SEQUENCE OF OPERATIONS FOR THE MATHEMATICAL SYNTHESIS OF OPTIMAL CONTROL OF A LOCAL OBJECT IN CONTINUOUS CHEMICAL PRODUCTION

V. Ya. KARTASHOV, R. N. LYUBLINSKY, N. M. OSKORBIN

(Presented by the scientific and technical seminar of the Department of Automation and Telemechanics)

At the present stage of organizing production control, a decentralized automated process-control system (APCS) is often used, in which the decomposition of the control tasks of the entire technological complex leaves, for certain local subsystems, part of the tasks of controlling objects. In this case the APCS is represented as a hierarchical system that includes a system for the operational control of the complex (the central system) and \(m\) (\(m \gg 1\)) local-control subsystems, and uses a sufficiently powerful computer.

The most complex local-control subsystems are the subsystems of optimal control of individual units of the technological chain. The objects for which such subsystems can be created are characterized by a high degree of isolation within the overall technological scheme, which is often encountered in continuous chemical production. The isolation of the apparatuses in such production is also intensified by the presence of intermediate vessels for input and output products. All this makes it possible to consider the object being optimized autonomously, while taking into account the connection of this apparatus with other apparatuses of the scheme by introducing, for example, cost functions for input and output products. In addition, the central system can set a number of constraints for the given subsystem and use information about the functioning of the autonomous apparatus to control the remaining apparatuses.

Depending on the nature of the disturbance flow, local optimal-control systems may perform:

1) control based on static models;
2) control based on a combination of static and dynamic models.

In the present work, a procedure is proposed for developing, at the APCS design stage of a chemical-technological scheme, the algorithm of a local optimal-control system functioning on the basis of a combination of static and dynamic models.

In the present work it is assumed that the general performance criterion of the apparatus and the form of the cost functions for intermediate products are known. Schematically, the sequence for developing the optimal-control algorithm at the design stage is shown in Fig. 1. To determine the optimal-control algorithm, it is necessary to solve the following problems:

1) construction of the structural diagram and solution of the observability problem (3, 5);

2) the problem of object identification for the static model (6, 7, 8, 10) and for the dynamic model (12, 13, 14, 15);

Fig. 1

3) formation of the criterion of efficiency of the operation of the technological scheme and, on its basis, formation of the criterion of optimal control (4, 18);

4) static optimization (11);

5) the problem of controllability of the object (16)

6) the problem of dynamic optimization and obtaining the optimal control algorithm (17, 19).

After solving all the enumerated problems, it is necessary to test the algorithm obtained, which is achieved by modeling the given object and the control system on a computer.

Let us make several clarifications concerning the scheme presented (Fig. 1).

1. A chemical system is a set of occurring physicochemical processes and the apparatuses in which they take place. Every system has inputs and outputs; it is usually subject to disturbances, and control actions are used to compensate for them [1].

2. The object under study is conventionally divided into a number of links. In chemical-technological systems, these are usually taken to be sections that are either repeating elements of the apparatus design or differ from other sections by the type of limiting process. The problem of rationally decomposing the object into links, whose degree of detail may vary, is associated with the problem of adopting assumptions aimed at simplifying and justifying the accepted structural scheme of the object [2].

3. The observability problem consists in reconstructing the current (or initial) state of the object from known output functions; that is, let \(\overline{X}(t) = (x_1(t), \ldots, x_n(t))\) characterize the state of the object at time \(t\), and not all coordinates are available for measurement. Information on the current state of the object is contained in the vector function \(\overline{Z}(t) = (z_1(t), \ldots, z_m(t))\), \(m \leq n\), related to \(\overline{X}(t)\) by the relation

\[ \overline{Z}(t) = \overline{h}(\overline{X}(t), t) \]

It is required, from the known \(\overline{z}(t)\), to determine \(\overline{X}(t)\) at time \(t\). R. Kalman [3] posed the problem in another form: what is the minimum number of output variables \(Z(t)\) necessary to make it possible to judge \(X(t)\).

10 and 15. The problem of the adequacy of a mathematical model to the process being studied serves to determine the random (nonrandom) nature of the discrepancies between the solutions of the equations of statics and dynamics and the recorded data. Since the present work presupposes the construction of models by an experimental-analytical method [2], statistical criteria may be used to check adequacy.

6 and 12. The static equations of individual links are compiled on the basis of material and energy balances, using additional relationships specific to the given process. The analytical derivation of the dynamic equations is based on the fact that, to a first approximation, the rate of change of the output coordinate of a link is proportional to the difference between the flow rates of the incoming and outgoing streams of matter and energy, with subsequent transition to the output coordinates of interest (pressure, concentration, level, temperature) [2].

8 and 14. The compilation and analysis of the static and dynamic equations of the entire object are carried out on the basis of matching the material and energy flows between links. If a continuous phase is present in the given object, then some models with distributed parameters may be replaced by a model with lumped parameters [4] by introducing delay times into the model. The static model is used as a forecasting model in the object-optimization problem, while the dynamic model is corrective, allowing the process to be brought optimally to a specified regime [5].

1. Static optimization, in the simplest case, is one of the methods of machine-aided control, consisting in determining a new, best technological regime when the need for this is caused by a change in external conditions. Such optimization

assumes that the process is in a steady state and can be instantaneously transferred to a new state [1].

1. The controllability problem can be formulated as follows: given two vectors \(\overline{X}_0\), \(\overline{X}_1\), it is required, by means of the control \(\overline{u}(t)\), to transfer the object, under evaluation by some quality functional, from \(\overline{X}_0\) to \(\overline{X}_1\). The solution of this problem answers the question of whether there exists at least one control that transfers the system from the initial state \(\overline{X}_0\) to the final state \(\overline{X}_1\) [3].

2. The problem of the existence of optimal controls can be formulated as follows: among the admissible controls, does there exist an optimal control that would impart an extremal value to the selected criterion of optimal control.

3. Dynamic optimization consists in developing such a control algorithm that would make it possible to transfer the object optimally in time from one regime to another, and also to compensate for substantial disturbances. The control problem at each instant of time is reduced to such a choice of the set of control actions that, for some set of disturbing actions, the optimal behavior of the object is achieved in the sense of the selected criterion.

4. The criterion of optimal control is constructed from the criterion of static optimization, taking into account the dynamic properties of the control and of the object, and is reduced, with some correction, to one of the criteria used in the theory of optimal control.

The control algorithm is obtained as a result of the joint or separate solution of static and dynamic optimization, taking into account the prescribed class of constraints. The synthesis of the algorithm may be carried out analytically, or by determining certain constants within a prescribed a priori structure.

REFERENCES

1. V. V. Kafarov. Methods of Cybernetics in Chemistry and Chemical Technology. Moscow, “Khimiya,” 1968.

2. E. G. Dudnikov, V. S. Balakirev, V. N. Krivsunov, A. M. Tsirlin. Construction of Mathematical Models of Chemical-Technological Objects. Leningrad, “Khimiya,” 1970.

3. R. Kalman. On the General Theory of Control Systems, Proceedings of the 1st International Congress of IFAC, Vol. 2. Moscow, Publishing House of the Academy of Sciences of the USSR, 1961.

4. V. V. Kafarov, A. I. Boyarinov. Optimization Methods in Chemical Technology. Moscow, “Khimiya,” 1969.

5. L. Lippitt David, A. Swann Dale. Industrial process control by digital computer. ISA Trans., 4, No. 4, 1965.