameters of AdIOIMC and

AuIOIMC on the control of the evaporator are also investigated in this section. Concluding remarks are provided in section 6. 2. Input-Output Internal Model Control Consider a MIMO minimum-phase nonlinear process P modeled by nonlinear model M that is available for nonlinear controller synthesis as shown below m

x3 ) f(x) +

gj(x)uj ∑ j)1

y ) hi(x) i ) 1, ..., m

(1)

where the definitions of the symbols in eq 1 can be found in the differential geometric literature.5,19 By assuming that model M is I/O linearizable with a globally invertible characteristic matrix, nonlinear state feedback control laws can be designed for I/O linearization and decoupling of the nonlinear plant P.8,20 If the model M in (1) is the perfect description of the nonlinear plant P (i.e., M ≡ P) and all state variables that are required for feedback are available, then the nonlinear state feedback control law would induce a set of decoupled SISO linear systems with desired dynamics as shown below ri

dky˜ i

k)0

dtk

∑ βˆ ik

) vi i ) 1, ..., m

(2)

The scalar values βˆ and v are, respectively, the design parameters and the external references of the nonlinear

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Figure 3. Augmented IOIMC (AuIOIMC) structure. Table 1. Design and Tuning Parameters for the Nonlinear Control Structures output

βˆ i0

βˆ i1

KPi

τIi (h)

τci (h)

ki

γi (h)

h1 h2 h3 h4 F4

21.8 12.3 10.6 9.62 1.55

1 1 1 1 10

1.0 1.0 1.0 1.0 4.0

0.046 0.081 0.094 0.104 6.452

1.0 1.0 1.0 1.0 2.5

5.0 5.0 5.0 5.0 5.0

5.0 5.0 5.0 5.0 5.0

state feedback control law. Because the nonlinear plant is assumed to be minimum-phase, the I/O system in eq 2 is internally stable. In IOIMC, a SISO IMC structure is applied to each of the I/O pairs of the linear system in eq 2. Consider the ith I/O pair of eq 2 in the Laplace transform domain.

Gmi(s) )

y˜ i(s) vi(s)

)

1 βˆ irisri + βˆ i(ri-1)sri-1 + ... + βˆ i1s + βˆ i0

1, serves as the underlying structure for enhancing the nonlinear control of the evaporator. The input-output process in Figure 1 represents the actual feedbacklinearized process and is denoted by GP in the discussion that follows. The IMC-based tuning for PI controllers requires that the closed-loop transfer function (CLTF) be the same as the filter in eq 6. This leads to the design equation for PI controller of a first-order system, after algebraic simplification, as

GCi(s) ) [τcisGPi(s)]-1

By substituting eq 3 with ri ) 1 into eq 7, one can show that the IMC-based tuning rules for the PI controller of MIMO GLC are given by

KPi )

βˆ i1 τci

(8)

τli )

βˆ i1 βˆ i0

(9)

(3) The model-inverse controller that provides “perfect” tracking of the ith output yi is given by1

G/Ci(s)

)

vi(s) ei(s)

ri

ri-1

) βˆ iris + βˆ i(ri-1)s

+ ... + βˆ i1s + βˆ i0 (4)

where ei ) ysp ˜ i) is the error of the modeli - (yi - y inverse controller. The controller in eq 4 is not proper, and a filter is included in the controller design such that it is implementable.

GCi(s) ) G/Ci(s) fi(s)

(5)

The filter has the following form:21

fi(s) )

1 (τcis + 1)i

(6)

where τci is the desired closed-loop time constant of the ith I/O pair and i is a positive integer that is selected to make the controller transfer function GCi(s) proper (i.e., the order of the denominator is equal to or higher than the order of the numerator). The main advantage of IOIMC is that the IMC structure can be indirectly applied to open-loop unstable, minimum-phase nonlinear processes to take advantage of its robustness properties.18 The IOIMC structure, as shown in Figure

(7)

