Correlation of Process Nonlinearity with Closed-Loop Disturbance

Given tunings optimized for low-frequency output disturbance rejection, a bioreactor .... for tuning parameters can be, at most, three-dimen- sional, ...
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Ind. Eng. Chem. Res. 2003, 42, 4611-4619

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Correlation of Process Nonlinearity with Closed-Loop Disturbance Rejection Nicholas Hernjak† and Francis J. Doyle, III*,‡ Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716, and Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106

In this work, correlations between control-relevant nonlinearity and the achievable performance of a variety of control structures are investigated for two processes. Each process’ degree of open-loop nonlinearity is assessed using a numerical nonlinearity measure and then compared to the performance results for a set of controllers of varying complexity in disturbance rejection. The results for an experimental quadruple-tank process indicate an extremely low degree of control-relevant nonlinearity, agreeing with the open-loop nonlinearity assessment and the mild performance specification. Given tunings optimized for low-frequency output disturbance rejection, a bioreactor model demonstrates a mild degree of control-relevant nonlinearity when output disturbances are considered and a relatively high degree of control-relevant nonlinearity for input disturbance rejection. The bioreactor thus demonstrates the need to assess properly control-relevant nonlinearity in terms of desired performance and to consider the effects of controller tuning on the nonlinearity of the closed-loop operators. Introduction One framework for the characterization of chemical processes1 advocates that systems with similar controlrelevant characteristics should be controlled effectively using similar strategies. Of the three characteristics considered in the framework (dynamics, interactions, and nonlinearity), the degree of nonlinearity remains an active area of research. Early nonlinearity characterization work was concerned with studying the linearizing effect of feedback2 and practical, on-line techniques for assessing nonlinearity.3 A number of additional measures of open-loop nonlinearity have been proposed, each with its own focus, including differences in steadystate gains,1 norm-based comparisons to best linear approximations,4 geometric measures of steady-state map curvature,5 norm-based measures using a novel inner product definition,6 and the extension of an existing measure to the analysis of nonstationary processes.7 Differences have been drawn between the degrees of open-loop and control-relevant nonlinearity of systems using the optimal control structure (OCS).8 Controlrelevant nonlinearity was shown to be a function of the inherent system dynamics, the region of operation, and the performance objective. Results using the OCS showed that the severity of the open-loop versus controlrelevant nonlinearity might not always be the same, thus indicating that some systems that appear highly nonlinear can be optimally controlled using linear control techniques and vice versa. Therefore, in terms of process characterization for control, it is the controlrelevant nonlinearity that must be assessed. The OCS was employed further9 to identify classes of nonlinear * To whom correspondence should be addressed. Tel.: 805893-8133. Fax: 805-893-4731. E-mail: doyle@engineering. ucsb.edu. † University of Delaware. ‡ University of California, Santa Barbara.

behavior that are relevant to the design of a controller. Techniques for assessing control-relevant nonlinearity utilizing the inverse of the steady-state map5 and linear internal model control10 have also been investigated. The degree of control-relevant nonlinearity places limitations on the level of performance that can be obtained from a given control scheme. For example, a process with a high degree of control-relevant nonlinearity would not be expected to be controlled effectively using a simple linear technique. Furthermore, one must be specific in the definition of performance being considered. Classical linear control theory, for example, carefully differentiates between regulatory and servo performance. It is easily shown that designing a controller on a linear process with the objective of optimizing regulatory response will result in suboptimal servo performance because of the following relationship (for SISO systems)11

where G(s) is the process transfer function, Gc(s) is the controller transfer function, S(s) is the closed-loop sensitivity transfer function, and T(s) is the closed-loop complementary sensitivity transfer function. Equation 1 shows that designing Gc(s) to improve the frequency characteristics of S(s) or T(s) will necessarily result in a loss of desirable dynamic features in the other transfer function. For nonlinear systems, the relationship between servo and regulatory performance is not as clear. In developing general closed-loop stability results, nonlinear systems-theoretic approaches treat reference inputs and disturbances together as exogenous inputs, without exploiting the inherent differences caused by the point of entry to the closed loop.12 It is the objective of this work to investigate the relationship between control-relevant nonlinearity and

