ROBUST CONTROL OF WIENER SYSTEMS: APPLICATION TO A pH NEUTRALIZATION PROCESS

Abstract

In this paper, the robustness of a typical control scheme for Wiener systems is studied. These systems consist of the cascade connection of a linear time invariant system and a static nonlinearity. To control this kind of systems, several approaches were discussed in the literature. Most of these control schemes involve transformation of the measured variable as well as the setpoint, by the inverse of the nonlinear gain. The approach followed in this work uses the inverse model of the static nonlinear gain, while the uncertainty in the Wiener model is treated as a partitioned problem. The linear block is considered as a parameter-affine-dependent model and, on the other hand, the nonlinear block uncertainty is analyzed as a conic-sector. The robustness analysis is performed using µ-theory. The results are evaluated on the basis of a simulation of a pH neutralization process.

Keywords:
Wiener Models; Process Control; Uncertainty; Robustness

INTRODUCTION

Many contributions for controller design are based on the assumption of a linear model of the system. However, in some cases it is difficult to represent a given process using a linear model. This situation takes place, for example, when a highly nonlinear system undergoes operating point changes along a wide region. For this reason, in the last decades, there has been much interest in nonlinear model-based control within the chemical engineering community. A critical step in the application of these methods is the development of a suitable model of the process dynamics. In this sense, Sjöberg et al. (1995) describe several approaches for model development and this approach can be specially appealing as regards control process applications. In particular, the Wiener model (WM) can be mentioned (Pearson and Pottmann, 2000) due to its wide diffusion and applicability in control. The WM consists of a cascade connection of a linear time invariant (LTI) system followed by a static nonlinearity.

The use of these models has been treated in the literature in various contexts. WMs have proved to be useful for applications in several fields, such as in chemical processes (Kalafatis et al., 1995; Pajunen, 1992; Pearson and Pottmann, 2000; Zhu, 1999), biological processes (Korenberg, 1973; Hunter and Korenberg, 1986), communications (Kang et al., 1998; Kang et al., 1999; Cheong et al., 2005) and control (Norquay et al., 1998; Gerkšič et al., 2000; Lussón Cervantes et al., 2003; Biagiola et al., 2004).

A highly cited paper regarding Wiener model identification is the contribution by Gómez and Baeyens (2004), in which noniterative algorithms for the identification of both multivariable Hammerstein and Wiener systems were developed. Rational orthonormal bases were considered for the representation of the linear subsystem in the case of the Wiener model.

In most of the control applications of Wiener Models, the underlying strategy involves the inverse of the nonlinearity, such as in the works by Gerkšič et al. (2000), Bloemen et al. (2001a) and Gómez and Baeyens (2000, 2004).

Although much research effort has been dedicated to Wiener models, to the best of our knowledge, a systematic robustness analysis for this scheme under uncertainty has not been developed in the literature.

Since Wiener models could be approximations of the real process, the robustness of the designed control structure may deserve analysis. In order to apply robust control theory, one needs not only a nominal process model, but also a suitable description of the modeling errors, which are typically in the form of some bounds of parameter variations (Wang and Romagnoli, 2003). The classical robust control analysis and design methodologies are based on linear models (Doyle, 1982). As regards techniques for robust nonlinear control, they usually consist of covering the nonlinearity by an affine convex hull and, then, to perform the analysis on it (Popov, 1962; Bloemen et al., 2001b). However, a dedicated controller design and robustness analysis should be developed for processes approximated by Wiener models.

In this article, an analysis of the robustness of closed loop Wiener Systems is performed. The uncertainty in the Wiener model is treated as a partitioned problem. The linear block is considered as a parameter-affine-dependent model, which is suitable for Lyapunov-based analysis. Therefore, the stated stability problem can be dealt with as a sector bounded uncertainty problem and easily converted to a linear fractional uncertainty model. On the other hand, the nonlinear block uncertainty is analyzed as a conic-sector. The robustness analysis is performed using µ-theory.

