IRIA

laboria

laboratoire de recherche
en informatique
et automatique

<figure><img src="image_1.png" /></figure>

CONTROL AND IDENTIFICATION
OF INDUSTRIAL PROCESSES

W. Fred RAMIREZ

Rapport de Recherche N° 252

Août 1977

Institut de Recherche
d'Informatique
et d'Automatique

Domaine de Voluceau
Rocquencourt
B. P. 105 78150 - Le Chesnay
France
Tél.: 954 90 20
CONTROL AND IDENTIFICATION
OF INDUSTRIAL PROCESSES

W. Fred RAMIREZ

Fulbright Research Fellow

IRIA / LABORIA

Resumé :

Le présent rapport présente de nouveaux concepts et techniques nécessaires au contrôle de
processus par mini et micro-ordinateurs. Les méthodes de contrôle feedforward sont présentées
en vue d'une implémentation sur micro-ordinateur. Le mini-ordinateur permet une implémen-
tation économique des lois du contrôle optimal pour des systèmes multivariables résultant de
l'optimisation dynamique de fonctions-objectifs exactes. On montre que le filtrage récursif est
efficace pour l'estimation des variables d'état nécessaires au contrôle optimal.

Abstract:

This report discusses new process control methodologies and concepts needed to exploit the
recent computer advances of mini and micro processes. Feedforward control strategies are
presented for micro processor implementation. Mini computers allow the economic implement-
ation of optimal control law for multivariable systems resulting from the dynamic optimizat-
ion of precise objective functions. On-line sequential filtering is shown effective for estimation
of the state variables needed for optimal control.
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With the introduction of the mini computer in the late 1940's and the micro computer in the 1970's, a wide range of new relatively cheap computer hardware has become available. Although there has been great advances in computer systems, we have not experienced the same degree of advancement in the application of new methodologies and concepts needed to utilize the computer for the control, optimization and identification of industrial processes. With today's need for material and energy conservation, it is important to develop the methodologies which can lead to improved process performance and operation.

## I. Definition of the Process Control Problem.

In general, a process control problem can be visualized as shown in Figure 1. There are four distinct process variables. There are:

1. Manipulated variables - those input variables which can be directly controlled or set

2. Disturbance Variables - those input variables which cannot be controlled.

3. Performance variables - those output variables which are used to compute or specify the performance of the process.

4. Intermediate variables - those output or state variables which do not directly enter the specification of the process performance.

The manipulated and disturbance variables are the input variables of the problem formulation. The performance and intermediate variables are the output or state variables.

<figure><img src="image_3.png" /><figcaption>Figure 1 - The Process Control Problem.</figcaption></figure>
We define the objective of process control as the adjustment of manipulated variables to achieve well-defined process objectives in spite of disturbances using measurements of performance, intermediate and disturbance variables.

Exactly how to implement this definition is the topic of process control. In the industrial world today, a keyword is productivity. By improving the quality of process control we directly increase productivity.

## II. SINGLE LOOP FEEDBACK CONTROL.

Most industrial control systems in operation today were designed based on the classical concepts of single variable, single loop feedback control. As an example, let us consider the heating of a stirred-tank as shown in Figure 2. In this system we want to control the output tank temperature, T, at a desired value $T_s$, the set point temperature. We measure the actual tank temperature and manipulate the final control element or heater duty.

<figure><img src="image_4.png" /><figcaption>Figure 2 - Feedback control of heating of a stirred-tank</figcaption></figure>

The block diagram for this control system is shown in Figure 3. It shows that there are three major dynamic elements to the feedback control system: the process model, the measurement model, and the control law.
<figure><img src="image_5.png" /><figcaption>Figure 3 - Block diagram.</figcaption></figure>

In order to design the process controller, we must first consider the process dynamics of the uncontrolled system. To develop a process model we need to perform a process simulation. A complete discussion of process simulation is given in the book Process Simulation by W.F. Ramirez (D.C. Heath and Co., 1976). To model this system we use the principle of the conservation of energy for macroscopic or lumped systems.

The undsteady state energy balance becomes :

$$ \frac{\text{Rate of accumulation of energy}}{\text{Rate of energy into the system}} = \frac{\text{Rate of energy out of the system}}{\text{Rate of energy out of the system}} $$

$$ \rho c_p V \frac{dT}{dt} = W c_p (T_i - T_r) + g - W c_p (T - T_r) \quad (1) $$

where : $\rho$ = fluid density

$c_p$ = fluid heat capacity

$V$ = volume of tank

$W$ = mass flow rate entering or leaving tank

$T_r$ = reference temperature

$T_i$ = inlet temperature

$T$ = outlet or tank temperature

$g$ = heater duty.