10.1021/ie020910k CCC: $25.00 © 2003 American Chemical Society Published on Web 06/17/2003

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the obtainable level of performance for a number of common control algorithms. Control-relevant nonlinearity is assessed by examining controller performance on two processes: an experimental quadruple-tank system and a bioreactor model. A set of controllers of increasing complexity is applied to each of the processes, with each controller tuned to obtain the best possible quantitative performance for a single disturbance. Performance is then further assessed for larger output and/or input disturbances under the same tunings. The two processes are also assessed for degree of open-loop nonlinearity using a numerical nonlinearity measure. The results are inspected for correlations between the degree of open-loop nonlinearity and the degree of complexity of the controller that yields the largest performance improvement over the controller prior to it in the spectrum. As the nonlinearity measure can be computed as a function frequency, correlations between the frequency characteristics of the closed-loop operators and the system nonlinearity are also considered. In the following sections, the nonlinearity measure and its computational techniques are introduced, a summary of the various control algorithms is provided, and an overview of the results is presented.

where A and Ω are the sets of input signal amplitudes s and frequencies, respectively, being considered. χU N is thus defined as the lower bound (LB) on eq 2 and usually lies within 10-15% of the best value obtained by using the optimization method discussed above.

Nonlinearity Measure

Control Algorithms

The numerical nonlinearity measure proposed by Allgo¨wer and co-workers4,7 is used for nonlinearity characterization

The controllers investigated in this work span a range of complexity. The lower end of the complexity spectrum includes proportional (P), proportional-integral (PI), and proportional-integral-derivative (PID) controllers. The standard digital realization of these controllers is utilized

φU N ) inf sup G∈G u∈U

||G[u] - N[u]||Py ||N[u]||Py

(2)

where N: U f Y is the system operator and G: U f Y is a linear approximation to N. U is the space of considered input signals, Y is the space of admissible output signals, and G is the space of linear operators. φU N is a number between 0 and 1, where a value of 0 indicates the existence of a linear approximation to the system whose output matches the output of the original system over the set of inputs being considered. A value close to 1 indicates a highly nonlinear system. As relationship 2 represents an infinite-dimensional optimization problem, approximate computational techniques must be utilized to compute the measure. A general computational technique involves selecting a representative set of inputs and then building a linear approximation composed of a weighted sum of linear basis functions, e.g. Nl

y(s) ) G[u(s)] ) wou(s) +

wi

u(s) ∑ i)1 τ s + 1

(3)

i

where wi represents the weights on the basis functions, τi represents the functions’ time constants, and Nl is the number of basis functions chosen. An optimization routine is then employed to find the set of wi to complete the infimum operation across the considered input set. It has been shown4 that the search for the optimal set of wi is convex. A less rigorous but more computationally efficient lower bound on eq 2 can be obtained by limiting the space of admissible inputs to sinusoids of varying amplitude and frequency. Provided that the nonlinear system in question preserves periodicity, the output

after any transients have decayed can be represented by a Fourier series ∞

ys ) Ao +

∑ Ak sin(kωt + φk)

(4)

k)1

By choosing the norm

||y(t)|| ) lim

Tf∞

xT1∫ y (t) dt T 2

(5)

0

it can be shown that eq 2 becomes4

s χU N

) sup

1-

a∈A,ω∈Ω

{

x

A12(ω,a)

∆t

c(k) ) Kc e(k) +

k

2Ao2(ω,a) +

∑ Ak2(ω,a)

k)1

τd

∑e(i) +∆t[e(k) - e(k - 1)] τ i)1 i

(6)