To illustrate the proposed control strategy, a pH neutralization process is selected herein. This process has been widely recognized in the literature as a challenging problem due to the highly nonlinear and time-varying dynamic nature. From the perspective of system identification, pH processes have often been considered in the literature as having a Wiener structure (Kalafatis et al., 1995). Different Wiener representations have been favorably used for the purpose of pH neutralization process control. Among these works, we mention the paper by Kalafatis et al. (2005a), in which they evaluated the control of pH processes based on the Wiener model structure, where the static nonlinearity was assumed to represent the titration curve. Moreover, assessment of the conditions under which the pH process behaves like an exact Wiener system was accomplished. A simple linearizing feedforward controller was proposed based on an estimation of the inverse titration curve.

Along the same line, Gómez and Baeyens (2004) illustrated the suitability of the proposed methods in the identification of the pH neutralization process. They recognized this dynamic system as a benchmark drawn from the process control literature.

More research work on this subject is due to Mahmoodi et al. (2009). They employed Laguerre filters and polynomial functions as the linear and nonlinear blocks of the Wiener model, respectively. The so-called Wiener-Laguerre model was used to evaluate identification and nonlinear model predictive control of a pH neutralization process.

It must be remarked that, although the study herein developed is in the context of a pH neutralization reactor control, the conclusions can be directly extended to any other application.

The paper is organized in the following way. In the next section the process description is given. A Wiener model (and the related uncertainty) is then developed. The controller is designed and the simulation results are presented. The paper concludes with some final remarks.

PROCESS DESCRIPTION

The control of a pH neutralization processes is a relevant topic in several industries, such as wastewater treatment, pharmaceuticals production, bioprocesses plants, and chemical processing. It is often difficult to achieve a high performance and robust pH control due to their time-varying and severe nonlinear characteristics (Henson and Seborg, 1994). Therefore, pH control is frequently conceived as the unavoidable case study for the assessment of novel modeling and control strategies. Actual research work confirms that this process is still interpreted as a control benchmark (Mahmoodi et al., 2009; Wang and Zhang, 2011; Kim et al., 2012).

The process consists of the neutralization reaction between a strong acid (HA) and a strong base (BOH) in the presence of a buffer agent (BX) as described by Galán et al. (2004). The neutralization takes place in a continuous stirred tank reactor (CSTR) with a constant volume V. In this reactor, an inlet acidic solution of composition x_1i(t) and a time-varying volumetric flow q_A(t) is neutralized using an alkaline solution of volumetric flow q_B(t) and known composition made up of base x_2i and buffer agent x_3i. The nominal values for q_A and q_B are 1 and 0.5 L/min, respectively. Due to the high reaction rates of the acid-base neutralization, chemical equilibrium conditions are instantaneously achieved. Moreover, under the assumptions that the acid, the base and the buffer are strong enough, then the total dissociation of the three compounds takes place.

The process dynamic model can be obtained by considering the electroneutrality condition (which is always preserved) and through mass balances of equivalent chemical species (known as chemical invariants) that were introduced in Gustafsson and Waller (1983). For this specific case, under the previous assumptions, the dynamic behavior of the process can be described considering the state variables: x₁=[A^-], x₂=[B⁺] and x₃=[X^-]. Therefore, the mathematical model of the process can be written in the following way (Galán, 2000):

where ξ=10-pH. K_w and K_x are the dissociation constants of the buffer and the water, respectively. The parameters of the system represented by Equations (1)-(4) are addressed in Table 1. Equation (4) was deduced by McAvoy et al. (1972), and it takes the standard form of the widely used implicit expression that connects pH with the states of the process.

Table 1
Neutralization Parameters.

Parameter	Value
X 1 i	0.0012 mol HCl/l
X 2 i	0.0020 mol NaOH/l
X 3 i	0.0025 mol NaHCO3/l
Kx	10-7 mol/l
Kw	10-14 mol2/l2
qA	1 l/m
V	2.5 l

Now, making x̃= x₂-x₁, it is possible to reduce the process model to:

WIENER MODEL

Figure 1 depicts a Wiener model. It consists of a LTI system described in a state space form (this description for the linear block was used by, for example, Bruls et al. (1999), Westwick and Verhaegen (1996) and Lussón Cervantes et al. (2003)) (A,B,C) followed by a static nonlinearity N(.). That is, the linear model maps the input signal q_B(t) into the intermediate variable v(t), and the overall model output is y(t)=N(v(t)). In this scheme, the bar over q_B means deviation: i.e., q_B = q_B - q_B,s, where q_B,s stands for the steady-state value of this physical variable.

