**5. Description of the indirect (sliding mode) centralized recurrent neural control with I-term**

The block-diagram of the control system is given on Fig.5. It contained a recurrent neural identifier RTNN 1, and a Sliding Mode (SM) controller with entries – the reference signal R, the output error Ec, and the states X and parameters A, B, C, estimated by the neural identifier RTNN-1. The total control is a sum of the SM control and the I-term control, computed using (41).

Centralized Distributed Parameter Bioprocess

which yields:

in the SM control law (55).

Identification and I-Term Control Using Recurrent Neural Network Model 187

(rectangular system is supposed), the matrix product (CB) is nonsingular with rank M, and the plant states X(k) are smooth non- increasing functions. Now, from (51)-(54), it is easy to obtain the equivalent control capable to lead the system to the sliding surface

( ) ( ) ( 1) ( 1) ,

*U k CB CAX k R k E k i Of*

Here the added offset Of is a learnable M-dimensional constant vector which is learnt using a simple delta rule (see Haykin, 1999, for more details), where the error of the plant input is obtained backpropagating the output error through the adjoint RTNN model. An easy way for learning the offset is using the following delta rule where the input error is obtaned from

*Of k O* 1 1 () *f k Of k CB E k*

If we compare the I-term expression (41) with the Offset learning (57) we could see that they are equal which signifyed that the I-term generate a compensation offset capable to eliminate steady state errors caused by constant perturbations and discrepances in the reference tracking caused by non equal input/output variable dimensions (rectangular case systems). So introducing an I-term control it is not necessary to use an compensation offset

The SMC avoiding chattering is taken using a saturation function inside a bounded control

( ), if ( )

The proposed SMC cope with the characteristics of the wide class of plant model reduction neural control with reference model, and represents an indirect adaptive neural control,

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:

*Uk Uk U*

*eq eq*

 

( ) ; , if ( ) . ( )

*eq*

*Uk U*

*eq i*

the output error multiplying it by the same pseudoinverse matrix, as it is:

level Uo, taking into account plant uncertainties. So the SMC has the form:

0

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

*U k UU k*

*eq*

*U k*

*eq*

given by (Baruch et al., 2007a; Baruch et al., 2007b).

1

(55)

. *T T CB CB CB CB* (56)

. (57)

0

0

(58)

*i*

1

*P*

Fig. 5. Block diagram of the indirect adaptive SM control with I-term containing RTNN identifier and SM controller

The linearization of the activation functions of the local learned identification RTNN-1 model, which approximated the plant, leads to the following linear local plant model:

$$AX(k+1) = AX(k) + B\mathcal{U}(k), \mathcal{Y}(k) = \mathcal{C}X(k) \tag{51}$$

where L > M (rectangular system), is supposed. Let us define the following sliding surface with respect to the output tracking error:

$$E(k+1) = E(k+1) + \sum\_{i=1}^{p} \gamma\_i E(k - i + 1) \; ; \; \mid \; \gamma\_i \; | < 1; \tag{52}$$

where: S() is the sliding surface error function; E() is the systems local output tracking error; i are parameters of the local desired error function; P is the order of the error function. The additional inequality in (52) is a stability condition, required for the sliding surface error function. The local tracking error is defined as:

$$E(k) = R(k) \text{ - } Y(k);\tag{53}$$

where R(k) is a L-dimensional local reference vector and Y(k) is an local output vector with the same dimension. The objective of the sliding mode control systems design is to find a control action which maintains the systems error on the sliding surface assuring that the output tracking error reached zero in P steps, where P<N, which is fulfilled if:

$$S(k+1) = 0.\tag{54}$$

As the local approximation plant model (51), is controllable, observable and stable, (Baruch et al., 2004; Baruch et al., 2008), the matrix A is block-diagonal, and L>M

Fig. 5. Block diagram of the indirect adaptive SM control with I-term containing RTNN

The linearization of the activation functions of the local learned identification RTNN-1 model, which approximated the plant, leads to the following linear local plant model:

where L > M (rectangular system), is supposed. Let us define the following sliding surface

i=1 ( 1) ( 1) ( - 1) ; | | 1; *P Sk Ek Ek i i i* 

where: S() is the sliding surface error function; E() is the systems local output tracking error; i are parameters of the local desired error function; P is the order of the error function. The additional inequality in (52) is a stability condition, required for the sliding

where R(k) is a L-dimensional local reference vector and Y(k) is an local output vector with the same dimension. The objective of the sliding mode control systems design is to find a control action which maintains the systems error on the sliding surface assuring that the

As the local approximation plant model (51), is controllable, observable and stable, (Baruch et al., 2004; Baruch et al., 2008), the matrix A is block-diagonal, and L>M

output tracking error reached zero in P steps, where P<N, which is fulfilled if:

*X k AX k BU k Y k CX k* ( 1) ( ) ( ), ( ) ( ) (51)

 

*Ek Rk Yk* ( ) ( ) - ( ); (53)

*S k*( 1) 0. (54)

(52)

identifier and SM controller

with respect to the output tracking error:

surface error function. The local tracking error is defined as:

(rectangular system is supposed), the matrix product (CB) is nonsingular with rank M, and the plant states X(k) are smooth non- increasing functions. Now, from (51)-(54), it is easy to obtain the equivalent control capable to lead the system to the sliding surface which yields:

$$\mathcal{U}L\_{eq}(k) = \left(\mathcal{C}B\right)^{+} \left[ -\mathcal{C}AX(k) + R(k+1) + \sum\_{i=1}^{P} \gamma\_i E(k-i+1) \right] + \mathcal{O}f,\tag{55}$$

 1 . *T T CB CB CB CB* (56)

Here the added offset Of is a learnable M-dimensional constant vector which is learnt using a simple delta rule (see Haykin, 1999, for more details), where the error of the plant input is obtained backpropagating the output error through the adjoint RTNN model. An easy way for learning the offset is using the following delta rule where the input error is obtaned from the output error multiplying it by the same pseudoinverse matrix, as it is:

$$\text{O}\,\text{O}\,\text{f}\,\text{(}k+1\text{)}=\text{O}\,\text{f}\,\text{(}k+1\text{)}=\text{O}\,\text{f}\,\text{(}k\text{)}+\eta\text{(}\text{CB}\,\text{)}^{+}\,\text{E}\,\text{(}k\text{)}.\tag{57}$$

If we compare the I-term expression (41) with the Offset learning (57) we could see that they are equal which signifyed that the I-term generate a compensation offset capable to eliminate steady state errors caused by constant perturbations and discrepances in the reference tracking caused by non equal input/output variable dimensions (rectangular case systems). So introducing an I-term control it is not necessary to use an compensation offset in the SM control law (55).

The SMC avoiding chattering is taken using a saturation function inside a bounded control level Uo, taking into account plant uncertainties. So the SMC has the form:

$$\mathcal{U}\mathcal{U}\left(k\right) = \begin{cases} \mathcal{U}\_{eq}(k), & \text{if } \left\| \mathcal{U}\_{eq}(k) \right\| < \mathcal{U}\_0\\ -\mathcal{U}\_0 \mathcal{U}\_{eq}(k), & \text{if } \left\| \mathcal{U}\_{eq}(k) \right\| \ge \mathcal{U}\_0. \end{cases} \tag{58}$$

The proposed SMC cope with the characteristics of the wide class of plant model reduction neural control with reference model, and represents an indirect adaptive neural control, given by (Baruch et al., 2007a; Baruch et al., 2007b).