∞

}

(7)

where e(k) is the set-point error, c(k) is the controller output, Kc is the proportional gain, τi is the integral time, τd is the derivative time, and ∆t is the sampling time. Arguably, the next most complex variety of controller is linear internal model control (IMC).13 The models used in the IMC designs are obtained from linearizations of the full system models. Note that linear IMC designs can be reduced to equivalent PID controllers in either standard form or augmented with filters for low-order process models. The difference in the design approaches is manifested in the tuning procedures that are necessitated. For PID designs, the search for tuning parameters can be, at most, three-dimensional, whereas the IMC tuning procedure typically involves a one-dimensional search for the filter time constant. The upper end of the complexity spectrum includes both linear and nonlinear model predictive control (MPC). MPC algorithms include a process model and use optimization routines to generate control moves that minimize a selected performance objective (see Prett and Garcıa14 for an overview of MPC algorithms). The specific linear MPC algorithm used was standard dynamic matrix control (DMC).15 The DMC objective function is given as p

min { ∆u(k)

Γy[yˆ (k + i|k) - r(i)]2 + Γu[∆u(k)]2} ∑ i)1

(8)

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where p is the length of the prediction horizon; r(i) is the reference signal; Γy and Γu are the error and move suppression weights, respectively; yˆ is the predicted output; and ∆u is the change in the input over a sampling time. For this work, the step response models used by the DMC algorithm to generate the predictions are obtained from the same linear approximations as used in the IMC algorithms. The nonlinear MPC (NMPC) algorithm differs from DMC in that the full nonlinear system equations are utilized to generate predictions (see Allgo¨wer et al.16 for an overview of NMPC algorithms). The NMPC algorithm uses the same quadratic performance objective (eq 8) as DMC. The nonlinear optimization is performed using MATLAB’s unconstrained optimization algorithm fminunc. At each time step, the full nonlinear system model is integrated to yield predictions of the output. These output trajectories along with the corresponding inputs are used to compute the argument of eq 8. The fminunc algorithm is programmed to perform a parametrized search for the optimal input sequence. A key difference between the MPC algorithms and the other controllers investigated is the allowance in eq 8 for penalties on input moves. This direct penalty on excessive input moves is a benefit of the MPC algorithm, as a similar objective can be realized in PID control and IMC only through a complex translation of the respective tuning parameters. Although commonly used in practice, this feature of MPC is not exploited in this work because the performance objective for the control trials is minimization of integral squared tracking error, which leads to a selection of Γu ) 0. All of the controllers, except for NMPC, are based on output feedback. The NMPC controller utilizes full state feedback to obtain the initial conditions necessary to integrate the model state equations accurately. In doing so, it is assumed that all states are available for feedback to avoid the use of estimators, which would introduce additional degrees of freedom into the design procedure. All of the control algorithms are unconstrained, and the experiments are performed in regions away from physical constraints. Therefore, all of the algorithms, except for NMPC, are truly linear. This choice is made so that the only nonlinearity affecting the results is that of the process and, for NMPC, the process model. The tunings for the controllers are obtained by varying the appropriate controller tuning parameters to minimize the integral squared error (ISE) result for one small disturbance. ISE results are then collected for a variety of other disturbances of different magnitudes and sign under the same tunings. Following common practice14 to avoid excessive numbers of computations, the MPC algorithms use prediction horizons of 30 coefficients and control move horizons of 10. Note that, by fixing the prediction horizons and by assuming that the length of the prediction horizon should be equal to the time to approximately 99% of steady state (based on the linearized model), the controller step size is set. All of the controllers are run with the same time step size to ensure that each controller makes the same number of moves during the trial time. Trial lengths, unless otherwise noted, are taken to be equal to the length of the MPC prediction horizons. System Analysis Two systems are analyzed in this paper: an experimental quadruple-tank apparatus and a bioreactor

Figure 1. Schematic of the quadruple-tank process.

