Re: Process Dynamics And Control Seborg Solution Manual 3rd
Kirby Apodaca
The theory of classic control is based on the design of linear controllers for systems described by linear models or nonlinear models linearized around an operating point. However, there exist some situations where it is not recommended to use a linear controller. One of those situations arises when the magnitude of the process gain experiences a dramatic variation within the operating range of interest. In this situation, the use of a fixed linear controller can lead to a poor performance of the closed loop system and even to its loss of stability. A classical example of a chemical process where this situation happens is the pH control around the neutralization point in a continuous stirred tank. In this control problem, the titration curve - which represents the system's input-output mapping - presents a highly nonlinear behavior in response to addition of acid or base. This behavior is amplified even more if the reagents are strong acid and/or base.
The finite L2-gain can be viewed simply as the maximum amplification, for every time T > 0, on the variable z(t), measured in terms of its energy (Euclidean) norm, caused by a energy-limited external input w(t). For a linear SISO system, the finite L2-gain turns back to be the usual system open-loop gain.
Definition 2 (Nonlinear H-infinity controller validity region). The region of the state space of Eq. (1) that, subject to the nonlinear state feedback law from theorem 1, simultaneously satisfies the HJI inequality and guarantees asymptotic stability of the worst-case disturbance of Eq. (5) in closed-loop system, is referred to as the validity region corresponding to the controller of Eq. (4). Any region that is a subset of this state-space region is referred to as an estimate of the validity region.
The experimental apparatus considered in this work is a reactor (a two-liters glass reactor) and pH, temperature and flow sensors, coupled with a control and monitoring unit. The reactor operates at atmospheric pressure and environment temperature, being continuously stirred at the 500 rpm. The reactor vessel is fed with two input streams provided with peristaltic pumps. The acid stream contains 0.1 M HCl, and the basic stream 0.1 M NaOH. The amount of fluid given by the sum of these streams is removed from the reactor. As a consequence, the reactor behaves like a typical CSTR (Continuous Stirred Tank Reactor) with constant volume. In this work, the acid stream flow rate is considered as the disturbance variable and the base stream flow rate as the manipulated variable. A sketch of that control system is shown in Fig.1.
The process model used in this work considers the change of coordinate proposed by (Narayanan et al., 1998): η = [H+] - [OH-]. The relation between the variable η and the original variable (pH) is given by Eq. (7).
where VR is the reactor volume, FA is the acid stream flow rate, FB is the basic stream flow rate, CA0 is the acid concentration in the acid stream, CB0 is the base concentration in the basic stream and KW = 10-14 is the equilibrium constant of water dissociation.
Fig. 2 presents the comparison between the phenomenological model of Eqs. (7)-(8) and the experimental data. The highly nonlinear behavior of the pH system can be easily seen in this figure. The modeling error in the basic region, due to the acid characteristic of the available water used in the experiments (pH ranging from 5 to 7), can also be seen in this figure.
In order to obtain a description whose steady state of interest is the origin, the following new variables were defined: , , , and The sub-index SS denotes the steady state values for the variables. For the neutral pH considered in this work, the values of these variables are: and .
In order to simplify the shape of the controller validity region, it was assumed that β1 = β2. In addition, it was arbitrarily chosen that the maximum and minimum levels of attenuation (γ) are 1.0 and 1.5, respectively. The solution for the problem of Eq. (13) is given by:
To implement the controller of Eq. (15) in the experimental plant described in section 3, it was developed an interface to connect the plant to a remote computer. This interface was written using the Matlab/Simulink environment.
However, not only the model mismatch affects the performance in those conditions but, mainly, the input constraints (magnitude and speed saturation) also worsen the performance at high pH values. This is an experimental drawback that does not occur in the simulations. This could be explained because the control law was obtained without considering the input saturation. Thus, the control law will only have a valid association with the controller within a range where these constraints are not violated. Regarding the experimental system of this work, this valid region is constrained to pH values ranging from approximately 6 to 8.
Furthermore, because of the input saturation (in magnitude) it is not possible any kind of implementation for pH above 9.5 and below 4.5. In these regions, other more conservative control laws should be used. This is why no experimental tests for pH higher than 9 or lower than 5 were considered. A possible ad-hoc solution for this problem could be found by choosing different controller's settings for each operating condition, characterizing a gain scheduling procedure. However, this approach was not considered here because it implies loss of optimality.
This section shows the relation between the solution of the optimization problem 1, solved for a quadratic form, and the theorem 1. To make this relation clear, the optimization problem is simplified and applied to design a disturbance attenuation controller for an IA system.
Now, it is easy to see that, if , then the HJI inequality, H*(x) < 0, implies in the same state space region. As a consequence, the optimization problem 1 is greatly simplified, because it is not necessary anymore to verify the negativity of the time derivative to assure the asymptotic stability for the worst case disturbance, w*(t).
Note that the huge difference among the gain of the controller of Eq. (22) and Eq.(15) is due to the different magnitude between pH and η. Also observe that equation set (22) furnishes the amount of base to be fed inside the reactor on each sample time by the peristaltic pump. The base stream flow rate, FB(t), may be obtained by dividing VB(pH) by 5 seconds (the sample time used in the experimental coupled unit).
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