Figure 1
The Wiener model structure.

In this case, the relation between the input and the output is represented by the following model:

where the linear description of the process results from a linearization around the steady state y(t) is the deviation variable for the measured pH. Note that Equation (7) can be rewritten as the following third-order polynomial:. The output variable

Then, the resulting linear model is:

where

Note that computation of the following linear model matrix in the steady-state

involves the entries of the matrices (A,B,C) that must be evaluated for q_B=q_B,s, q_A=q_A,s, x_1i= x_1i,s, x_2i= x_2i,s, x_3i= x_3i,s, pH= pHs.

The nominal linear model is computed at q_B,s=0.5. Then, to determine the values for the static nonlinear gain, the values of q_B are varied in the range [0,1]. This also implies that the linear model will differ from the nominal one. To cope with these changes, the following proposals are considered: a) the linear model includes uncertainties in the entries of the matrices (A,B,C) and b) the nonlinear gain is characterized (i.e., no uncertainty is present in it).

The influence of q_B values on the LTI model parameters was determined and the results are shown in Figure 2.

Figure 2
Parameters of the linear model as a function of qB.

As regards the characterization of the static nonlinear gain, the titration curve is approximated by means of a Piecewise Linear (PWL) function. PWL functions have proved to be a very powerful tool in the modeling and analysis of nonlinear systems (Chua and Ying, 1983). The general formulation of PWL functions allows us to represent a continuous nonlinear function through a set of linear expressions, each of them valid in a certain operation region. To make this approximation, the range of input variables v(i.e., ℵ) is partitioned into a set of σ non-empty regions ℵ¹, such that . In each of these regions, the non-linear function is approximated using a linear (affine) representation. These functions allow the development of a systematic and accurate treatment of the approximation.

It can be proved (Julián et al., 1999) that any continuous nonlinear function N(v): ℜ^m→ℜ¹ can be uniquely represented using the PWL functions as

where β_i are given parameters that define the partition of the domain of v, and Λ are functions that involve nested absolute values (Julián et al., 1999).

Figure 3 depicts the real curve for the nonlinear gain, as well as the PWL approximation. The parameters identification of the PWL model in Eq. (10) is accomplished using the pH data and the v(t) data obtained with Eq. (8). Therefore, for this pH neutralization model description the identified parameters are: β^N=[-3, -1.5, -1.2, 0.4, 1, 3]^T, where the superscript N means that this partition is used to represent the gain N.

Figure 3
Nonlinear gain.

WIENER SYSTEM CONTROL

Wigren (1990) presented a structure for the control of Wiener systems. In this scheme (see Figure 4), two static nonlinearities are included in the loop. Under the hypothesis that N(.) is invertible, the natural selection for the controller nonlinear functions is f(.)N^-1(.). Note that this is the case of the pH neutralizer.

Figure 4
The closed loop scheme for Wiener Model.

Now, the controller design involves two steps: a) the inversion of the nonlinear gain and, b) the computation of a LTI controller in order to compensate for the linear block model of the process.

To compute f(.) we approximate it using a PWL function (Lussón Cervantes et al., 2003); Figure 5 shows this function. The partition of the f-domain is defined as β^f= [2.5, 3.8, 6.5, 8, 10]_T, which corresponds to the partition of the pH range.

As mentioned above, the linear controller K should be designed to compensate for the behavior of the linear dynamic block of the process model. This controller could be designed using any of the classical techniques found in the literature. In our case, we use the H_∞ methodology (Gahinet et al., 1995). Let us consider the reduced loop (i.e., the nonlinearity is excluded) of Figure 6. In this case, the design signals are u_c=[u], y_c= ẽ, w_c=[v_r n]^T and z_c=[e u]_T. The design transfer function is then

Figure 5
Inverse of the nonlinear gain.

Figure 6
The closed loop scheme for the linear block.

