**3. Discrete PID controller tuning using Pole Assignment technique**

Suppose conventional feedback control loop with discrete PID controller (7) and controlled system described by nominator *B*(*z*-1) and denominator *A*(*z*-1) – see Fig. 4.

Fig. 4. Feedback control loop with discrete PID controller

Then, Z – transfer function of closed control loop is

$$\frac{Y(z^{-1})}{R(z^{-1})} = \frac{B(z^{-1})Q(z^{-1})}{A(z^{-1})P(z^{-1}) + B(z^{-1})Q(z^{-1})}\tag{8}$$

Denominator of Z – transfer function (8) is the characteristic polynomial

$$D(z^{-1}) = A(z^{-1})P(z^{-1}) + B(z^{-1})Q(z^{-1})\tag{9}$$

It is well known that dynamics of the closed loop behaviour is defined by the characteristic polynomial (9). It has three tuneable variables which are PID controller parameters *q*0, *q*1, *q*2. The roots of the polynomial (9) are responsible for control dynamics and one can assign those roots (so called poles) (see Fig. 5) by suitable tuning of the parameters *q*0, *q*1, *q*2.

Thus, discrete PID controller tuning using Pole Assignment means choosing desired control dynamics (desired definition of characteristic polynomial) and subsequent computing of discrete PID controller parameters.

Let us show an example: suppose we need control dynamics defined by characteristic polynomial (10), where *d*1, *d*2, … are integers (there are many ways how to choose those parameters, one of them is introduced in the case study at the end of this contribution).

$$D(z^{-1}) = 1 + d\_1 z^{-1} + d\_2 z^{-2} + \cdots \tag{10}$$

So we have to solve Diophantine equation (11) to obtain all controller parameters.

$$1 + d\_1 z^{-1} + d\_2 z^{-2} + \cdots = A(z^{-1})P(z^{-1}) + B(z^{-1})Q(z^{-1})\tag{11}$$

If any solution exists, it provides us expected set of controller parameters. Comprehensive foundation to pole assignment technique is described in (Hunt, 1993).

#### **4. Continuous linearization using artificial neural network**

The tuning technique described in section 3 requires linear model of controlled system in form of Z – transfer function. If controlled system is highly nonlinear process, linear model has to be updated continuously with operating point shifting. Except some classical techniques of continuous linearization (Gain Scheduling, Recurrent Least Squares Method, …), there has been introduced new technique (Doležel et al., 2011), recently. It is presented in next paragraphs.
