**6. Description of the centralized optimal control with I-term**

The block-diagram of the optimal control system is given on Fig.6. It contained a recurrent neural identifier RTNN 1, and an optimal controller with entries – the reference signal R, the output of the I-term block, and the states X and parameters A, B, C, estimated by the neural identifier RTNN-1. The optimal control algorithm with I-term could be obtained extending the linearized model (51) with the model of the I-term (41). The extended model is:

Centralized Distributed Parameter Bioprocess

1,

2,

z=0.4H, c) z=0.6H, d) z=0.8H

sinusoids as:

**7.1 Simulation results of the system identification** 

Identification and I-Term Control Using Recurrent Neural Network Model 189

The RTNN-1 performed real-time neural system identification (parameters and states estimation) of 18 output plant variables, which are: 4 variables for each collocation point z=0.2H, z=0.4H, z=0.6H, z=0.8H of the fixed bed as: X1 (acidogenic bacteria), X2 (methanogenic bacteria), S1 (chemical oxygen demand) and S2 (volatile fatty acids), and the following variables in the recirculation tank: S1T (chemical oxygen demand) and S2T (volatile fatty acids). For lake of space we shall show some graphical results (see Fig. 7-9) only for the X1 variable. The topology of the RTNN-1 is (2, 20, 18), the activation functions are tanh(.) for both layers. The learning rate parameters for the L-M learning are as follows: the forgetting factor is =1, the regularization constant is ρ*=*0.001, and the initial value of the P matrix is an identity matrix with dimension 420x420. For the BP algorithm of learning the learning constants are chosen as =0, =0.4. The simulation results of RTNN-1 system identification are obtained on-line during 400 days with a step of 0.5 day in four measurement points using BP and L-M learning. The identification inputs used are combination of three

<sup>3</sup> 0.5 0.02sin 0.1sin 0.04 cos

5 8 0.5 0.1sin 0.1sin 0.1cos

*S ttt in*

Fig. 7. Graphical simulation results of the neural identification of the plant output X1 vs. RTNN output in four measurement points for the total time of L-M learning : a) z=0.2H, b)

*S tt t in*

100 100 100

100 100 100

(63)

(64)

 

Fig. 6. Block diagram of the real-time optimal control with I-term containing RTNN identifier and optimal controller

$$X\_{\varepsilon}(k+1) = A\_{\varepsilon}X\_{\varepsilon}(k) + B\_{\varepsilon}\mathcal{U}(k)\tag{59}$$

Where: Xe = [X|V] T is a state vector with dimension (L + N) and:

$$A\_c = \begin{vmatrix} A & 0 \\ -(CB)^+CA & I \end{vmatrix}, B\_c = \begin{vmatrix} B \\ -I \end{vmatrix} \tag{60}$$

The optimal I-term control is given by:

$$\mathbf{U}L(k) = -[\mathbf{B}\_e^\ \mathbf{P}\_e \mathbf{B}\_e + \mathbf{R}]^{-1} [\mathbf{B}\_e^\ \mathbf{P}\_e \mathbf{B}\_e] \mathbf{X}\_e(k) \tag{61}$$

Where the Pe is solution of the discrete Riccati equation:

$$P\_{\varepsilon}(k+1) = A\_{\varepsilon} \,^T [P\_{\varepsilon}(k) - P\_{\varepsilon}(k)B\_{\varepsilon}(B\_{\varepsilon} \,^T P\_{\varepsilon}(k)B\_{\varepsilon} + R)^{-1}B\_{\varepsilon} \,^T P\_{\varepsilon}(k)]A\_{\varepsilon} + Q \tag{62}$$

The given up optimal control is rather complicated and here it is used for purpose of comparison.