model. These systems are discussed in detail in the following subsections. Quadruple-Tank System. The first system studied is an experimental quadruple-tank apparatus.17 A schematic of the process is given in Figure 1. The control objective of this system is to regulate the liquid levels in tanks 1 and 2 by manipulating the speeds of pumps 1 and 2. The multivariable and dynamic characteristics of the system can be varied by adjusting the fraction (γi) of fluid that enters the bottom tanks directly. The nonlinear state equations for the system are given by

a3 γ1k1 dh1 a1 v ) - x2gh1 + x2gh3 + dt A1 A1 A1 1 dh2 a2 a4 γ2k2 ) - x2gh2 + x2gh4 + v dt A2 A2 A2 2 a3 (1 - γ2)k2 dh3 ) - x2gh3 v2 dt A3 A3 (1 - γ1)k1 dh4 a4 v1 ) - x2gh4 dt A4 A4

(9)

where hi is the liquid level in the ith tank, vi is the speed (as a percentage of the maximum) of pump i, and the remaining parameters are system constants. In obtaining the system linearization, parameters values are taken from a previously identified model17 but with the drainage areas (ai) reduced to 2.0 cm2 to account for recent modifications to the system. To focus attention on the nonlinear characteristics of the system and away from its multivariable characteristics, the control objective is limited to the single-input/ single-output control problem of regulating the level in tank 2 using pump 2. The disturbance in question is a step increase in the speed of pump 1 that affects the level in tank 2 through tank 4. Note that all results in this section are the averages of three trials to compensate for any spurious results due to differing levels of system noise. The sampling rates for this system are taken to be 20 s. The nonlinearity of the experimental system is measured around the steady state of v1 ) v2 ) 55% using eq 6. Pump 2 is set to provide input sinusoids of amplitudes between 5 and 20% at frequencies ranging

4614 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 1. PID Tuning Parameters for the Quadruple-Tank System P PI PID

Kc

τi (s)

τd (s)

32.0 12.6 9.12

46.5 22.2

2.5

from 0.1 to 4.5 rad/min. Data are collected when the system reaches an oscillatory steady state and the Fourier coefficients are determined using a fast Fourier transform algorithm. The obtained value of the LB is 0.03, indicating an essentially linear system. Because the system nonlinearity is mainly the effect of the square roots, the low nonlinearity result is expected. Controller tunings are determined from trial-anderror experiments based on minimizing the ISE for a step increase in the speed of pump 1 of 5%. In the case of the PID controllers, the P-only controller is tuned first, and the obtained value of the proportional gain is used as the starting point in the search for the PI parameters. Likewise, the values from the PI trials are used as the starting point in locating the PID parameters. Table 1 lists the obtained PID tuning parameters. Both the IMC and DMC algorithms use the following model of the system obtained by linearization of the system model in eq 9 with the constant disturbance v1

G(s) )

0.1331 57.05s + 1

Figure 2. ISE results for the quadruple-tank system given step changes in the speed of pump 1. Each data point is the average of three trials.

(10)

where the gain has units of cm/% and the time constant has units of s. For the IMC design, a first-order filter is implemented with time constant λ. Tuning is performed by a trial-and-error search to find the λ value that yields the smallest ISE for the +5% change in the speed of pump 1, resulting in the choice of λ ) 13 s. It is found for DMC that the tuning that minimizes the ISE corresponds to very large Γy values relative to Γu, essentially corresponding to the limiting case of Γy/Γuf ∞, as further increases in Γy result in no improvement. This result is expected because taking Γu ) 0 in eq 8 reduces the performance criterion to minimization of the ISE. NMPC is not investigated for this system because of the long computational times required by the particular algorithm used in this work. Unpublished NMPC simulation results indicate that NMPC performs, at best, only marginally better than the corresponding DMC controller. As can be noted in Figure 2, for the PID and IMC algorithms, there is a general trend of decreasing ISE with increasing controller complexity for all of the investigated disturbances. Note that, because the IMC algorithm uses a first-order model and filter, it is equivalent to a PI algorithm with tuning values of Kc ) 32.9 and τi ) 57.1, suggesting that the PI parameters in Table 1 found by trial-and-error searching are not optimal. In this case, the DMC algorithm performs similarly to the PID and IMC algorithms but does not exceed their performance as one might expect given the optimization capabilities of DMC. The reason for this is likely the method by which the DMC algorithm handles disturbances. At a given time step, DMC estimates the value of the output disturbance and assumes it to be constant across the prediction horizon. In the case of a simple step disturbance in the output, this assumption is accurate. For this set of trials, the effect of the step increase in the speed of pump 1 is filtered through tank

