**4. Stability assurance**

#### **4.1 Internal stability**

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

Fig. 13. The structure of the control system with the correction of the non-linear part *N* The linear part L has the input-state-output model (60).

$$\begin{aligned} \mathbf{x}\_{L1} &= A\_{L1}\mathbf{x}\_{L1} + b\_{L1}\mathbf{K}\_{\text{CNA}}\mathbf{x}\_{a2} + \frac{1}{2}b\_{L1}\mathbf{K}\_{\text{CNA}}\tilde{d}u\_{d} \\\\ \mathbf{x}\_{d1} &= -\frac{2c\_{de}}{\hbar}\mathbf{K}\_{\text{CAN}}\mathbf{c}\_{L1}^{T}\mathbf{x}\_{L1} - \frac{2}{\hbar}\mathbf{x}\_{a1} + \frac{2c\_{de}}{\hbar}w \\\\ \mathbf{x}\_{d2} &= \frac{1}{\hbar}\tilde{d}u\_{d} \\\\ \tilde{e} &= -c\_{e}\mathbf{K}\_{\text{CAN}}\mathbf{c}\_{L1}^{T}\mathbf{x}\_{L1} + c\_{e}w \\\\ \tilde{d}e &= -\frac{2c\_{de}}{\hbar}\mathbf{K}\_{\text{CAN}}\mathbf{c}\_{L1}^{T}\mathbf{x}\_{L1} - \frac{2}{\hbar}\mathbf{x}\_{a1} + \frac{2c\_{de}}{\hbar}w \end{aligned} \tag{60}$$

With the new compound variable (61)

$$
\stackrel{\sim}{1}\infty = \begin{bmatrix} 1 & 1 \end{bmatrix} \stackrel{\sim}{y} = \stackrel{\sim}{e} + \stackrel{\sim}{de} \tag{61}
$$

there may be introduced a new function of the compound variable <sup>~</sup> *xt* and parameter <sup>~</sup> *de* (62).

$$\mathcal{K}\_N(\stackrel{\sim}{\underset{\mathcal{X}}{\times}};\stackrel{\sim}{de}) = \frac{\stackrel{\sim}{f\_N(\stackrel{\sim}{e},\stackrel{\sim}{de})}}{\stackrel{\sim}{\underset{\mathcal{X}}{\times}}}, \text{pt.}\stackrel{\sim}{\underset{\mathcal{X}}{\times}} \neq 0 \tag{62}$$

The families of characteristics ~~ ~ *dud f* (;) *x de <sup>t</sup>* present the sector property to be placed only in the quadrants I and III and they are inducing the consideration of the relation (63).

$$0 \le K\_N(\tilde{\mathbf{x}}\_l; \tilde{de}) \le K\_M \tag{63}$$

Tuning Fuzzy PID Controllers 187

( ) ( ( ), ( )) *<sup>y</sup> y t f xt wt*

*f* (, ) *e de* is considered introduced in *f*x. According to [14], we may write the following conditions: *x*=0 is a stable point of equilibrium with *w*=0*,* and *f*x(0, 0)=0, *t*0; *x*=0 is a global equilibrium point of the system;

Jacobian matrix *fx* /*x*, evaluated for *w*=0, and *fx* /*w* are global limited; *f*y(*t*, *x*, *w*),

*yt wt*

A fuzzy control system, as it is in the example, has the block diagram from Fig. 14. A fuzzy PI controller RF- is used in a speed control system of an electrical drive with the following elements: MCC - DC motor, CONV – power converter, RG-I – current controller, RF- speed controller, Ti – current sensor, T - speed sensor, CAN, CNA - analogue to digital

The fuzzy controller has the structure from Fig. 15. It is a quasi-fuzzy PI controller with summation at the output, with an internal fuzzy block BF with the structure presented at the beginning, and a correction circuit to insure stability. The controller has also an anti wind-

0 0 sup ( ) sup ( ) *t t*

.

( ) ( ( ), ( )) *<sup>x</sup> x t f xt wt* (65)

( ,0) *<sup>x</sup> x fx* (66)

<sup>123</sup> *f* (, ) *xw k x k u k* (67)

, there are the constants >0 şi 3

(68)

 (, ) 0 *k*

.

where the non-linear part ~ ~

global, for *k*1, *k*2, *k*3>0. Then, for any *x*(0)

**5. Control system example** 

and digital to analogue converters.

Fig. 14. The block diagram of the fuzzy control system

satisfies:

such as:

up circuit.

The characteristic of the non-linear part has null intervention, due to the limitations placed at the inputs of the fuzzy block. To the fuzzy blocks we may attach a fuzzy relation of which characteristic is placed only in the quadrants I and III.

