Polyoptimal Multiperiodic Control of Complex Systems with Inventory Couplings Via the Ideal Point Evolutionary Algorithm

The paper is devoted to the polyoptimal control of a complex system with inventory couplings, which transfer the by-products of some subsystems to other subsystems as their input components or energy carriers. The cooperation of the subsystems on the recycling may enhance desired ecological features of complex production processes reducing the waste stream endangering the natural environment. The consideration of such systems is connected with the tendency of the rearrangement of complex industrial production systems from an open loop form with many waste products to a closed loop form guaranteeing their beneficial utilization (Ignatenko et al., 2007; Salmiaton & Garforth, 2007; Tan et al., 2008; Tatara et al., 2007; Yi & Luyben, 1996). The networks of interconnected chemical or biochemical reactors can be mentioned as the examples of systems discussed (Diaconescu et al., 2002; Russo et al., 2006; Smith & Waltman, 1995). The recycling problem is analyzed for various operation modes of the networks. Because of the flexible couplings the subsystems have high autonomy degree and can be operated in their own mode. In particular the following three nested operation kinds of the flexibly coupled network can be distinguished (Skowron & Styczeń, 2009): the steady state process (the low intensity production process), the periodic process (the increased intensity production process with the same operation period for all subsystems), and the multiperiodic process (the high intensity production process with different operation periods for the subsystems adjusted to their particular dynamic properties).


Introduction
The paper is devoted to the polyoptimal control of a complex system with inventory couplings, which transfer the by-products of some subsystems to other subsystems as their input components or energy carriers.The cooperation of the subsystems on the recycling may enhance desired ecological features of complex production processes reducing the waste stream endangering the natural environment.The consideration of such systems is connected with the tendency of the rearrangement of complex industrial production systems from an open loop form with many waste products to a closed loop form guaranteeing their beneficial utilization (Ignatenko et al., 2007;Salmiaton & Garforth, 2007;Tan et al., 2008;Tatara et al., 2007;Yi & Luyben, 1996).The networks of interconnected chemical or biochemical reactors can be mentioned as the examples of systems discussed (Diaconescu et al., 2002;Russo et al., 2006;Smith & Waltman, 1995).The recycling problem is analyzed for various operation modes of the networks.Because of the flexible couplings the subsystems have high autonomy degree and can be operated in their own mode.In particular the following three nested operation kinds of the flexibly coupled network can be distinguished (Skowron & Stycze ń, 2009): the steady state process (the low intensity production process), the periodic process (the increased intensity production process with the same operation period for all subsystems), and the multiperiodic process (the high intensity production process with different operation periods for the subsystems adjusted to their particular dynamic properties).
Since each of the subsystems has its own objective function composed of the product value, the recycled loop cost, and the waste neutralization cost, the polyoptimal (multiobjective) formulation of the control problem for the complex dynamic network comes to mind.The importance of the search of a compromise solution for a set of conflicting objectives has been widely emphasized in the literature (Huang & Yang, 2001;Sanchis et al., 2008;Sawaragi et al., 1985;Zitzler & Thiele, 1999).The extension of the admissible control processes may yield an essential improvement of the optimal objective function.To this end the periodic dynamic processes of the subsystems (the cycles) are represented by the finite-dimensional vectors encompassing their periods, their initial states, their local controls, and their inventory interactions.The three nested control problems are considered, namely the polyoptimal steady-state control problem, the polyoptimal periodic control problem, and the polyoptimal multiperiodic control problem.The evolutionary optimization algorithm is proposed, which finds the approximations of the polyoptimal ideal point for steady-state, periodic, and multiperiodic processes.The proper dominance for such three polyoptimal points in the objective function space is analyzed.The algorithm is generalized to the case of the improper dominance of the approximated nested ideal points.It uses increased aspiration levels of the objective functions for suitably chosen subsystems.
The application of the evolutionary algorithm to the nested polyoptimization has the advantage of the searching for a globally optimal solutions on each nested stage of the system operation.The easiness of the incorporation of various side constraints including the stability conditions for the optimized control process should also be emphasized (Skowron & Stycze ń, 2006).On the other hand the algorithm proposed is time consuming.It deals with dynamic interconnected processes, it evaluates the objective function of the complex recycled systems implementing the globalized Gauss-Newton method for the finding of periodic control processes of the subsystems, and it reconstructs both the averaged control constraints as well as the interaction constraints.This complicates its application for advanced polyoptimal approaches requiring a broad scanning of the Pareto set, the niching technique or the nondominated sorting technique (Audet et al., 2008;Sarkar & Modak, 2005;Tarafder et al., 2005;2007;Zhang & Li, 2007).In this context the ideal point method shows the advantage of the moderate extent of the computations necessary for the determination of a nested polyoptimal solution.
The theoretical and algorithmic developments are illustrated by the illustrative example of the nested polyoptimization of production processes performed in systems of cross-recycled chemical reactors.

Polyoptimal multiperiodic control problem for recycled systems
Consider the following polyoptimal multiperiodic control (POMC) problem for systems composed of N subsystems with the inventory couplings (IC): minimize the vector objective function composed of the τ i -averaged objective functions of the particular subsystems and subject for i = 1, 2, ..., N to the τ i -periodic state equations of the subsystems to the resource constraints to the stability constraints to the box constraints and to the averaged inventory interaction constraints where τ i ∈ R + is the operation period of the i-th subsystem, is its averaged level of the resource availability, and is the control process of the IC system.The objective functions G i (z i ) for the particular systems are combined from such quantities as, for example, the averaged yield of the major products and the by-products, the averaged selectivity of the production process, and the averaged energy consumption or its dissipation.
The dynamics of the subsystems is governed by the τ i -periodic state equations (3), the periods of which can be chosen independently for each of the subsystems according to their dynamic properties.This is guaranteed by the flexible inventory couplings between the subsystems, which enable to stock up on some output products of the subsystems to recycle them in a complex production system.The inequalities (7) restrict the averaged outflows of the inventory couplings by their averaged inflows.
The constraints (4) mirror the averaged availability of the resources used in the process operation.The relationships (5) are responsible for the local asymptotic stability of periodic control processes for particular subsystems.The constraints for the local stability F-levels are important to ensure practical applicability of optimized processes.
To depict the ideal point evolutionary algorithm implementable for the POMC problem we apply the time scaling t := τ i t independently to each subsystem.We reduce this way the IC system to the computationally convenient unit time interval [0, 1].We convert the continuous-time control of the i-th subsystem u i (t) and its inventory interaction v i (t) to the discrete-time form ũk i and ṽk We assume that the normalized nonlinear state equation of each subsystem has the uniquely determined solution x i (t, τ i , x i (0), u i , v i ) for every optimization argument the resolvable variables x i (t, z i ) found by high accuracy integration procedures for nonlinear differential equations with the given initial state and the input functions.
We convert this way the POMC problem to the following normalized and disretized form: minimize the vector objective function composed of the normalized objective functions of the subsystems and subject for i = 1, 2, ..., N to the normalized process periodicity constraints to the normalized and discretized resource constraints to the normalized stability constraints to the normalized box constraints and to the normalized and discretized inventory interaction constraints where an additional dense time grid {t l } L l=1 is used in ( 14) to approximate sufficiently exactly the state constraints within the normalized control horizon, and is the discrete representation of a controlled cycle of the i-th subsystem encompassing its period, its initial state, its discretized control, and its discretized inventory interaction, while is the normalized discretized control process of the IC system.
Let Z be the set of all the admissible solutions of the POMC problem, i.e. the set of all the multiperiodic cycles z satisfying the constraints ( 11)-( 15).We determine the ideal point values of the objective functions of the particular subsystems for the multiperiodic operation of the IC system: We define the compromise multiperiodic solution z * for the POMC problem as the solution minimizing the distance to the ideal point where the distance is defined with the help of the uniform norm The multiperiodic control process of the IC system may ensure high productivity of particular subsystems and it may be characterized as the high intensity production process with different operation periods for the subsystems adjusted to their particular dynamic properties.On the other hand its implementation is connected with increased requirements for inventory capacities and their maintenance.It may also have low stability margins for the periodic state trajectories, which involve the need of the design of high quality stabilizing loops for the subsystems.
For such reasons we consider the nested control processes "sitting inside" the multiperiodic control process i.e. the periodic control process and the static control process.
The periodic control process may be interpreted as the synchronized operation mode of the IC system.It requires moderate inventory capacities and facilitates the balancing of the inventory interactions.
The POMC problem is converted to the polyoptimal periodic control (POPC) problem by the setting τ i = τ (i = 1, ..., N).Such a choice of the operation periods may be convenient for the balancing of the inventory interactions.It reduces, however, the set of admissible solutions to the set Z of all the periodic cycles z satisfying the constraints ( 11)-( 15) with the equal periods τ i = τ.We determine the ideal point G * .
=( G * 1 , G * 2 , ..., G * N ) in the objective space of the POPC problem by the computation of N optimal values of the objectives functions connected with the τ-periodic operation of particular subsystems: We define the compromise periodic solution z * for the POPC problem as the solution minimizing the uniform distance to the ideal point Fixing in time all the process variables leads to the simplified system with direct interconnections and without inventories.The steady-state control processes may be implemented with the help of simple stabilization loops, for example, of relay type.However, such processes ignore the optimization potential underlying in the process dynamics.
The POPC problem is converted to the polyoptimal steady-state control (POSS) problem by the fixing in time all the process variables, which is equivalent to the minimization of the vector steady-state objective function and subject for i = 1, ..., N to the steady-state constraints where zi .
= xi , ūi , vi is the steady-state control process of the i-th subsystem, and is the steady-state control process for the IC system.
Let Z be the set of all the admissible solutions of the POSS problem, i.e. the set of all the steady-state processes z satisfying the constraints ( 18)-( 22).We determine the ideal point Ḡ * .=( Ḡ * 1 , Ḡ * 2 , ..., Ḡ * N ) in the objective space of the POSS problem by the computation of N optimal values of the objectives functions connected with the steady-state operation of particular subsystems: Ḡ * i = min z∈ Z Ḡi ( z)( i = 1, 2, ..., N).We define the compromise steady-state solution z * for the POSS problem as the solution minimizing the distance to the ideal point Definition 1: The triple of compromise nested control processes z * .
=( z * , z * , z * ) is said to be • strongly proper if it satisfies the relationships • partially strongly proper if it satisfies the relationships • proper if it satisfies the relationships • weakly proper if it satisfies the relationships Industrial Waste www.intechopen.com • improper if it satisfies the relationship where the vector inequality means that the inequality ≤ holds for all the components with the strict inequality for some of them, and ⋚ means that higher level compromise solutions may improve objective functions for some subsystems at polyoptimal static solution, but deteriorate for other subsystems at this solution.
We are aimed at the comparison of the ideal point compromise solutions of the POMC problem for the steady-state processes, for the periodic processes, and for the multiperiodic processes.The finding of the strongly proper nested triple z * means the uniform improvement of the ideal point compromise solutions between all the levels of nested optimization problem.
It may be the basis for the application of the compromise multiperiodic control process.
The other types of the nested triple z * determine weaker possibilities of the polyoptimal nested optimization of the IC system.The practitioner choosing a definitive process for the implementation takes into account the degree of the improvement of the objective functions for the subsystems between the nested compromise solutions.

Ideal point evolutionary polyoptimal multiperiodic optimization
The general scheme of the ideal point evolutionary algorithm for the nested polyopimal multiperiodic optimization can be stated as follows : Algorithm 1: Finding of the ideal point compromise solution for the POMC problem.
Step 1: Choose randomly an initial steady-state control process population z0 .
Step 3: Choose randomly an initial periodic control process population z0 =(τ 0 , x0 i , ũ0 i , ṽ0 Step 4: Choose randomly an initial multiperiodic control process population z 0 =( τ 0 , x 0 i , u 0 i , v 0 i ) N i=1 and apply the evolutionary global optimization (EGO) algorithm to solve N single objective multiperiodic optimization problems in the objective functions space of the POMC problem.Apply the EGO algorithm to find the ideal point compromise solution for the POMC problem Step 5: Determine the properness of the determined nested triple z * on the basis of the Definition 1.
If the nested triple z * turns out to be improper the following regularization may improve its properness.
Algorithm 2: Combined the ideal point compromise solution and aspiration levels approach for the POMC problem.
Step 1: Modify the set of admissible solutions for the POPC problem as follows: where N⊂{ 1, 2, ..., N} is the set of indices of the subsystems, the objective functions of which are deteriorated by the ideal point compromise solution of the POPC problem at the point z * , and the corrections ∆i > 0 determine the aspiration levels for the subsystems with deteriorated objective functions on the periodic optimization level.
Step 2: Apply the EGO algorithm to find the combined ideal point compromise and aspiration levels solution z * =(τ * , x * i , ũ * i , ṽ * i ) N i=1 for the POPC problem as Step 3: Modify the set of admissible solutions for the POMC problem as follows: where Ñ⊂{ 1, 2, ..., N} is the set of indices of the subsystems, the objective functions of which are deteriorated by the ideal point compromise solution of the POMC problem at the point z * , and the corrections ∆i > 0 determine the aspiration levels for the subsystems with deteriorated objective functions on the multiperiodic optimization level.