Note that the IMC-based tuning rules for the PI controller are based on a first-order transfer function model as each of the outputs of the evaporator has relative order of 1. The tuning rules in eqs 8 and 9 were used for the design of the PI controllers of MIMO GLC that is used as the benchmark for comparisons in this paper. 3. Adaptive I/O Internal Model Control In AdIOIMC, model errors e˜ are fed to a set of parameter adaptation laws (i.e., block A in Figure 2) that adjusts the parameters of the I/O model G ˜ m, and the model-inverse controller G ˜ C, as shown in Figure 2. It was shown that each output of the four-effect evaporator has a relative order of 1.12 The nominal I/O dynamics, in state-space form, for the ith output of the evaporator is

βˆ i1y˜˙ i + βˆ i0y˜ i ) vi

(10)

Assume that the following I/O model is used to capture the actual I/O dynamics:

βˆ i1y˜˙ i + Rˆ iβˆ i0y˜ i ) vi

(11)

where Rˆ i is the adaptation parameter for the ith I/O


Figure 4. Comparison of closed-loop responses of the controlled outputs by MIMO GLC, AdIOIMC, and AuIOIMC. Table 2. Calculated ITAEs of the Outputs of the Evaporator (70 h)

Rˆ i )

outputs

MIMO GLC

AuIOIMC

AdIOIMC

h1 h2 h3 h4 F4

0.0475 0.0511 0.0035 0.0122 6.0019

0.0049 0.0052 0.0007 0.0013 0.6776

0.0187 0.0096 0.0009 0.0018 0.0133

model, G ˜ mi, the Laplace transform domain model of eq 11. Thus, the ith model-inverse controller is

G ˜ Ci ) fiG ˜ -1 mi

βˆ i1 βˆ i0

( ) ]( )

1 1 1 (y - y˜ i) v γi i βˆ i1 i y˜ i

(13)

such that the ith predicted output tracks the ith actual output with the following desired first-order dynamics:

γiy˜˙ i + y˜ i ) yi

(14)

Proof: Consider the ith I/O dynamics in eq 11 and the adaptation parameter Rˆ i is updated by eq 13. By substituting eq 13 into eq 11, one obtains

(12)

Note that the time constant and the gain of the I/O model are βˆ i1/(Rˆ iβˆ i0) and 1/(Rˆ iβˆ i0), respectively. Therefore, by updating Rˆ i, the time constant and the gain of the model are updated simultaneously with the same factor in order to force the model error (i.e., yi - y˜ i) to zero. It should be noted that the adjustments of the model’s gain and time constant at the same rate do not imply that the same properties of the feedback-linearized process GPi change at the same rate. Theorem 1: For the AdIOIMC structure, there exists an update law

( )[( )

βˆ i1y˜˙ i +

( )[( ) βˆ i1 βˆ i0

( ) ]( )

1 1 1 (y - y˜ i) v - βˆ i0y˜ i ) vi (15) γi i βˆ i1 i y˜ i

By further algebraic manipulation, eq 15 can be shown to reduce to eq 14. Remark: The update law in eq 13 is designed for firstorder I/O dynamical system. For higher order I/O systems, similar update laws can be derived based on their I/O models. Note that the external input vi, the model output y˜ i and the output yi in eq 13 are the actual values, instead of their deviation variables to avoid division by zero.

660


Figure 5. Manipulated inputs corresponding to the closed-loop responses in Figure 4.

Note that γi is a tuning parameter, which ensures that y˜ i f yi asymptotically at an adjustable rate set by γi. It can be seen from eq 14 that, as long as γi > 0 and the output yi is bounded, the tracking dynamics in eq 14 is stable. Note that a small value of γi causes fast tracking of the predicted output to the actual output (or fast error convergence dynamics), whereas a large value of γi causes sluggish response in the error convergence dynamics. The update law in eq 13 is similar to the parameter estimation methodology of Tatiraju and Soroush22 except that it is for an SISO linear system. Property 1: The value Rˆ i converges to

( ) vi βˆ i0yi

asymptotically in the limit as the parameter γi goes to zero. Proof: In the limit that γi f 0, from eq 14, y˜ i f yi asymptotically and exponentially. From eq 13, it can be shown that, as y˜ i f yi, the adaptation parameter Rˆ i converges to