Standard analog controllers are either pneumatic or electronic.

Proportional, proportional-integral, and proportional-integral-derivative control algorithms are available. Proportional-integral-derivative (PID) controllers operate on the design equation,
$$B = B_{ss} + K_p (E + \frac{1}{\tau_I} \int Edt + \tau_D \frac{dE}{dt}) \quad (2)$$

where : B = controller output

$B_{ss}$ = steady state controller output

$E$ = error signal (difference between set point and actual process variable, $C_{set} - C$)

$\tau_I$ = integral or reset time

$\tau_D$ = derivative time

$K_p$ = proportional gain constant.

Typical controller tuning rules (Luyben, 1969) for determining value for the controller constants $K_p, \tau_I$ and $\tau_D$, are based on the study of the response of second order systems which can exhibit underdamped, overdamped or critically damped behavior.

### III. FEED FORWARD CONTROL

Feedback control systems are very effective in making corrective or regulator control action for systems in which load disturbances cannot be measured. These unmeasured disturbances are allowed to propagate through the system and are detected by changes in the desired output variable. Based upon these changes in the output variable, control action is formulated. Often, however, input process disturbances can be measured. When this is the case it is desirable to develop a control policy which takes advantage of this type of information. Feedforward control is the term used to describe control algorithms formulated so that changes in the manipulated variables are based upon values of measured disturbances. The main advantage of feedforward system over feedback systems is the fact that measured disturbances are not allowed to propagate through the system before control action is initiated.

The open loop transfer function for a typical process is shown in Figure 4(a) and is denoted as $G_m$. The input to this system is the manipulated variable m. The disturbance, d, is propagated through the system according to the dynamics associated with the transfer function $G_D$. The final output response, C, is the sum of the effects from the manipulated and disturbance variables.
<figure><img src="image_6.png" /><figcaption>a) Open loop system</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>b) Feedforward control system</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>c) Combined feedforward-feedback control</figcaption></figure>

**Figure 4 - Control system block diagrams**

The structure of a feedforward control scheme for this system is shown in Figure 4(b). The feedforward controller, F, is shown to be based upon the direct knowledge or measurement of the input disturbance, d. If the object of the feedforward controller is to maintain a constant output, then the desired value of C is zero. The system output is
$$C = G_p \frac{d}{m} + G_m \frac{m}{d} \quad (3)$$

Setting $C=0$ gives

$$F = \frac{m}{d} = - \frac{G_D}{G_m} \quad (4)$$

Equation 4 is therefore the design equation for the feedforward controller. Two important aspects of a feedforward controller are the fact that it requires the measurement of the process disturbance and also requires knowledge of the process dynamics. An accurate model of process dynamics must be available for feedforward control to be successful. Also, since process dynamics vary from process to process, there is no such thing as a standard feedforward controller. It must be designed especially for the system under consideration. This latter fact makes feedforward control ideal for the application of micro-computer technology. With a micro-computer one has a standard piece of hardware which can be programmed for the specific application. This is an easier and cheaper task than designing and fabricating specific analog circuits for each feedforward application.

Since most physical systems experienced both measurable and unmeasurable disturbances and also since our knowledge or process dynamics is often approximate, a combined feedforward-feedback control policy is usually implemented instead of a pure feedforward system. The block diagram for such a combined system is shown in Figure 4(c).

#### IV. SUPERVISORY COMPUTER CONTROL

The most common application of computer control is associated with supervisory control. In this mode of control, conventional plant analog feedback control systems are responsible for the actual control, however, a digital supervisory system performs a steady-state process optimization that prescribes optimal set-point values for the analog controllers. An information flow diagram for a typical supervisory control system is shown in Figure 5. Although the actual control loops have traditionally employed analog equipment using feedback concepts, it is now possible to install remote micro-
computer feedforward-feedback control loops. The advantages of remote
micro-computer installations include ease of interfacing to the main
digital supervisory computer and the improved dynamic response of
feedforward control.

<figure><img src="image_9.png" /><figcaption>Figure 5 - Supervisory computer control</figcaption></figure>

In order to perform process optimization by a supervisory computer, an
adequate steady-state process model must be developed. The engineering
cost of model development is usually a substantial portion of the total
project cost of implementing supervisory control. Optimization strategies
operating on the process model will use measured information on process
states and disturbances as well as management directives and market con-
dition. The results from the model optimization are used to adjust con-
troller set-points. Any supervisory computer control system will also
provide useful logging data and records which are helpful in forming
the basis for engineering improvements and management decisions.