Figure 3. Performance measure results for the quadruple-tank system given step changes in the speed of pump 1. η computed from averaged ISE measurements.

4 before affecting tank 2. Therefore, the disturbance entering tank 2 is more closely first-order, leading to suboptimal disturbance predictions. This inherent limitation in the DMC algorithm has been discussed previously18 and can be addressed by using more accurate disturbance models. By defining a performance measure, η, as

η)1-

ISE ISEOL

(11)

where ISEOL is the value of the ISE calculated for a given disturbance under open-loop conditions for the same length of time as used in the closed-loop trials, one obtains a useful measure of the trends in performance for the individual controllers across the set of disturbances. Note that a value of η ) 1 indicates complete rejection of the disturbance, whereas η ) 0 indicates that the controller is not aiding in rejecting the disturbance. In this work, the theoretical discontinuity in η given a disturbance of magnitude zero is ignored. As Figure 3 shows, all of the controllers yield η values greater than 0.96, indicating generally excel-

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4615

lent performance. Note that, when comparing η between positive and negative disturbances for a given controller, the slightly higher value for the positive disturbance is likely due to positive changes in the pump 1 level yielding smaller overall changes in the tank 2 height than negative changes as a result of the system nonlinearity in both tanks 2 and 4. With regard to the degree of system nonlinearity, the performance assessment results show agreement with the results of the open-loop nonlinearity assessment. Because the best performance is yielded by the IMC algorithm, it is shown that the linear approximation of the system performs very well in the closed-loop architecture. It is evident that the optimization capabilities of the DMC algorithm are limited by the inability to handle the disturbance accurately, but note that the differences among all of the controllers are very slight. Therefore, the control-relevant nonlinearity of the system is very low. The results in this subsection demonstrate, first, the inherent limitation of an MPC algorithm in handling unmeasured input disturbances. As will be seen in the bioreactor results, for an unmeasured step output disturbance, MPC algorithms will outperform any of the classical control techniques because of the optimization capabilities of these algorithms. Second, these results provide a basis of comparison for a system, such as the bioreactor discussed in the next subsection, that has much higher control-relevant nonlinearity. Bioreactor. The second system analyzed is a continuous version of a fed-batch bioreactor model used to simulate the fermentation of whey lactose to lactic acid by Lactobacillum bulgaricus19

Figure 4. Step responses for the continuous bioreactor around the steady-state operating condition of u ) 0.08 L/h.

ux1 dx1 ) µ(x)x1 dt V µ(x)x1 dx2 u ) (Sf - x2) dt V YX/S Figure 5. Bioreactor open-loop nonlinearity as a function of frequency as characterized by the LB for 0.02 e u(t) e 0.14 L/h.

ux3 dx3 ) [Rµ(x) + β]x1 dt V µ(x) )

µm(1 - x3/Pm)x2 Km + x2 + x22/Ki

(12)

where x1 is the biomass concentration, x2 is the substrate concentration, x3 is the product concentration (output), and u is the substrate feed rate to the reactor. The model steady-state conditions are chosen from a point along an optimal trajectory found by use of differential flatness techniques20 and are reported, along with the model parameters, in Table 2. As will be seen, the time scales for this system are on the order of hundreds of hours, which is a rather impractical magnitude. In the original fed-batch design, a typical batch length was roughly 12 h, thus justifying that method of operation. The continuous version is explored purely for clarity of results. As can be seen in Figure 4, step responses for the bioreactor indicate that the steady-state behavior in the operating region in question is essentially linear while the dynamic behavior demonstrates significant nonlinearity. These observations are confirmed by considering the LB nonlinearity measure for the bioreactor as a function of frequency (where eq 6 is computed at each frequency individually), as shown in Figure 5. At low