( ) vi βˆ i0yi

Remark: The update law in eq 13 inherently includes a low-pass filter whose input is the measured output yi. Whereas y˜ i f yi asymptotically in the limit γi f 0, a low value is not desirable if the measurement of yi is noisy. On the other hand if γi f ∞, the tracking dynamics in eq 14 reduces to

y˜˙ i ) 0

(16)

This implies that, as γi f ∞, convergence of the model output y˜ i is guaranteed under the update law in eq 13. However, convergence of the model output to the actual output is not guaranteed as the tracking dynamics is not forced by the actual output. It should also be noted that the proposed AdIOIMC scheme cannot discriminate between parametric uncertainty and presence of unmodeled dynamics. The effects of increasing γi of AdIOIMC on the four-effect evaporator are investigated in section 5.2. The nonconvergence of the model output to the actual output under the update law in eq 13 with a large value of γi is also shown.


Figure 6. Responses of the liquor temperatures corresponding to the closed-loop responses in Figure 4.

by setting ki ) 0 in eq 19) that

4. Augmented I/O Internal Model Control The AuIOIMC structure, as shown in Figure 3, consists of an IOIMC (in an equivalent form) and an additional loop where the process/model mismatch e˜ is fed back through gain k and added to the input to the original linear model-inverse controller GC. The AuIOIMC structure in Figure 3 is similar to the AuIMC structure that was proposed by Hu and Rangaiah6 for nonlinear processes, with the exception that the controller GC is linear and manipulates the input to the feedback-linearized process. Note that if k ) 0, the AuIOIMC structure reduces to the IOIMC structure in Figure 1. If the nonlinear process P is approximately linear and decoupled after I/O linearization and decoupling, the ith I/O pair of the feedback-linearized system can be approximated by transfer function GPi. From Figure 3, it can be shown that the transfer function relating the ith controlled output yi to the error ei is given as

GAi(s) )

GCiGPi(1 + kiGCiGmi) 1 + kiGCiGPi

(17)

It can be shown that the error ei is related to the output error (i.e., ysp i - yi) via

ei )

( )

1 (ysp - yi) 1 - fi i

(18)

By substituting eq 18 into eq 17, it can be shown that the CLTF, GRi Ai (s), is given by -1 -1 -1 GRi Ai (s) ) (1 + kiGCiGmi)[1 + ki + GPi (GCi - Gmi)] (19)

If GRi i (s) is the CLTF for IOIMC, it can be shown (e.g.,

-1 -1 -1 GRi i (s) ) [1 + GPi (GCi - Gmi)]

(20)

Theorem 2: If the original IOIMC structure is stable, then there always exists ki > 0 such that AuIOIMC is stable. Proof: The stability of the original IOIMC implies that Gmi, GPi, and GCi are stable. By substituting eq 20 into eq 19, it can be shown that the CLTF of AuIOIMC can be written as Ri Ri -1 (21) GRi Ai (s) ) Gi (s)(1 + kiGCiGmi)[1 + kiGi (s)]

If GRi i (s) ) Z(s)/N(s), then the characteristic equation of eq 21 can be written as

Ni(s) + kiZi(s) ) 0

(22)

Because the IOIMC is stable and minimum-phase, the roots of Ni(s) are on the left half of the s plane. The root locus of eq 22 with ki as the parameter starts from the left half of the s plane and will cross over to the right half of the s plane at point ki ) ki*. Therefore, the AuIOIMC must be stable as long as ki < ki*. Remark: The values ki* cannot be determined unless the bound on the extent of the modeling error is known. In an actual system, this bound cannot be easily defined, and the tuning of ki requires trial and error at implementation or during simulation studies. If ki can be selected sufficiently large without forcing the roots to cross over to the right-hand side of the s plane, then we will have -1 Ri Ri ) fi GRi Ai (s) f Gi (s)(kiGCiGmi)[kiGi (s)]

(23)

The fact that fi is the CLTF of the nominal IOIMC structure18 indicates that the resulting performance of AuIOIMC on the uncertain nonlinear process P is