V. MULTIVARIABLE AND OPTIMAL CONTROL

There are two main drawbacks or shortcomings to classical single loop
control. The first is the fact that most chemical process control pro-
blems involve multivariable systems where the control variables interact
with both system state variables as well as other control variables.
Also, there is vagueness in the specification of the exact goal or mathe-
matical definition of the dynamic control problem. In order to handle
these two drawbacks a more general theory of control must be used which
can handle multivariable problems and the dynamic optimization of precise
mathematical objective functions.

To model multivariable systems a state variable formulation of the system dynamics is used. The state of a system is defined by the set of variables, called state variables, which if known at a given time along with the input to the system can then be determined at any time. Therefore the state variables are those variables that characterize the dynamic behavior of the system under investigation. A large class of physical system of practical interest can be described by ordinary differential equations. In terms of the state variables we have

$$ \dot{\underline{x}} = \underline{f}(\underline{x}, \underline{u}, \underline{d}, t) \qquad (5) $$

where

$\underline{x}$ = state variable vector

$\underline{u}$ = control variable vector

$\underline{d}$ = disturbance variable vector

$\underline{f}$ = set of non linear time varying function

The outputs or measurements $\underline{y}$ are related to the states and the inputs by the time varying functions

$$ \underline{y} = \underline{c}(\underline{x}, \underline{u}, \underline{d}, t). \qquad (6) $$

Performance measures are used to quantify the quality of the control. The performance index is a mathematical expression which when minimized indicates that the system is performing in a desirable manner. It is important that the index be chosen so that for the system under investigation the minimization results in an implementable control law.

A general form for the performance index is

$$ J = h(\underline{x}(t_f), t_f) + \int_{t_o}^{t_f} F(\underline{x}, \underline{u}, z)dt \qquad (7) $$

This type of performance index is a function of the final state $\underline{x}(t_f)$ and the integral of the system dynamic behavior from the initial time to the final time.
In order to minimize functions such as the performance index of Equation 7, we are looking for the minimum of a functional. A functional is therefore a function which is a function of another variable (in this case time). Variational calculus gives methods for finding extremum of functionals. The classical work of Euler has been extended by Pontryagin and others to handle control and state variable constraints. Of particular importance is the work of Kalman (ASME Trans. J. of Basic Engr., 1960) who studied the linear regulator problem and showed that for the linear system

$$ \dot{\underline{x}} = \underline{A}\underline{x} + \underline{B}\underline{u}, \qquad (8) $$

the optimal control that minimizes the quadratic performance index

$$ J = \frac{1}{2} \underline{x}^T \underline{H} \underline{x} + \frac{1}{2} \int_{t_o}^{t_f} (\underline{x}^T \underline{Q} \underline{x} + \underline{U}^T \underline{R} \underline{u}) dt \qquad (9) $$

where $\underline{H}$ and $\underline{Q}$ are real symmetric positive semi-definite matrices and $\underline{R}$ is a real symmetric positive definite matrix,

$$ \underline{u} = - \underline{K}(t) \underline{x}(t) \qquad (10) $$

where $\underline{K}(t)$ is a computable time varying proportional state feedback control matrix. The computation of the state feedback control matrix via Ricciti differential equation is standard due to the availability of VASP (Variable Automatic Synthesis Program). A complete description of the VASP FORTRAN computer program is given in the NASA Technical Memo TM-X-2417 (1971).

Systemshaving measurable and unmeasurable disturbances can be formulated after algebraic transformation as a Kalman problem (Anderson, IEE, 116, 1969, Johnson, IEEE AC-13, 1968)

To date many industrial process optimal control problems have been formulated and solved mathematically. These include :

1) Dynamic Control of a Power Boiler - Anderson, IEE, 116, N° 7, 1257 (1969).
2) Optimal Operation of Batch Cooling Crystallizer - Jones, Chem. E. Sci. 29, N° 5, 1075 (1974).

3) Optimal Batch Reactor Temperature - Pommersheim and Chandra
AI Ch. E. J. 21, N° 5, 1029 (1975).

4) Real-time Control of Air Pollution - Kyan and Seinfeld, AI Ch. E.J. 19, N° 3, 579 (1973).

