3.2.2. Deduction of the variable structure parameter θ<sup>v</sup>

∂½ �¼ Nmð Þs λ1ð Þs np � 1: (24)

ð Þs ½ � vð Þt , (25)

<sup>5</sup>: (26)

ð Þ<sup>s</sup> <sup>θ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>v</sup>ð Þ<sup>t</sup> � �, (27)

ð Þs ½ � vð Þt : (28)

� �, (29)

<sup>1</sup> <sup>þ</sup> <sup>ξ</sup><sup>T</sup><sup>ξ</sup> (31)

<sup>5</sup>: (33)

(32)

<sup>1</sup> ¼ �ecea (30)

The Hurwitz polynomial L(s) is chosen such that the transfer function Hm(s)L(s) becomes SPR. The degree of the L(s) is <sup>∂</sup>[L(s)] = n\*m�1. If the L(s) is Hurwitz polynomial, then L�<sup>1</sup> is stable.

, <sup>ξ</sup> <sup>¼</sup> <sup>L</sup>�<sup>1</sup>

ξn ξy ξp 3 7

<sup>1</sup> ξ1ec

1

h i

2 6 4 ξu ξyp ξp

vð Þt

3 7

ð Þs θ<sup>g</sup> þ θ<sup>v</sup> � �<sup>T</sup>

2 6 4

Therefore, the λ1(s) polynomial is a design component.

v ¼

v<sup>1</sup> ¼

vu vyp yp r

> vu vyp yp

3 7 <sup>5</sup>, <sup>ξ</sup><sup>1</sup> <sup>¼</sup>

2 6 4

ea <sup>¼</sup> <sup>θ</sup><sup>T</sup><sup>ξ</sup> � <sup>L</sup>�<sup>1</sup>

<sup>ξ</sup>ðÞ¼ <sup>t</sup> <sup>L</sup>�<sup>1</sup>

ec <sup>¼</sup> <sup>e</sup><sup>0</sup> <sup>þ</sup> Hmð Þ<sup>s</sup> L sð Þ <sup>K</sup>1ea � <sup>ξ</sup><sup>T</sup>

and the on-line gradient adjustable parameter K1 depends only by the augmented error:

K o

<sup>g</sup> ¼ �γgsign Kp

<sup>ξ</sup> � <sup>L</sup>�<sup>1</sup>

� �ecξð Þ<sup>t</sup>

ð Þs ½ � vð Þt ; where ξ ¼

3.2. The parametric adjustment laws for the compound adaptive control

θ o

ea ¼ θ<sup>g</sup> þ θ<sup>v</sup> � �<sup>T</sup>

<sup>ξ</sup>ðÞ¼ <sup>t</sup> <sup>L</sup>�<sup>1</sup>

The parameters vectors v, ξ∈ ℜ2np consist of

126 Adaptive Robust Control Systems

The v1, ξ1∈ ℜ2np � <sup>1</sup> are defined as follows:

The auxiliary error is computed on-line:

The augmented error is defined as:

The gradient law [2–4] is expressed as:

where:

3.2.1. Gradient

The parameter θ<sup>v</sup> can be deducted by using the following [2]:

$$
\theta\_v = \overline{\theta}\_v \left[ \left( e^{\mathbb{K} \epsilon\_i \xi} - 1 \right) / \left( e^{\mathbb{K} \epsilon\_i \xi} + 1 \right) \right] \text{sign} \left( \mathbb{K}\_p \right) \tag{34}
$$

$$
\stackrel{\circ}{\overline{\Theta}}\_{\upsilon} = -\lambda \overline{\Theta}\_{\upsilon} - \gamma\_{\upsilon}|\xi \mathbf{e}\_{\mathfrak{C}}|.\tag{35}
$$

The block diagram of obtaining the augmented error is depicted in Figure 8, in which Φ is the vector of the parameter estimation errors

$$
\Phi = \theta - \theta^0 \tag{36}
$$

The vector of the parameter θ is obtained by using the compound structure: θ = θ<sup>g</sup> + θv.

In the adaptive control, there is a commutation function; usually the signum function conducts toward a sliding mode regime such that the evolution to the equilibrium point is very fast. Therefore, the compound adaptive law is used:

$$\mathbf{u}(t) = \left(\boldsymbol{\theta}\_{\mathcal{S}} + \boldsymbol{\theta}\_{v}\right)^{T} \mathbf{v} + \left[-\boldsymbol{\gamma}\_{p} \text{sign}\left(\mathbf{K}\_{p}\right) \boldsymbol{\xi} e\_{0} + \frac{\stackrel{\circ}{\boldsymbol{\Theta}}}{\boldsymbol{\theta}^{K\mathcal{E}\boldsymbol{\varepsilon}\_{0}}} \frac{\boldsymbol{e}^{K\boldsymbol{\xi}\boldsymbol{\varepsilon}\_{0}} - 1}{\boldsymbol{e}^{K\boldsymbol{\xi}\boldsymbol{\varepsilon}\_{0}} + 1}\right]^{T} \boldsymbol{\xi}. \tag{37}$$

The adaptive control provides robust characteristics to external disturbances and to unmodelled dynamics.