The controller (A_K, B_K, C_K) is designed to reduce the norm of the transfer function between wc and zc, then the integral action (1/_s) is included in the controller expression. Note that in this formulation we are minimizing the H_∞ norm between the set point input (_vr) and the measurement noise (_n) to the weighed integrated error (e) and the manipulated variable (_u) signals. This minimization is performed using the function hinfsyn (Gahinet et al., 1995) that implements the algorithm defined by Doyle et al. (1989). It is important to note the significance of considering the measurement noise because its existence in the complete scheme (which includes the static nonlinearity) could produce a mismatch between the PWL sector of N and the PWL sector of f.

The robustness of this controller is tested against all the uncertainty sources: the linear model parameters variation (Fig. 2), the conic sector of the nonlinear gain model (Fig. 3) and the conic sector related to the controller (Fig. 5). The system used for robustness analysis is described in Figure 7. This analysis is performed using µ-tools (Gahinet et al., 1995) with Δ=diag{ΔA, ΔB, ΔN, Δf} (the uncertainty Δf in the forward line does not affect the stability) and the following M-Δ structure:

In these expressions, the nominal values Ñ (and f̃) are computed as the average of the conic sector of Fig. 3 and Fig. 5, and the magnitude of the uncertainties depends on the variation of the entries of A, B and the respective conic sectors. Under these definitions, the closed loop becomes robustly stable (µ=0.74).

Figure 7
The closed loop scheme for robustness analysis.

Figure 8 shows the simulation results for setpoint changes. Note that the system follows the setpoint with smooth changes in the manipulated variable, even under the wide excursion of the reference signal. An interesting point is that, when we try to follow the same setpoint using only the linear controller (i.e., without considering the block f in the feedback loop), the system becomes unstable due to the significant nonlinearity of the process.

Two additional tests are performed; first we consider the effect of the measurement noise. A Gaussian noise of variance 0.05 is added to the pH output. The simulation results are shown in Figure 9. It is important to mention that, when noise variance increases, the performance of the controller deteriorates and the control system could become unstable.

Figure 8
Closed loop simulations.

Figure 9
Closed loop simulations for noisy measurement.

In what follows, the effect of perturbations is evaluated. When load changes are applied to the process, a fixed Wiener model no longer represents adequately the process. For example, the titration curve changes drastically (Kalafatis et al., 2005a; 2005b). To perform a robustness analysis in this case, we consider that q_A varies between 0.8 and 1.2 and that x_1i varies between 0.0008 and 0.0014. The effects of these changes on the LTI model parameters and the nonlinear gain are shown in Figure 10 and Figure 11, respectively. From these plots, it is clear that the model uncertainties increase. For this level of uncertainty the system is no longer robustly stable (µ=1.147).

A simulation to evaluate the influence of lower perturbations is shown in Figure 12. In this case, q_A is reduced to 0.9 at t=10 min and x_1i is increased to 0.0014 at t=30 min.

Figure 10
Parameters of the linear model as a function of q_B for different values of q_A and x_1i.

Figure 11
Nonlinear gain for different values of q_A and x_1i.

Figure 12
Closed loop simulations under perturbation.

CONCLUSIONS

In this article, controller design as well as robustness analysis of Wiener systems were considered. Wiener systems modeling and control were treated in the context of the more realistic case in which different sources of uncertainty are present. For this purpose, PWL approximating functions were introduced. As regards the modeling and control approaches proposed in this work, PWL functions proved to be an appropriate and simple tool for uncertainty inclusion, nonlinearity modeling and nonlinearity inversion.

A design technique for the controller synthesis was also proposed. It makes use of well-known tools based on H_∞ theory, and it was shown to be a suitable method for the uncertain feedback structure proposed herein. Stability aspects of the closed loop system under uncertainty were also dealt with using µ-theory. The different topics developed were tackled together in an application example of significant complexity.

• *
  To whom correspondence should be addressed
• 1
  Also with Comisión de Investigaciones Científicas de la Pcia. de Bs. As. (CIC)

ACKNOWLEDGMENTS

This work was financially supported by the Agencia Nacional de Promoción Científica y Tecnológica, CONICET, CIC and Universidad Nacional del Sur, Bahía Blanca, Argentina.

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Publication Dates

• Publication in this collection
  Jan-Mar 2016

History

• Received
  18 July 2013
• Reviewed
  06 Feb 2015
• Accepted
  21 Feb 2015