**1. Introduction**

PID is regarded as the standard control structure of classical control theory. PID controllers are used successfully for single-input single-output (SISO) and linear systems due to their good performance and can be easily implemented. The control of complex dynamic systems using classic PID controllers is considered as a big challenge, where the stabilization of these systems requires applying a more robust controller technique. Many studies have proposed to develop a new hybrid PID controller with ability to provide better and more robust system performance in terms of transient and steady-state responses over the standard PID controllers. Lotfollahzade et al. [1] proposed a new LQR-PID controller to obtain an optimal load sharing of an electrical grid. The presented hybrid controller is optimized by Particle Swarm Optimization (PSO) to compute the gain parameters of the PID controller. A new hybrid control algorithm was presented by Lindiya et al. for power converters [2]. They adopted a conventional multi-variable PID and LQR algorithm for reducing cross-regulation in DC-to-DC converters. Sen et al.

introduced a hybrid LQR-PID controller to regulate and monitor the locomotion of a quadruped robot. The gain parameters of the hybrid controller is tuned using the Grey-Wolf Optimizer (GWO) [3]. In [4] a new PID and LQR control system was proposed to improve a nonlinear quarter car suspension system.

*aD<sup>α</sup> t* ¼

non-integer order fundamental operator which is denoted by *aD<sup>α</sup>*

*f t*ð Þ *dt<sup>α</sup>* <sup>¼</sup> lim *h*!0

commonly used definitions for general fractional differ-integral *aD<sup>α</sup>*

continuous differ-integral operator (*aD<sup>α</sup>*

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

*aD<sup>α</sup>*

where ½ � *<sup>x</sup>* is integer part of *x, x* <sup>¼</sup> *<sup>t</sup>*�*<sup>a</sup>*

coefficients, its expression is given by:

**Riemann-Liouville definition:**

used for realization of control problem algorithm: **Grunwald – Letnikov (GL) definition:**

*<sup>t</sup> f t*ðÞ¼ *<sup>d</sup><sup>α</sup>*

*α j* � �

*<sup>t</sup> f t*ðÞ¼ <sup>1</sup>

*Γ*ð Þ *n* � *α*

*Γ*ð Þ¼ *X*

*aD<sup>α</sup>*

Gamma function, which its defination is given by:

*L aD<sup>α</sup>*

*2.2.2 Fractional order controller*

**19**

Laplace transform of differ-integral operator aD<sup>α</sup>

*<sup>t</sup> f t*ð Þ � � <sup>¼</sup> *<sup>s</sup>*

Fractional order PID controller denoted by PI<sup>λ</sup>

*L aD<sup>α</sup> <sup>t</sup> f t*ð Þ � � <sup>¼</sup>

*dα*

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

8 >>>><

>>>>:

Ð*t α*

the operation limits and *α α*ð Þ ∈ **R** is the order of the operation. The formula of

*dt<sup>α</sup> <sup>α</sup>* <sup>&</sup>gt;<sup>0</sup> 1 *α* ¼ 1

ð Þ *dt* �*<sup>α</sup> <sup>α</sup>* <sup>&</sup>lt;<sup>0</sup>

Fractional order calculus is a generalization of differentiation and integration to

*h*�*<sup>α</sup>*X ½ � *x*

*j*¼0

<sup>¼</sup> *α α*ð Þ � <sup>1</sup> … … *::*ð Þ *<sup>α</sup>* � *<sup>j</sup>* <sup>þ</sup> <sup>1</sup>

*dn dtn*

where *n* ∈ þ. The condition for above equation is *n* � 1< *α*<*n*, *Γ*ð Þ*:* is called

∞ð

*z<sup>X</sup>*�<sup>1</sup> *e* �*z*

0

ð<sup>∞</sup> 0 *e* �*staD<sup>α</sup>*

*m*¼0

Where *F s*ðÞ¼ *L ft* f g ð Þ is the normal Laplace transformation and *n* is an integer number that satisfies *n* � 1< *α*≤*n* and *s* ¼ *jw* denotes the Laplace transform variable.

Podlubny [9] in 1997. It is an extension of traditional PID controller where *λ* and *μ*

*<sup>s</sup>*ð Þ �<sup>1</sup> *<sup>j</sup>*

*<sup>α</sup>F s*ðