*2.2.2 Fractional order controller*

Fractional order PID controller denoted by PI<sup>λ</sup> D<sup>μ</sup> was proposed by Igor Podlubny [9] in 1997. It is an extension of traditional PID controller where *λ* and *μ* have non-integer fractional values. **Figure 1** shows the block diagram of the fractional order PID controller. The integer-differential equation defining the control action of a fractional order PID controller is given by:

$$u(t) = K\_p e(t) + K\_i D^{-\lambda} e(t) + K\_d D^{-\mu} e(t) \tag{9}$$

the full state feedback concept. Therefore, using the LQR controller to stabilize the 3DOF helicopter system based on the assumption that the system states are considered measurable. LQR approach includes applying the optimal control effort:

*A Hybrid Control Approach Based on the Combination of PID Control with LQR Optimal Control*

Where *K* is the state feedback gain matrix, that will enable the output states of

the system to follow the trajectories of reference input, while minimizing the

Where *Q* and *R* are referred to as weighting state and control matrices. The controller feedback gain matrix can be determined by using below equation:

*<sup>K</sup>* <sup>¼</sup> *<sup>R</sup>*�<sup>1</sup>

*ATP* <sup>þ</sup> *PA* � *PBR*�<sup>1</sup>

*k*<sup>11</sup> *k*<sup>12</sup> *k*<sup>13</sup> … *k*1*<sup>n</sup> k*<sup>21</sup> *k*<sup>22</sup> *k*<sup>23</sup> … *k*2*<sup>n</sup> : : : :: km*<sup>1</sup> *km*<sup>2</sup> *km*<sup>3</sup> … *kmn*

Where *P* is *(nxn)* matrix deterrmined from the solution of the following Riccati

For *nth* order system with mth input, the gain matrix and control input are given by:

Based on the above expression, the control effort *u t*ð Þ of the system stated in Eq. (11) can be written as in Eq. (15). For the purpose of simplicity of control problem the weighting matrices *Q* and *R* are chosen as the diagonal matrices:

*<sup>Q</sup>* <sup>¼</sup> *blkdig q*11, *<sup>q</sup>*22, *<sup>q</sup>*33, ……… , *qnn*Þ, *<sup>R</sup>* <sup>¼</sup> *blkdig r*ð Þ 11,*r*22,*r*33, … ,*rmm* �

*k*<sup>11</sup> *k*<sup>12</sup> *k*<sup>13</sup> … *k*1*<sup>n</sup> k*<sup>21</sup> *k*<sup>22</sup> *k*<sup>23</sup> … *k*2*<sup>n</sup> : : : :: km*<sup>1</sup> *km*<sup>2</sup> *km*<sup>3</sup> … *kmn*

<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*11*u*<sup>1</sup>

Where *q*11, *q*22, *q*33, … *::*, *qnn* and *r*11,*r*22,*r*33, … *::*,*rmm* denote the weighting elements of *Q* and *R* matrices respectively. The optimal control approach LQR is highly recommended for stabilizing complex dynamic systems as it basically looks

<sup>2</sup> <sup>þ</sup> … *::* <sup>þ</sup> *rmmum*<sup>2</sup> � �*dt* (16)

<sup>2</sup> <sup>þ</sup> *<sup>r</sup>*22*u*<sup>2</sup>

so that the cost function Eq. (12) can be reformulated as in Eq. (16).

*J* ¼ ∞ð

0

following the cost function:

*DOI: http://dx.doi.org/10.5772/intechopen.94907*

matrix equation:

*J* ¼ ð ∞

**21**

0

*q*11*x*<sup>1</sup>

<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*11*x*<sup>2</sup>

K ¼

*u t*ðÞ¼�

<sup>2</sup> <sup>þ</sup> … *::* <sup>þ</sup> *qnnxn*

*u t*ðÞ¼�*Kx t*ð Þ (11)

*<sup>x</sup>T*ð Þ*<sup>t</sup> Qx t*ðÞ� *uT*ð Þ*<sup>t</sup> Ru t*ð Þ � �*dt* (12)

and *u t*ðÞ¼

*x*1 *x*2 *x*3 *: xn*

(15)

*BTP* (13)

*BTP* <sup>þ</sup> *<sup>Q</sup>* <sup>¼</sup> <sup>0</sup> (14)

*u*1 *u*2 *u*3 *: um*

Based on the above equation, it can be expected that the FOPID controller can enhance the performance of the control system due to more tuning knobs introduced. Taking the Laplace transform of Eq. (9), the system transfer function of the FOPID controller is given by:

$$\mathcal{G}\_{\text{FOPID}}(\mathbf{s}) = \mathcal{K}\_p + \frac{\mathcal{K}\_i}{\mathbf{s}^l} + \mathcal{K}\_d \mathbf{s}^\mu \tag{10}$$

Where λ and *μ* are arbitrary real numbers. Taking *λ* ¼ 1 and *μ* ¼ 1 a classical PID controller is obtained. Thus, FOPID controller generalizes the classical PID controller and expands it from point to plane as shown in **Figure 2(b)**. This expansion provides the designer much more flexibility in designing PID controller and gives an opportunity to better adjust the dynamics of the control system. This increases robustness to the system and makes it more stable [10]. A number of optimization techniques can be implemented for getting the best values of the gain parameters of the controller.