From the relation (, ) *BF e de f x x* , which is describing the fuzzy block, a source of nonlinearity is there made by the membership functions. If the block will work on the universe of discourse [-1, 1], its characteristic will only be in the sector [*K*1, *K*2], 0<*K*1<*K*2. By introducing the saturation elements with a role of limitation at the inputs of the fuzzy block, the non-linear

part <sup>~</sup> *N* is placed in a sector [0, *K*]. To accomplish the sector condition, necessary for the stability insurance, a correction is used to the non-linear part. It consists in summation at the output *du*d of the fuzzy block of the quantity du:

$$\boldsymbol{\delta}\_{du} = \boldsymbol{K}\_c[\boldsymbol{\tilde{\{e-e\}}} + (\boldsymbol{\tilde{\{de-de\}}})] = \boldsymbol{K}\_c(\boldsymbol{\tilde{\{x\_l-x\_t\}}}) \tag{64}$$

The value *K*c>0 will be chosen in a way that the nonlinearity <sup>~</sup> *Nc* characteristic is to be framed in an adequate sector [*K*min, *K*max].

The design method in order to obtain the value for the gain coefficient is presented as it follows:

The method recommended for stability insurance is as it follows:

1. For a certain fuzzy block type, the minimum value of *K*m and the maximum value of *K*<sup>M</sup>

are chosen from the curve families *K*Nc=*f*( ~ *x* t), or *du*dc=*f*( ~ *<sup>x</sup>* t), with <sup>~</sup> *de* as a parameter.


#### **4.2 External stability**

To assure external BIBO stability (Khalil, 1991) the following relation may be taken in consideration:

~ ~

The characteristic of the non-linear part has null intervention, due to the limitations placed at the inputs of the fuzzy block. To the fuzzy blocks we may attach a fuzzy relation of which

From the relation (, ) *BF e de f x x* , which is describing the fuzzy block, a source of nonlinearity is there made by the membership functions. If the block will work on the universe of discourse [-1, 1], its characteristic will only be in the sector [*K*1, *K*2], 0<*K*1<*K*2. By introducing the saturation elements with a role of limitation at the inputs of the fuzzy block, the non-linear

*N* is placed in a sector [0, *K*]. To accomplish the sector condition, necessary for the stability insurance, a correction is used to the non-linear part. It consists in summation at the

> ~~ ~ *du c*[( ) ( )] ( ) *<sup>t</sup> K e e de de K x x c t*

The design method in order to obtain the value for the gain coefficient is presented as it

1. For a certain fuzzy block type, the minimum value of *K*m and the maximum value of *K*<sup>M</sup>

~

2. The value of incremental coefficient of the command variable is limited by the capacity

3. The incremental coefficient of the command variable may be determined with the

7. In the choosing of *c*du we must take account to the maximum values of *K*M of the

To assure external BIBO stability (Khalil, 1991) the following relation may be taken in

4. The maximum value of the command variable cannot overpass a maximum value. 5. At an incremental step, on a sampling period *h*, for the incremental of the command variable, a value is not recommended. For this, there may be chosen maximum a value

6. The values of coefficients *cdu* and *K*c may be chosen to insure sector stability.

8. The chosen of *K*c is done by taking account on the rapport *r*k=*K*min/*K*max.

*x* t), or *du*dc=*f*(

in the quadrants I and III and they are inducing the consideration of the relation (63).

*dud f* (;) *x de <sup>t</sup>* present the sector property to be placed only

(64)

~

*<sup>x</sup>* t), with <sup>~</sup>

*Nc* characteristic is to be

*de* as a parameter.

0 (;) *K x de K N M <sup>t</sup>* (63)

The families of characteristics ~~ ~

part <sup>~</sup>

follows:

characteristic is placed only in the quadrants I and III.

output *du*d of the fuzzy block of the quantity du:

are chosen from the curve families *K*Nc=*f*(

of the incremental coefficient of *c*duM.*K*M.

**4.2 External stability** 

consideration:

relation that is describing the digital integration.

superior limit of the nonlinearity of the fuzzy block.

framed in an adequate sector [*K*min, *K*max].

The value *K*c>0 will be chosen in a way that the nonlinearity <sup>~</sup>

The method recommended for stability insurance is as it follows:

of control system to furnish the command variable to the process.

$$
\dot{\mathbf{x}}(t) = f\_{\chi}(\mathbf{x}(t), w(t))\tag{65}
$$

$$
\mathbf{y}(t) = f\_{\chi}(\mathbf{x}(t), w(t))
$$

where the non-linear part ~ ~ *f* (, ) *e de* is considered introduced in *f*x.

According to [14], we may write the following conditions: *x*=0 is a stable point of equilibrium with *w*=0*,* and *f*x(0, 0)=0, *t*0; *x*=0 is a global equilibrium point of the system;

$$\dot{\hat{\mathbf{x}}} = f\_{\chi}(\mathbf{x}, \mathbf{0}) \tag{66}$$

Jacobian matrix *fx* /*x*, evaluated for *w*=0, and *fx* /*w* are global limited; *f*y(*t*, *x*, *w*), satisfies:

$$\left\| f(\mathbf{x}, w) \right\| \le k\_1 \left\| \mathbf{x} \right\| + k\_2 \left\| \mu \right\| + k\_3 \tag{67}$$

global, for *k*1, *k*2, *k*3>0. Then, for any *x*(0) , there are the constants >0 şi 3 (, ) 0 *k* such as:

$$\sup\_{t\geq 0} \|y(t)\| \leq \sup\_{t\geq 0} \|w(t)\| + \beta \tag{68}$$
