**9. References**

200 Recurrent Neural Networks and Soft Computing

Fig. 25. Three dimensional plot of the I-term optimal control results of the plant output X1 in four measurement points of L-M learning : z=0.2H, z=0.4H, z=0.6H, z=0.8H

The given on Fig. 23-25 graphical results of I-term optimal control showed smooth

The paper proposed a new neural identification and control methodology for distributed parameter bioprocess plant. The simplification of the DPS given by PDEs is realized using the orthogonal collocation method in four collocation points, converting the PDE plant description in ODE one. The system is identified using RTNN model and BP and L-M learning, where a high precision of convergence is achieved (the final MSE% for both BP and L-M learning algorithms is of order of E-4 in the worse case). The comparative results showed a slight priority in precision and convergence of the L-M over the BP which could be seen in Figures 8, 11, and Tables 2, 3 (the worse case MSE for the L-M RTNN learning is 2.5476E-4 vs. BP RTNN learning which is 2.8282E-4). The obtained comparative simulation results of centralized adaptive direct, indirect SM and optimal control with I-term exhibited a good RTNN convergence and precise reference tracking. The MSE% of plant outputs tracking for the three considered methods of control is of order of E-5 in the worse case. The graphical simulation results showed that all control methods with I-term could compensate constant plant input noises and the I-term removal caused a system outputs deviation from the reference signals (see Fig. 21). The MSE study ordered the control methods used as: optimal, direct, and indirect, but the difference between them is little (see Tables 4.5.6 where worse case final MSE for DANC is 1.7568E-5; for SMC is 2.1347E-5; for the optimal control it

exponential behavior, fast convergence and the removal of the constant noise terms.

**8. Conclusion** 

is 1.4949E-5).


**10** 

*Russia* 

Mikhail S. Tarkov

**Optimization of Mapping Graphs of Parallel** 

**Systems by Recurrent Neural Network** 

*A.V. Rzhanov's Institute of Semiconductor Physics Siberian Branch, Russian Academy of Sciences* 

**Programs onto Graphs of Distributed Computer** 

A distributed computer system (CS) is a set of elementary computers (ECs) connected by a network that is program-controlled from these computers. Each EC includes a computing module (CM) (processor with a memory) and a system unit (message router). The message router operates under CM control and has input and output ports connected to the output and input ports of the neighboring ECs, correspondingly. The CS structure is described by the graph *G (V ,E ) sss* , where *Vs* is the set of ECs and *E =V V s ss* is the set of connections

The topology of a distributed system may undergo changes while the system is operating, due to failures or repairs of communication links, as well as due to addition or removal of ECs (Bertsekas, Tsitsiklis, 1989). The CS robustness means that failures and recoveries of the ECs bring only to increasing and decreasing time of a task execution. Control on resources and tasks in the robust distributed CS suggested solution of the following problems (Tarkov, 2003, 2005): the CS optimal decomposition to connected subsystems; mapping parallel program structures onto the subsystem structures; static and dynamic balancing computation load among CMs of the computer system (subsystem); static and dynamic message routing (implementation of paths for data transfer), i.e. balancing communication load in the CS network; distribution of program and data copies for organization of fault tolerant computations; subsystem reconfiguration and redistribution of computation and

As a rule, all these problems are considered as combinatorial optimization problems (Korte & Vygen, 2006), solved by centralized implementation of some permutations on data structures distributed on elementary computers of the CS. The centralized approach to the problem solution suggests gathering data in some (central) EC, solving optimization problem in this EC with the following scattering results to all ECs of the system (subsystem). As a result we have sequential (and correspondingly slow) method for the problem solution with great overhead for gathering and scattering data. Now a decentralized approach is significantly developed for solution of problems of control resources and tasks in computer

communication load for computation recovery from failures, and so on.

**1. Introduction** 

between the ECs.