662


Figure 7. Closed-loop responses of the controlled outputs for various values of γi for AdIOIMC. Table 3. Calculated ITAEs of the Outputs under AdIOIMC with Various Values of γi AdIOIMC outputs

MIMO GLC

γi ) 50

γi ) 5000

h1 h2 h3 h4 F4

0.0475 0.0511 0.0035 0.0122 6.0019

0.0685 0.0088 0.0010 0.0019 0.0172

0.0132 0.0078 0.0011 0.0016 0.0129

similar to the performance of IOIMC on the nonlinear process in the case of no plant/model mismatch. It also indicates that model uncertainty reduction capability of AuIOIMC increases as ki increases. It should be noted that the effect of the feedback loop in AuIOIMC structure is similar to the influence of high-gain feedback control. The model uncertainty reduction properties of AuIOIMC with increasing ki on the four-effect evaporator will be evaluated through simulation. 5. Application to the Evaporator The closed-loop simulations of the proposed AuIOIMC and AdIOIMC structures on the four-effect evaporator are presented in this section. The evaporator consists

of a falling film and three forced circulation effects. Two models, M1 and M2, were developed for the evaporator for control studies. Each model consists of 12 state equations with 5 inputs and 5 outputs. Further details of the evaporator and the models can be found elsewhere.12,23 In the simulation studies, model M1 was used for the simulation of the industrial evaporator, and model M2 was used for the synthesis of the nonlinear state feedback control. It was shown that there were significant mismatches between the two models.23 A reduced-order observer was used for estimating the unmeasured liquor densities (i.e., F1, F2, and F3) from the available measurements of the controlled outputs (i.e., h1, h2, h3, h4, and F4), the secondary outputs (i.e., T1, T2, T3, and T4), and the inputs (i.e., QP1, QP2, QP3, ˘ S4). The reduced-order observer and the QP4, and m observer-based nonlinear state feedback control law were designed based on the methodology presented in Kam and Tade´.12 Two simultaneous unmeasured disturbances were used for the closed-loop simulations. They were a 2 m3/hr increase in the liquor feed flow Qf to the first effect and a 15% reduction in the overall heat transfer rates (i.e., UA’s) of all of the heaters.


Figure 8. Model error dynamics of AdIOIMC with various values of γi.

5.1. Comparison of Nonlinear Control Strategies. In this section, the regulatory control performances of MIMO GLC, AdIOIMC, and AuIOIMC on the evaporator are compared. The design parameters for the nonlinear control structures are given in Table 1. It should be noted that the design parameters βˆ i0 and βˆ i1 were selected according to the design rules given by Kam and Tade´.23 The tuning parameter γi for AdIOIMC and ki for AuIOIMC were selected arbitrarily as the purpose of the case study was to demonstrate the advantages of AdIOIMC and AuIOIMC over MIMO GLC. Note that the PI tuning parameters of MIMO GLC (i.e., KPi and τIi) were obtained using the corresponding values of τci. It can be seen from the Appendix that the performance of MIMO GLC with IMC-tuned PI controllers is equivalent to those of IMC. The effects of tuning parameters γi and ki on the evaporator control are given in the next section. In this paper, the outputs are plotted in terms of percentage deviations from their respective set points, whereas the inputs are expressed in terms of percentage of their respective operating ranges. The closed-loop responses of the controlled outputs are given in Figure 4. Note that the liquor levels are shown only for the first 10 h (although the total simulation time was 70 h) to

clearly indicate the differences in the transient responses between the three nonlinear control strategies. Although not shown in Figure 4, the liquor levels approach their set points after 10 h. The calculated ITAEs of the controlled outputs are given in Table 2. It can be noted from Table 2 that AdIOIMC and AuIOIMC improved the control of all outputs. For example, the disturbance and the settling time of the liquor density under AdIOIMC and AuIOIMC, as shown in Figure 4e, are significantly reduced. The improvement in control can also be seen from the significant reduction in the ITAE of the liquor density in Table 2. The improved control on the liquor density provided by AdIOIMC was superior to that provided by AuIOIMC. However, when the liquor levels in Figure 4 are compared, the performance of AdIOIMC was worse than that of AuIOIMC. Similar results can also be seen from the calculated ITAE of the outputs in Table 2. The observed differences in the performance between the AuIOIMC and AdIOIMC were largely due to the tuning of the gains and the parameter update laws, respectively. Nevertheless, the important observation to be made is that both of the proposed strategies with arbitrary tuning parameters improved the control of the evaporator.