Step 4: Apply the EGO algorithm to find the combined ideal point compromise and aspiration levels solution for the POMC problem as Of course it may suffice to solve one of the corrected problems POPC or POMC.
to the box constraints to the stability constraints and to the inventory interaction constraints where the reactions obey the power law with the exponents p ij .The optimization goal is equivalent to the maximization of the averaged yield of the useful product for each of the reactors.We compare the nested polyoptimal steady-state, periodic, and multiperiodic control processes for such cross-recycled reactors.
The evaluation of the initial advantageous duration of the operation periods for the subsystems can be found with the help of the π-test.Assuming the unit mean value of the inventory interactions and the subsystem parameters n i = 3, m i = 2, p i1 = 2, p i2 = 1, p 13 = 1.5, κ 11 = 40, κ 12 = 12, κ 13 = 10, κ 21 = 30, κ 22 = 15, κ 23 = 2 we obtain the π-curves for the subsystems (Fig. 1) with the suboptimal operation periods τ 1 = 2.5, τ 2 = 3.1.The form of the π-curves shows that optimal operation periods for the case considered should be searched within the intermediate frequencies.1.The values of the objective functions obtained with the help of evolutionary algorithm Table 1 shows the results which were achieved with the aid of the evolutionary algorithm described Skowron and Stycze ń (Skowron & Stycze ń, 2009).It is easy to notice that applying the periodic control for the considered system of two continuous stirred tank reactors cooperated with the help of the inventory interactions significantly improves the productivity of the system in comparison to the steady-state approach.Comparing ideal points for POSS and POPC problems we can see that productivity of the first system is improved about 116%.Much greater improvement is observed for the second system.Applying the periodic control improves that the productivity of the second about 366%.
The results confirm also that efficiency of the system can be increased by applying multiperiodic control.The ideal point of the first system is improved about 1.2% and for the second system we see improvement equal 0.009%.The improvement after applying the multiperiodic control is not such spectacular like for the case when the steady-state control is replaced by the periodic control.But for some systems improvement of the efficiency of the process about 1-2% can give very huge economical profits.
Encouraged by observed improvement after applying multiperiodic control we calculated also the compromise solution (Table 1, Fig. 2-5).For the first system the improvement is about 25% and for the second system is about 0.46%.We see that received compromise solution for POMC problem is strongly proper according Definition 1. Thus these results confirm that for the considered system of two continuous stirred tank reactors cooperated with the help of the inventory interactions it is wise to apply multiperiodic approach.

Conclusion
The polyoptimal multiperiodic control problem for complex systems with the inventory couplings was analysed.The ideal point evolutionary algorithm was proposed for the solving of this problem.It has been shown that the multiperiodic operation of the complex cross-recycled chemical production systems may ensure the uniform improvement of the vector objective function as compared with the steady-state operation, and with the periodic operation.Such polyoptimal solution may be preferred by practitioners.The method applied shows the advantage of the moderate extent of the computational effort necessary for the finding of a best compromise solution.The solution obtained this way may be further exploited as the starting point for the implementation of some improved nested multiobjective optimization based, for example, on the verification of the attainability of the given aspiration levels for particular objective functions.
in the objective space of the POMC problem by the computation of N optimal 216 Industrial Waste www.intechopen.com N ) in the objective functions space of the POSS problem.Apply the EGO algorithm to find the ideal point compromise solution for the POSS problem z * = arg min z∈ Z |G( z) − Ḡ * | ∞ .

Fig. 1 .
Fig. 1.The results of π-test (z i ) endowed with the norm |s i | ∞ .= max j=1,2,...,n i |s ij |, and α i ∈ R + is the local stability F-level of the i-th subsystem, and satisfying the constraints (6).Thus we can treat the states of the subsystems as 215 Polyoptimal Multiperiodic Control of Complex Systems with Inventory Couplings Via the Ideal Point Evolutionary Algorithm www.intechopen.com Fig. 2. The optimal control ũ * ij (t) and the inventory interaction ṽ * i (t)(i, j = 1, 2) for POPC problem 223 Polyoptimal Multiperiodic Control of Complex Systems with Inventory Couplings Via the Ideal Point Evolutionary Algorithm www.intechopen.com