Table 2. Parameter and Steady-State Values for the Bioreactor Model symbol

state/parameters

value

u x1 x2 x3 V Sf Km Ki Pm YX/S R β µm

substrate feed rate biomass concentration substrate concentration product concentration volume substrate feed concentration substrate saturation constant substrate inhibition constant product inhibition constant cell mass yield growth-assoc. prod. yield nongrowth-assoc. prod. yield maximum specific growth rate

0.08 L/h 1.80 g/L 10.5 g/L 48.7 g/L 9.92 L 15.0 g/L 1.20 g/L 22.0 g/L 50.0 g/L 0.40 g/g 2.20 g/g 0.20 1/h 0.48 1/h

frequency, corresponding to the steady-state behavior of the system, the nonlinearity is quite low. It is in the mid-frequency range that the nonlinearity passes through a maximum of approximately 0.34. At 0.34, this system is considered mildly open-loop nonlinear. For the control studies, step output disturbances are first considered. The controllers are tuned to minimize the ISE for a +0.10 g/L step change in output concentration. The PID tunings can be found in Table 3. The IMC and DMC algorithms include the following linear-

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Figure 6. Performance measure results for the bioreactor given step disturbances in the outlet concentration.

Figure 7. Performance measure results for the bioreactor given sinusoidal disturbances of frequency 0.004 rad/h in the outlet concentration.

Table 3. PID Tuning Parameters for the Bioreactor P PI PID

Kc

τi (h)

τd (h)

-0.0127 -0.007 52 -0.006 31

19.20 8.23

22.02

ized model of the system

G(s) )

-15.25(136.2s + 1) 423.5s2 + 13.21s + 1

(13)

where the steady-state gain has units of (g/L)/(L/h) and the time parameters have units of h. This transfer function represents an underdamped (damping coefficient ) 0.32) second-order system with a strong lefthalf-plane zero. Note that a pole-zero cancellation occurs in the linearization that reduces the transfer function to second order from the expected third order. The selected IMC filter was first-order with a time constant of λ ) 11.71 h, resulting in a controller equivalent to a PID algorithm preceded by a first-order filter. For both DMC and NMPC, the best tuning choice is found to be the limiting case of Γy/Γu f ∞, with a prediction horizon of 30 and a move horizon of 10. All controllers are operated at a sampling time of 13 h, yielding a trial length of 390 h. As can be seen in Figure 6, there is a clear trend of increasing performance with increasing controller complexity given step output disturbances. Because the disturbances were perfect steps, the MPC algorithms had exact predictions of the disturbances and did not experience the limitation seen in the quadruple-tank results and likewise achieved the highest performance values. Judging from these results, it can be concluded that the optimization capabilities of the MPC algorithms provided no significant advantage to these controllers as compared to IMC. Also, for NMPC, utilization of the full nonlinear model and full state feedback did not provide any advantage to the algorithm. The results suggest that, for a step output disturbance, the system demonstrates very low control-relevant nonlinearity. These results are in agreement with the open-loop nonlinearity assessment in Figure 5 that shows nonlinearity tending to 0 as frequency is decreased. To further assess the degree to which open-loop nonlinearity affects controller performance given output

Figure 8. Performance measure results for the bioreactor given sinusoidal disturbances of frequency 0.022 rad/h in the outlet concentration.