664


Figure 9. Responses of the adaptation parameters of AdIOIMC with various values of γi.

Oscillatory responses can clearly be seen in Figure 4, especially for the liquor levels of FT #1 and #2 (i.e., Figure 4a,b). These are not significant as the amplitudes of the oscillations are relatively small. Furthermore, the oscillations damped out quickly. The responses of the manipulated inputs of the evaporator that correspond to the controlled outputs in Figure 4 are given in Figure 5, and the responses of the liquor temperatures are given in Figure 6. It can be seen from Figure 5 that the three nonlinear controllers responded to the unmeasured disturbances rapidly by making large manipulations to the inputs initially. However, AuIOIMC and AdIOIMC required more aggressive control moves than did MIMO GLC. This can be seen from the fast settling of the inputs of AdIOIMC and AuIOIMC when compared to those of MIMO GLC. The responses of the liquor temperatures, as seen in Figure 6, are less oscillatory or faster settling under AdIOIMC and AuIOIMC than under MIMO GLC. This is clearly favorable in terms of the operation of the evaporator because of the decrease in the oscillatory responses in the state variables of the system. The effect of the tuning parameters γi in AdIOIMC on the error convergence dynamics is investigated in next section.

5.2. Robustness of AdIOIMC on the Evaporator. The study of the effects of the tuning parameters γi on the performance of AdIOIMC on the four-effect evaporator was carried out for four cases, namely, (1) γi ) 0.5, i ) 1, ..., 5; (2) γi ) 5.0, i ) 1, ..., 5; (3) γi ) 50.0, i ) 1, ..., 5; and (4) γi ) 5000, i ) 1, ..., 5. The closed-loop responses of the controlled outputs of the evaporator are given in Figure 7. Note that the responses of the outputs are only shown for the first 10 h, rather than the full simulation time (i.e., 70 h). This is to clearly indicate the differences in the transient responses for various values of γi. It can be seen from Figure 7 that the disturbance of the outputs and their settling times are significantly reduced as the values of γi get smaller. It is also interesting to note that the performances of cases 3 and 4 (i.e., γi ) 50.0, i ) 1, ..., 5 and γi ) 5000, i ) 1, ..., 5) were comparable (i.e., no significant difference in the output responses). This indicates that the parameter adaptations of AdIOIMC approached the steady-state parameter update laws for values of more than 50 for γi. The above observations also suggest that AdIOIMC with the steady-state parameter update law performed better than MIMO GLC. This can be seen from Table


Figure 10. Closed-loop responses of the controlled outputs for various values of ki for AuIOIMC.

3, where the calculated ITAE of the outputs under AdIOIMC (with γi ) 50 and 5000) are significantly lower than those of MIMO GLC. The responses of the manipulated inputs and the liquor temperatures are not given here in order to conserve space. It can be shown that their responses are satisfactory in terms of fast stabilization dynamics. The error dynamics of the predicted outputs are given in Figure 8. It can be seen that the model errors were reduced, and they converged to zero more rapidly as the tuning parameters were reduced. This indicates that the tracking dynamics of the I/O model outputs to the actual outputs were more vigorous as the tuning parameters were reduced. When the values of γi were large, there were steady-state offsets between the I/O model outputs and the actual outputs, especially for the liquor level of FT #1. The model error for the liquor level of FT #1 was more significant because of the parametric uncertainty of +2 m3/hr in Qf between models M1 and M2. The nonconvergence of the model outputs to the actual outputs at large value of the tuning parameter was consistent with the remark given after Theorem 1. The changes in the adaptation parameters of AdIOIMC with various values of γi are shown in Figure 9. It can be seen that, as γi f 0, the adaptations of Rˆ i are faster, as

