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A complex autonomous inventory coupled system is considered. It can take, for example, the form of a network of chemical or biochemical reactors, where the inventory interactions perform the recycling of by-products between the subsystems. Because of the flexible subsystems interactions, each of them can be operated with their own periods utilizing advantageously their dynamic properties. A multifrequency second-order test generalizing the p -test for single systems is described. It can be used to decide which kind of the operation (the static one, the periodic one or the multiperiodic one) will intensify the productivity of a complex system. An illustrative example of the multiperiodic optimization of a complex chemical production system is presented.

We consider complex autonomous inventory coupled (IC) systems. Such systems can take, for example, the form of a network of chemical or biochemical networks, where the inventory interactions perform the recycling of by-products or by-streams from some subsystems to other subsystems as their input components or energy carriers [

Notation:

Consider the following optimal multiperiodic control problem for IC systems (the problem M) composed of the set

being a scalar function of the

and subject to the resource-technological constraints of the subsystems

the state equations of the subsystems

the inventory constraints

and the box constraints for the process variables

where the extended control

while

The objective function (1) represents the global benefits from the multiperiodic operation of the IC system, which are determined with the help of the

Constraining the variables

where

Assumption 1: The functions

Assumption 2: The steady states

Let

for the reduced

We convert the problem

where

The

We approximate the controls

with the coefficients

with

We write the multi-trigonometric approximation

where the mappings

with

and

Assumption 3: The number of points

Lemma 1. The

Proof. The

Let

where

We exploit the finite-dimensional optimization theory avoiding regularity conditions discussed for nonlinear programing problems in [

Lemma 2. If

Let

and let

We abbreviate the (partial) derivatives evaluated at

Assumption 4: The matrices

This assumption eliminates the onset of free, and resonance oscillations in the subsystems.

Lemma 3. The s-process satisfies the FON conditions of the problem

Proof. The problem

Thus the FON conditions of the problem

where

Let

vector

parts (

its

The contradiction of the SON conditions for the problem

Theorem 1. The problem

holds, where

or in the structural version

and

Proof. Lemma 2 shows that the finite-dimensional optimal steady-state process satisfies the FON conditions with a nonzero Lagrange multiplier without regularity conditions. Lemma 3 shows that this process satisfies also the FON conditions of the optimal multiperiodic control problem regardless if it is local minimum or not. This means that such conditions do not allow to distinguish improving multiperiodic control processes. For this reason the attention is directed to the SON conditions, which take for multiharmonic control variations especially simple form connected with the generalized

The discussed second order test has the following distinctive features: it concerns the different (possibly incommensurate) basic operation frequencies

Let two continuously stirred tank reactors be coupled by the inventory interactions. In each of them the parallel chemical reactions

being a scalar function of the averaged outputs

with

and subject for

and to the interaction constraints

Thus

The variation of the optimal steady state solution

The positive component of the second order test generated by the steady state variation

The multifrequency second order test for the discussed complex system with the inventory interactions is shown on

The second order test obtained shows the diversified advantageous operation frequencies for particular subsystems

In this note, we formulated the optimal multiperiodic control problem for inventory constrained subsystems. It is aimed at the intensification of the productivity of complex processes. We proposed a multifrequency second-order test for complex multiperiodic systems including the boundary optimal steady-state process and an arbitrary large number of harmonics used to verify its improvement by the multiperiodic operation. We generalized the method of critical directions for single periodic systems [

This work has been supported by the National Science Center under grant: 2012/07/B/ ST7/01216.

Skowron, M. and Stycze?, K. (2016) Optimal Multiperiodic Control for Inventory Coupled Systems: A Multifrequency Second-Order Test. Open Journal of Optimization, 5, 91-101. http://dx.doi.org/10.4236/ojop.2016.53011