disturbances, sinusoidal output disturbances with frequencies of 0.004 and 0.022 rad/h are applied to the system. According to Figure 5, these disturbances correspond to frequencies of low and mild open-loop nonlinearity, respectively. Because these disturbances correspond to persistent excitations, the trial lengths in these cases are selected to be long enough to correspond to a point where η asymptotes to a constant value. The results in Figure 7 show that, as expected, the 0.004 rad/h output disturbance does not significantly excite the process nonlinearity. The relative rankings of the controllers first observed for the step output disturbances remain, and note also that most of the η trends remain linear across the range of amplitudes investigated. Aside from possibly altering the rankings, one would expect that significant nonlinearity would result in curvature of the performance trends as the disturbance amplitude is increased. The sharp drop in DMC performance near an amplitude of 2.0 g/L is a result of the controller approaching input constraints. Figure 8 displays the performance results given sinusoidal disturbances at a frequency of 0.022 rad/h. These results show a change in the relative ranking of

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4617

Figure 9. Magnitude of the output disturbance closed-loop transfer function for the linearized bioreactor system.

the controllers, with PID performing best and IMC performing worst. To understand these results, one must first consider the performance predicted by the linearized process model before attributing performance loss to process nonlinearity. Therefore, consider the closed-loop sensitivity transfer function (as shown in eq 1) for the linearized system. A plot of the magnitude of S(s) as a function of frequency is seen in Figure 9. In terms of relative degree of attenuation, Figure 9 shows predictions of the step disturbance and 0.004 rad/h results for the rational controllers. One can observe that, near 0.022 rad/h, the magnitude trends in Figure 9 intersect, resulting in IMC attenuating the least in parts of that frequency range. The results do not suggest the large difference in performance shown in Figure 8, implying that nonlinear effects are contributing. The presence of nonlinearity is further suggested by considering the results of the two predictive controllers. For a linear system, NMPC and DMC should give similar results, as observed in Figures 6 and 7. Furthermore, it is known that, in certain unconstrained realizations, DMC is equivalent to IMC.21 Therefore, Figures 6 and 7 further suggest mainly linear operation, as DMC and NMPC generally match both IMC and each other in terms of performance. For the 0.022 rad/h disturbance, DMC and NMPC perform better than IMC, and NMPC shows a performance benefit compared to DMC. These results suggest that the additional robustness characteristics of DMC relative to IMC give the algorithm a performance edge, which is then improved upon by NMPC because it contains the correct process model and can exactly compensate for the nonlinearity. To summarize the discussion so far for the bioreactor, the controller performance results for output disturbances have been shown to be predictable based on analysis of the linearized closed-loop transfer function and a frequency-dependent analysis of the open-loop nonlinearity. In general, the assessment of mild nonlinearity by the open-loop nonlinearity measure has been confirmed by only mild performance improvements by the NMPC and DMC algorithms. The next logical step is to consider how the controllers perform for an input disturbance under the same tunings. Figure 10 contains the η performance results for step input disturbances of varying magnitudes that still remain within the range of inputs used to charac-

Figure 10. Performance measure results for the bioreactor given step disturbances in the substrate feed rate.

Figure 11. Magnitude of the input disturbance closed-loop transfer function for the linearized bioreactor system.

terize the open-loop nonlinearity. The performance trends in this case show significant curvature and changes in relative ranking. The PI and NMPC algorithms yield negative performance results for the negative inputs indicating performance worse than openloop. The failure of the NMPC algorithm in this case can be attributed to the difference between the real and assumed (by the controller) inputs and the resulting difference in the location in the state space between the actual process and the model included in the controller. A similar cause is likely responsible for the DMC giving worse performance than IMC. In an attempt to explain as many of the results as possible using linear methods, consider the closed-loop transfer function for input disturbances

D(s) )

G(s) 1 + G(s) Gc(s)

(14)

The plot of the magnitude of this transfer function is found in Figure 11. As the step disturbance corresponds to the limiting case of a low-frequency input, it can be seen that the results are not predicted accurately by the linear transfer function except for the largest positive

4618 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

Figure 12. Degree of nonlinearity of the closed-loop output disturbance operator for the bioreactor system.