evidenced by the steep responses in Figure 9. Note the difference in the ultimate responses of Rˆ i between γi ) 50, 5000 and γi ) 0.5 in Figure 9a. This is due to the existence of model error at large value of γi. 5.3. Robustness of AuIOIMC on the Evaporator. The effect of the tuning parameter ki on AuIOIMC for improving control on the evaporator was evaluated, and the results are presented in this section. The values of the ki’s used in the investigation were 1.0, 5.0, and 50.0. The closed-loop responses of the controlled outputs of the evaporator are illustrated in Figure 10. It can be seen that the regulatory control of the outputs was improved as the values of ki were increased. This improved regulatory control of the outputs can be seen from the reduced effect of the disturbances on the outputs and the fast settling time of the outputs as the values of ki increase, as shown in Figure 10. The responses of the manipulated inputs and the liquor temperatures of the evaporator (not shown here for brevity) are satisfactory for all values of ki in term of fast stabilization dynamics. 6. Conclusions Two strategies have been proposed for enhancing the control performance of IOIMC of an industrial four-

666


Figure 11. Comparison of input responses by MIMO GLC with IMC-tuned PI controllers and IOIMC.

effect evaporator. The first method is adaptive IOIMC (AdIOIMC), which is composed of the original IOIMC structure with parameter adaptation based on the plant/ model mismatch. In AdIOIMC, a SISO parameter update law is used to update the parameter of the I/O model and the model-inverse controller of IOIMC. The update law guarantees the asymptotic tracking of the I/O model output to the process output with an adjustable rate that is set by a tuning parameter. The effects of the tuning parameter on the performance of the update law were also analyzed. The second strategy is augmented IOIMC (AuIOIMC), which is composed of an IOIMC structure and an additional SISO feedback loop whose input is the plant/model mismatch. The additional feedback loop simply multiplies the plant/ model mismatch with a gain, and the result is used to adjust the output error to the model-inverse controller of IOIMC. The stability of AuIOIMC is guaranteed if the IOIMC is stable. It was also shown that the performance of AuIOIMC on uncertain systems approaches the closed-loop characteristics of IOIMC in the case of no plant/model mismatch if the gain of the additional feedback loop is chosen to be sufficiently large. The proposed AdIOIMC and AuIOIMC structures are computationally simple because of the analytical

adaptation law and simple additional gain feedback loop, respectively. Simulation results for AuIOIMC and AdIOIMC on the evaporator were provided. Comparative studies on rejecting unmeasured disturbances on the evaporator indicate that AuIOIMC and AdIOIMC performed significantly better than MIMO GLC (with IMC-tuned PI controllers) in terms of disturbance reductions and fast settling time. Furthermore, both proposed strategies also provided faster stabilization dynamics for the liquor temperatures of the evaporator than MIMO GLC, providing better operation of the evaporator in terms of less oscillatory responses of the state variables of the process. Both strategies were shown to reduce the effect of plant/model mismatches. Therefore, both strategies are justifiable in terms of delivering better control performance. Appendix The calculated ITAEs of the controlled outputs under MIMO GLC with IMC-tuned PI controllers and the IOIMC are given in Table 4, and the input responses are shown in Figure 11. It can be seen from Table 4 and Figure 11, respectively, that the outputs and inputs

Ind. Eng. Chem. Res., Vol. 40, No. 2, 2001 667 Table 4. Calculated ITAEs of the Outputs under MIMO GLC with IMC-Tuned PI Controllers and IOIMC outputs h1 h2 h3 h4 F4