#### 3.3. The stability of the solution

The perturbation of the dc drive system can leads to the instability of the system. The signals in the variable structure law are bounded. Therefore, the adaptive control assures a global stability [4].

Figure 8. The block diagram of the augmented error determination ec.

#### 4. Numerical simulation results

Taken into account the dc machine from the conventional control, operating at the constant flux, a speed cycle is applied in order to test the compound adaptive control. The speed cycle contains the dynamic regimes (starting, braking, reversing) and the steady state regime (Figures 9–22).

0 5 10 15 20

speed of the reference model and of the DC drive output

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129

time[s]

load torque

0 5 10 15 20

augmented error

0 5 10 15 20

time[s]

time[s]


Figure 11. The speed cycle. At t = 12 s, the rated load torque is applied.

0

Figure 12. The load torque. At t = 12 s, the rated load torque is applied.


Figure 13. The augmented error. At t = 12 s, the rated load torque is applied.



ec

0

0.05

5

10

TL[Nm]

15

20


speed [rpm]

2000

4000

0

Figure 9. The Simulink implementation of the compound adaptive dc drive with supraunitary degree and with unknown gain.

Figure 10. The Simulink implementation of the augmented error deduction ea\_vu.

Figure 11. The speed cycle. At t = 12 s, the rated load torque is applied.

Figure 12. The load torque. At t = 12 s, the rated load torque is applied.

Figure 9. The Simulink implementation of the compound adaptive dc drive with supraunitary degree and with unknown

Figure 10. The Simulink implementation of the augmented error deduction ea\_vu.

gain.

128 Adaptive Robust Control Systems

Figure 13. The augmented error. At t = 12 s, the rated load torque is applied.

0 5 10 15 20

time[s]

Figure 17. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θyg gradient-

variable structure-process output

0 5 10 15 20

time[s]

Figure 18. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θyv variable

gradient-control

0 5 10 15 20

Figure 19. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θug gradient

thetarv

time[s]

gradient-process output

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131

0

1

2

0.5


0

5

10


0

20

40

process output.

structure process output.

control.

1.5

Figure 14. The adaptive control. At t = 12 s, the rated load torque is applied.

Figure 15. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θrg gradient reference.

Figure 16. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θrv variable structure.

Matlab-Simulink-Based Compound Model Reference Adaptive Control for DC Motor http://dx.doi.org/10.5772/intechopen.71758 131

Figure 17. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θyg gradientprocess output.

0 5 10 15 20

time[s]

0 5 10 15 20

thetarv

time[s]

Figure 15. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θrg gradient

variable structure-reference

0 5 10 15 20

time[s]

Figure 16. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θrv variable

gradient-reference

adaptive control


0

Figure 14. The adaptive control. At t = 12 s, the rated load torque is applied.


reference.

structure.

0



0

5

10

0.5

1

1.5

200

rotor voltage UA[V]

130 Adaptive Robust Control Systems

400

600

Figure 18. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θyv variable structure process output.

Figure 19. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θug gradient control.

5. Conclusions

Acknowledgements

Author details

Marian Găiceanu

References

Dumitrache, I.; 1993

The conventional control and the compound adaptive control have been investigated by using a dc drive system. The complete methodology of tuning the controller parameters for the conventional control is provided. Under the assumptions mentioned in this chapter, the dc drive system has been implemented in Matlab-Simulink software. The adequate numerical simulation results have been obtained (Figure 6) for. Therefore, the regulation capability of the PI controller is tested under a variation of the load torque for a starting. Both, the dynamic and steady state regimes are investigated. In Figure 6b, the maximum load torque of the dc motor is used for a starting under the 70% load conditions. The maximum limit is maintained during the dynamic regime, the torque decreases in steady state due to the armature current decreasing. The constant parameter values have been considered. In case of the gradient and variable structure laws, the adaptive system is more robust to parameter uncertain or to unmodelled dynamics of the dc drive, and the increased regulation performances are obtained. The

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133

supraunitary relative degree model reference adaptive control has been considered.

Research, CNDI–UEFISCDI, project number PN-II-PT-PCCA-2011-3.2-1680.

Address all correspondence to: marian.gaiceanu@ieee.org

Center, Dunarea de Jos University of Galati, Romania

This work was supported by a grant of the Romanian National Authority for Scientific

Integrated Energy Conversion Systems and Advanced Control of Complex Processes Research

[1] Didactical Pedagogical House, editor. Automatizări electronice. 1993rd ed. Bucuresti:

[2] Filipescu A. Robustness in compound adaptive controlling with sigmoid function for supraunitary relative degree of the plant model. In: The 9'th Symposium on Modelling

[3] Gaiceanu M. Embedded control of the DC drive system for education. In: Yildirim S, editor. Design, Control and Applications of Mechatronic Systems in Engineering. Croatia: InTechOpen; 2017. DOI: 10.5772/67461. Available from: https://www.intechopen.com/

and Identification systems, SIMSIS9'96; 1996; Galati. Romania; 1996. pp. 45-52

Figure 20. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θuv variable structure control.

Figure 21. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θpg gradientprocess output.

variable structure-process output

Figure 22. The adaptive mechanism of the parameter vector θ obtained by using the compound structure: θpv variable structure process output.