Connections to the performance results can be found by examining the nonlinearity magnitudes of the closedloop operators corresponding to the various frequencies investigated. For example, Figure 8 displays poor performance for IMC given the 0.022 rad/h output disturbance. Figure 12 shows that the closed-loop output disturbance operator for IMC is significantly nonlinear in that frequency region. For the input disturbances, which IMC handled easily, the plot in Figure 12 shows that the closed-loop input disturbance operator is essentially linear across all of the low frequencies. The results in this section suggest that the performance limitations encountered by controllers designed for a nonlinear process can be separated into two classes: those related to classical linear performance and those related to the process nonlinearity. It was shown that, although an assessment of open-loop nonlinearity is useful in guiding the design of controllers, open-loop nonlinearity is insufficient in predicting all of the potential closed-loop performance losses. Conclusions

Figure 13. Degree of nonlinearity of the closed-loop input disturbance operator for the bioreactor system.

disturbance. For example, Figure 11 does not predict the poor performance of PI for the majority of disturbances. Recall that, in tuning the controllers, optimal performance was implicity desired for low-frequency output disturbances. As is common knowledge in the linear control literature, tuning for output disturbances can result in a trade-off with performance for input disturbances in terms of dynamic behavior. A similar effect is evident in these results, but in this case, design and tuning for one disturbance is apparently affecting the relative nonlinearity of the closed-loop operators. To demonstrate this result, the LB (eq 6) was applied to the closed-loop systems at both the input and the output positions to characterize the nonlinearity of the closed-loop operators. The frequency-dependent results are shown in Figures 12 and 13 for the rational controllers. The results show that, as the level of complexity of the controllers (designed for output disturbance rejection) increases, the nonlinearity of the closed-loop output operator generally increases, whereas the nonlinearity of the input disturbance operator decreases. Note also that the frequencies at which the maximum nonlinearities occur do not necessarily correspond to the frequency of maximum open-loop nonlinearity.

Through analysis of two process systems, the relationship between control-relevant nonlinearity and controller performance was investigated. For the mildly nonlinear quadruple-tank system, performance assessment for disturbance rejection indicated that all of the controllers performed very well. Generally, the results suggest low control-relevant nonlinearity for this system, as would be expected given the low open-loop nonlinearity and the standard performance objective of minimizing the ISE. Analysis of the bioreactor yielded results that demonstrate the links between control-relevant nonlinearity and performance objectives and the potential differences between open-loop and control-relevant nonlinearity. For the output disturbance rejection problem, the control-relevant nonlinearity appeared generally to be low at most frequencies, with the DMC, NMPC, and IMC algorithms all performing similarly and with performance of the low-complexity controllers being predicted by the frequency characteristics of the linearized closed-loop operators. At frequencies near the system’s reciprocal time constant, performance loss for the IMC and MPC algorithms was related to a combination of linear dynamic effects and process nonlinearity as predicted by the open-loop nonlinearity assessment. In the case of input disturbance rejection, the wide range of performance demonstrated by the controllers was essentially unpredicted by the linearized closedloop operators, even with the addition of the open-loop nonlinearity results. A nonlinearity assessment of the closed-loop disturbance operators suggested relationships between the nonlinearity of the operators and the performance objectives of the controllers. In general, the results demonstrate the danger of considering only the open-loop nonlinearity of a system in controller design without clearly defining the intended performance specifications. The authors’ research interests include the study and development of control-relevant metrics of nonlinearity that aid in the prediction of the types of results discussed here. The OCS has been shown to be a useful tool for this analysis, but it is currently only welldeveloped for single-state systems. Analysis of multistate systems has been performed22 but has included the use of an observer. As the choice of observer design

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Received for review November 13, 2002 Revised manuscript received April 17, 2003 Accepted April 17, 2003 IE020910K