MIMO GLC

IOIMC

0.0475 0.0511 0.0035 0.0122 6.0019

0.0486 0.0523 0.0035 0.0125 6.0510

under the two nonlinear controllers are equivalent. As such, comparison of AdIOIMC and AuIOIMC to MIMO GLC with IMC-tuned PI controllers is equivalent to comparison to IOIMC. Literature Cited (1) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NY, 1989. (2) Garcia, C. E.; Morari, M. Internal Model Control: A Unifying Review and Some New Results. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 308. (3) Fisher, D. G. Process Control: An Overview and Personal Perspective. Can. J. Chem. Eng. 1991, 69, 5. (4) Henson, M. A.; Seborg, D. E. An Internal Model Contol Strategy for Nonlinear Systems. AIChE J. 1991, 37, 1065. (5) Kravaris, C.; Kantor, C. J. Geometric Methods for Nonlinear Process Control: Parts 1 and 2. Ind. Eng. Chem. Res. 1990, 29, 2295. (6) Hu, Q. P.; Rangaiah, G. P. Strategies for Enhancing Nonlinear Internal Model Control of pH Processes. J. Chem. Eng. Jpn. 1999, 32, 59. (7) Kravaris, C.; Chung, C. B. Nonlinear State Feedback Synthesis by Global Input/Output Linearization. AIChE J. 1987, 33, 529. (8) Kravaris, C.; Soroush, M. Synthesis of Multivariable Nonlinear Controllers by Input/Output Linearization. AIChE J. 1990, 36, 249. (9) Klatt, K. U.; Engell, S. Nonlinear Control of Neutralization Processes by Gain-Scheduling Trajectory Control. Ind. Eng. Chem. Res. 1996, 35, 3511. (10) Kurtz, M. J.; Henson, M. A. Input-Output Linearizing Control of Constrained Nonlinear Processes. J. Process Control 1997, 7, 3.

(11) Krothapally, M.; Juan, J. C.; Palanki, S. Sliding Mode Control of I/O Linearizable Systems with Uncertainty. ISA Trans. 1998, 37, 313. (12) Kam, K. M.; Tade´, M. O. Nonlinear Control of a Simulated Industrial Evaporation System Using a Feedback Linearization Technique with a State Observer. Ind. Eng. Chem. Res. 1999, 38, 2995. (13) To, L. C.; Tade, M. O.; Kraetzl, M. An Uncertainty Vector Adjustment for Process Modelling Error Compensation. J. Process Control 1998, 8, 265. (14) Wu, W.; Chou, Y. S. Robust Output Regulation for Nonlinear Chemical Processes with Unmeasurable Disturbances. AIChE J. 1995, 41, 2565. (15) Wong, Y. H.; Krishnaswamy, P. R.; Teo, W. K.; Kulkarni, B. D.; Deshpande, P. B. Experimental Application of Robust Nonlinear Control Law to pH Control. Chem. Eng. Sci. 1994, 49, 199. (16) Shukla, N. V.; Deshpande, P. B.; Ravi Kumar, V.; Kulkarni, B. D. Enhancing the Robustness of Internal-Model-Based Nonlinear pH Control. Chem. Eng. Sci. 1993, 48, 913. (17) Kosanovich, K. A.; Piovoso, M. J.; Rokhlenko, V.; Guez, A. Nonlinear Adaptive Control with Parameter Estimation of a CSTR. J. Process Control 1995, 5, 137. (18) Kam, K. M.; Tade´, M. O. An Input-Output Internal Model Control Strategy for Nonlinear Process Control. In Proceedings of CHEMECA ’99; Newcastle, Australia, 1999 (CD-ROM). (19) Isidori, A. Nonlinear Control Systems: An Introduction; Springer-Verlag: New York, 1995. (20) Henson, M. A.; Seborg, D. E. A Critique of Differential Geometric Control Strategies for Process Control. J. Process Control 1991, 1, 122. (21) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley & Sons: New York, 1989. (22) Tatiraju, S.; Soroush, M. Parameter Estimator Design with Application to a Chemical Reactor. Ind. Eng. Chem. Res. 1998, 37, 455. (23) Kam, K. M.; Tade´, M. O. Simulated Control Studies of FiveEffect Evaporator Models. Comput. Chem. Eng. 2000, 23, 1795.

Received for review February 8, 2000 Revised manuscript received July 24, 2000 Accepted October 13, 2000 IE000205G

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