Formalization of target invariants and designing an adaptive control system for the model of anaerobic biological wastewater treatment

A model of a two-step anaerobic fermentation in a bioreactor-mixer, whose mathematical description is a system of nonlinear mathematical equations, is discussed. The purpose is to design an algorithm of an energy-saving control relying on the principles of nonlinear adaptation on target manifolds with an attractive property, which are referred to as invariants or invariable laws of the target system behavior for the object under study. A necessary preliminary step is the formalization of possible target invariants in the form of prescribed (desired) laws of the controlled object behavior. An example is given of designing a vector regulator with a compensation of random disturbances over the control channel in a system of anaerobic biological waste water treatment.


Introduction
Several effective design and engineering solutions for the systems of anaerobic purification, based on an in-depth analysis of the features of anaerobic fermentation processes and mathematical modeling of the bio-methanogenesis, were advanced in [1][2][3][4][5][6].
More recently (based on [1][2][3][4]), an international IWA Anaerobic Digestion Task Group proposed a generalized model of the anaerobic fermentation kinetics ADM-1 [5,6], which today represents a tool providing the most complete and detailed description of the principal stages of conversion of organic pollutants and intermediates of biodegradation.
The purpose of this research is to formalize the invariants (laws of behavior of a weakly formalized object in a steady state) for the system of anaerobic biological wastewater treatment from the point of view of the possibilities of the synergetic control theory (SCT) [7], which implements the principles of physical control theory [8].

Formulation of the problem of setting invariants for organizing an adaptive control design for anaerobic biological wastewater treatment
Let the initial substance of volume V be fed into a bioreactor at rate Q(t), concentration of nutrient organic substances Sin(t) and biomass Bin(t), and let an equivalent amount of liquid be removed (see anaerobic fermentation model, ADM1). is the value of the parameter at the normal temperature of the biomass.

Figure 1.
Flowchart of conversion of organic wastewater pollutants in an anaerobic bioreactor.
Consider a system of equations of the anaerobic biological wastewater treatment dynamics for a twostage process (figure 1) of anaerobic fermentation in a bioreactor-mixer acquiring the following form [1][2][3][4][5][6][7]   The examples of anaerobic biological wastewater treatment invariants and the algorithm of nonlinear adaptation presented below demonstrate a possibility in principle of an analytical synthesis of an adaptive regulator with a compensation of disturbances, without being regarded as a final result.

Target invariants for anaerobic biological wastewater treatment
1. The main purpose of a bioreactor functioning in the wastewater purification system is to reduce the concentration of the initial organic pollution Sin to (or lower than) a specified guideline value S * . In so doing, it is desirable to convert the pollutants ad maximum to a biogas. If G * is the design amount of biogas, which could be generated from the set amount of the initial substance, then the formal model of the control target will be given by 2. The stability-indicating parameter of the process of anaerobic fermentation is the content of organic acids in the reaction medium, which are intermediate products of the process. The limiting concentration of organic acids ** , ( ) P P t P = can be determined for the anaerobic biomass, at which inhibition of the purification process starts. Therefore, the respective control targets will be as follows: 3. Anaerobic bioreactors, in addition to the systems of wastewater treatment, are used in biogas production units. Then the control target will be given by the following: 4. An increased amount of gas does not always improve the energy efficiency of the system due to higher energy consumption on heating the reactor.
Supplement the anaerobic biological wastewater treatment model with an energy efficiency index where EG is the amount of energy that can be potentially produced from biogas, Eheating is the energy spent on heating the bioreactor, determined in the general case from the thermal system equation. Then, the target invariant for the waste water treatment system will be given by and for the biogas plants is reasonable to use it in the form

Fundamental concepts and definitions of SCT
Fundamental concepts and definitions related to the algorithm of nonlinear adaptation on a target manifold used here (based on the formalism of system's invariants and SCT) are following.  The imaging point of the system in its state space is set by the values of the state vector coordinates at fixed time t.
Sets VR n are referred to as invariant sets with respect to the flow xf(t, x0, t0), if xf(t,x0,t0)V for any x0V for all t >t0, where xf:(x0,t0)→(x, t)V.
Set V is referred to as attractive, if it is a closed and invariant set.
Macrovariables are certain (user)-defined functions ( )  x of the object coordinates, whose equality determines the target (desirable) set of states.
A classical formulation of the problem of control on a target manifold includes 1) control object, set by a system of ODEs or a system of difference equations for the continuous and the discrete problems of control, respectively; 2) control aim in the form of an analytically formulated equation ( ) =0 The mathematical apparatus for the method of analytical design of aggregated regulators is based on the results of theoretical mechanics (e.g., [10]), in particular if the variational problem ( )   (7) is transparent: it reaches the target of control at a given velocity determined by the parameter w.

Implementation algorithm of target invariants for anaerobic biological wastewater treatment
Consider case (2), introducing the notations For the sake of readability of the algorithm for synthesizing control, let us denote the initial description (1) as ; , , , , , , , , , , .
Step 1. Phase space extension aimed at transforming the initial system into a closed system in order to compensate for disturbances; the description will be obtained in the following form: , , , , , , , , , , , const 0, 1, 2 From now on, system E S is referred to as the initial system for synthesizing a vector regulator.

Results of numerical simulation of the object control
Let us consider the example of application of the control design technique presented in Section 5 for a given target invariant ( * GG → ) by means of a temperature control set  in the bioreactor.
In this case, the control system will take the form      Thus, the constructed control for such a nonlinear multidimensional multi-connected object provides robust properties of the target system that operates under uncertainty. The numerical solution of algebraic equations (16) that arise in the process of regulator synthesis introduces an additional disturbance due to the existence of unstable limit states of the object (1).

Conclusion
The formulation of the problem of control, based on the description (1) and the reasonable invariants (2)-(7), deserves a separate consideration and a study of the issues related to each of them consisting in the following: • mathematical conditions of a possible linear synthesis a correctness of the physical values of all variables of the target system; • necessary and sufficient conditions of an existence of a single or multiple solution forms for the control variables; • a physical interpretation of the local functionals accompanying the regulator synthesis and a practical implementation of the resulting control systems.
Finally, a quality solution of the tasks of data assurance of the control over a complex object anaerobic biological wastewater treatment needs the information on the complete state vector of this object. In the object studied in this work not all coordinates are available for an immediate change; for part of the coordinates in real systems this is either technically difficult to implement or physical impossible, since the rate and quality of biochemical reactions are affected by multiple factors, not all of which are included into the model. Therefore, the presence of unmodelled dynamics not only is possible but can also play an undesirable role resulting in an unstable state of anaerobic biological wastewater treatment.