**3.5 Pseudo-equivalence of the fuzzy PD controller**

As in the case of the fuzzy PI controller, a quasi-continual form is obtained:

$$H\_{RF}(\mathbf{s}) = \frac{\mu(\mathbf{s})}{e(\mathbf{s})} = \tilde{c\_{\mu}} \left(\mathbf{c}\_{\varepsilon} + \mathbf{c}\_{dc}\mathbf{s}\right) = \frac{\tilde{c\_{\mu}}}{c\_{\varepsilon}} \left(1 + \frac{\mathbf{c}\_{dc}}{c\_{\varepsilon}}\mathbf{s}\right) \tag{48}$$

From the identification of the coefficients, the following relations of tuning result:

$$K\_{RG} = \frac{\stackrel{\sim}{c\_{\mu}}}{c\_{e}} \tag{49}$$

Tuning Fuzzy PID Controllers 185

~ *RG de D u*

(59)

*<sup>K</sup> c T c*

Fig. 13. The structure of the control system with the correction of the non-linear part *N*

1 11 1 2 2 *<sup>T</sup>* <sup>2</sup> *de de <sup>a</sup> CAN L L a c c x K cx x w h hh* 

. ~ 1 11 1 2 1

> . ~ 2

~ ~~~

~ ~ ~ ~ <sup>~</sup> ~ (, ) (;) , . 0 *<sup>N</sup> N t t t*

*x*

*f e de K x de pt x*

1 1 *T e CAN L L e e cK c x cw*

11 1 2 2 *<sup>T</sup>* 2 *de de CAN L L a c c de K c x x w h hh* 

*x A x b K x b K du L d L L L CNA a L CNA*

1 2

<sup>1</sup> *x du a d <sup>h</sup>* (60)

*xt* 1 1 *y e de* (61)

(62)

*xt* and parameter <sup>~</sup>

*de* (62).

For stability analysis, we are working with the structure from Fig. 13.

The linear part L has the input-state-output model (60).

.

~

With the new compound variable (61)

~

there may be introduced a new function of the compound variable <sup>~</sup>

**4. Stability assurance** 

**4.1 Internal stability** 

$$T\_{RG} = \frac{c\_{de}}{c\_e} \tag{50}$$

From these equations, the expressions of the scaling coefficients results:

$$\mathcal{L}\_e = \frac{\stackrel{\cdots}{c\_{\mathcal{U}}}}{\mathcal{K}\_{RG}} \tag{51}$$

$$
\mathcal{L}\_{dc} = \frac{T\_{RG}\stackrel{\sim}{c}\iota\_{\mu}}{K\_{RG}}\tag{52}
$$

#### **3.6 Pseudo-equivalence of the fuzzy PID controller**

As in the case of the fuzzy PI controller, there is obtained a quasi-continual form:

$$H\_{RF}(\mathbf{s}) = \frac{\mu(\mathbf{s})}{\mathbf{c}(\mathbf{s})} = H\_{RF}(\mathbf{z})\Big|\_{\mathbf{z}} \frac{\mathbf{1} + sh/2}{\mathbf{1} - sh/2} = \mathbf{c}\_{il} \left(\mathbf{c}\_{\varepsilon} + \mathbf{c}\_{i\varepsilon} \,/\, \mathbf{2}\right) \left[\mathbf{1} + \frac{\mathbf{c}\_{i\varepsilon}}{h(\mathbf{c}\_{\varepsilon} + \mathbf{c}\_{i\varepsilon} \,/\, \mathbf{2})\mathbf{s}} + \frac{\mathbf{c}\_{d\varepsilon}}{\mathbf{c}\_{\varepsilon} + \mathbf{c}\_{i\varepsilon} \,/\, \mathbf{2}}\mathbf{s}\right] \tag{53}$$

From the identification of the coefficients, the following relations of tuning are:

$$K\_{RG} = \tilde{c}\_{\mu} \left( c\_e + c\_{ie} \;/\; \mathcal{D} \right) \tag{54}$$

$$T\_I = \frac{h(c\_e + c\_{ie} \, / \, \text{2})}{c\_{ie}} \tag{55}$$

$$T\_D = \frac{c\_{de}}{c\_e + c\_{ie} / 2} \tag{56}$$

From these equations, the expressions of the scaling coefficients are:

$$\mathbf{c}\_e = \left(\frac{T\_I}{h} - \frac{1}{2}\right) \frac{hK\_{RG}}{\tilde{\mathbf{c}}\_\mu \, T\_I} \tag{57}$$

$$\mathcal{L}\_{i\mathcal{e}} = \frac{hK\_{RG}}{\mathcal{L}\_{\text{fl}}T\_I} \tag{58}$$

 <sup>~</sup> <sup>~</sup> ( ) ( ) <sup>1</sup>

~ *u*

*e*

*de*

*e*

~ *u*

*RG c*

~ *RG u*

*RG T c*

<sup>~</sup>

*RF RF sh <sup>u</sup> e ie <sup>z</sup> sh e ie e ie u s c c H s H z cc c <sup>s</sup> e s hc c s c c*

( ) ( ) ( ) /2 1 ( ) ( /2) /2

( /2) ( /2)

/ 2

*T hK*

~ *RG*

*hK*

*u I*

*c T*

*ie*

*c*

*<sup>h</sup> c T*

~ 1 2 *<sup>I</sup> RG <sup>e</sup>*

*u I*

*ie de*

*c*

(53)

*u s <sup>c</sup> <sup>c</sup> H s c c cs <sup>s</sup> e s c c*

*RG*

*RG*

*e*

*c*

*de*

*c*

As in the case of the fuzzy PI controller, there is obtained a quasi-continual form:

From the identification of the coefficients, the following relations of tuning are:

~

*K cc c hc c <sup>T</sup>*

*RG e ie u e ie*

*e ie*

1 /2 1 /2

*I*

*D*

*c*

From these equations, the expressions of the scaling coefficients are:

*<sup>c</sup> <sup>T</sup> c c*

*<sup>c</sup> <sup>T</sup>*

*<sup>c</sup> <sup>K</sup>*

*u de*

(48)

(54)

(55)

(56**)** 

*<sup>c</sup>* (49)

*<sup>c</sup>* (50)

*<sup>K</sup>* (51)

*<sup>K</sup>* (52)

*ie de*

(57)

(58)

*e e*

( )

From these equations, the expressions of the scaling coefficients results:

**3.6 Pseudo-equivalence of the fuzzy PID controller** 

*RF u e de*

From the identification of the coefficients, the following relations of tuning result:

$$\mathcal{L}\_{de} = \frac{\mathcal{K}\_{RG}}{\mathcal{L}\_{\mathcal{U}}} T\_D \tag{59}$$