5) Optimal Control of Batch Distillation Comumn - Robinson, Chem. E. Sci. 25, N° 6, 921 (1970).

6) Time Optimal Control of Stirred Tank Reactor - Nyquist and Ramirez, Chem. E. Sci. 26, N° 10, 1673 (1971).

7) Optimal Control of Distillation Comumn - Hu and Ramirez, AI Ch. E.J. 18, N° 3, 479 (1972).

## VI. ON-LINE FILTERING AND IDENTIFICATION

Although many industrial optimal control problems have been formulated and solved mathematically, few have been implemented. The major difficulty in implementation is the fact that optimal control laws require information on all the state variables of the system. It is indeed a rare chemical processing system for which measurements are available for all states. Therefore, to be able to estimate the state variables from available process measurements is of great practical importance. In order to determine optimal state estimates, we use knowledge about the degree of uncertainty in the measurements and the degree of uncertainty in the process model to determine the most likely values of the state variables at any instant of time.

The general state estimation problem is described in the block diagram of Figure 6. Here we want to determine the optimal state estimates $\hat{x}$, from measurements of the system $\hat{y}$, which are corrupted by stochastic noise of random fluctuation $\hat{v}$. The system itself is perturbed by the stochastic disturbances $\hat{w}$.
<figure><img src="image_10.png" /><figcaption>Figure 6 - State estimation problem</figcaption></figure>

One form for the optimal state estimation system is that of the Kalman extended filter. The Kalman extended filter considers the nonlinear system

$$ \hat{x}_{i+1} = f_i(x_i) + \Gamma_i w_i \quad (11) $$

with measurements

$$ y_i = h_i(x_i) + v_i \quad (12) $$

where $f_i$ and $h_i$ are non-linear function. The optimal state estimates which minimize the quadratic performance index

$$ J = \frac{1}{2} \{ (x - \bar{x})^T M^{-1} (x - \bar{x}) + (y - Hx)^T R^{-1} (y - Hx) \} \quad (13) $$

where

$\bar{x}$ = estimate of state before measurement

$M$ = state covariance matrix

$R$ = measurement covariance matrix

$H$ = measurement matrix = $\frac{\partial h}{\partial x}$

are

$$ \hat{x}_i = \bar{x}_i + K_i(y_i - h_i). \quad (14) $$

Here the filter gain, $K_i$ is computed from the recursive relation (Bryson & Ho, Applied Optimal Control, 1969).
$$ K_{::i} = P_{::i} \left( \frac{\partial \dot{h}_{::i}}{\partial \dot{x}_{::i}} \right)^T R^{-1} \quad (15) $$

$$ P_{::i} = M_{::i} - M_{::i} \left( \frac{\partial \dot{h}_{::i}}{\partial \dot{x}_{::i}} \right)^T \left[ \left( \frac{\partial \dot{h}_{::i}}{\partial \dot{x}_{::i}} \right) M_{::i} \left( \frac{\partial \dot{h}_{::i}}{\partial \dot{x}_{::i}} \right)^T + R_{::i} \right]^{-1} \left( \frac{\partial \dot{h}_{::i}}{\partial \dot{x}_{::i}} \right) M_{::i} \quad (16) $$

$$ M_{i+1} = \left( \frac{\partial f_{i}}{\partial \dot{x}_{i}} \right) P_{i} \left( \frac{\partial f_{i}}{\partial \dot{x}_{i}} \right)^{T} + \Gamma_{i} Q_{i} \Gamma_{i}^{T} \quad (17) $$

The Kalman filter has been used for state estimation in order to implement the real-time time-optimal control of a stirred-tank chemical reactor (Lynch and Ramirez Al Ch. E.J. 21, N° 4, 799, 1975). A General Data Corporation Nova 1210 minicomputer with 8K words of storage was used to implement the time-optimal control and on-line sequential state estimation algorithm. Simulation and experimental results showed that the Kalman Filter worked very well in estimating the unmeasured system states and filtering the measurement noise in the system.

## VII. CONCLUSION

With the current availability of computer hardware in the form of mini and micro computer coupled with the well developed state of optimal control and estimation theory we are now starting to experience rapid improvements and changes in industrial process control. The application of modern control theory results in the implementation of control algorithms unique to specific industrial systems under study. This is due to their dependency upon specific process models and performance criterion. The importance of creative, intelligent control engineering is therefore increased. The ability to consider multivariable systems subject to precise dynamic optimization yields economic returns in terms of increased productivity as well as material and energy conservation.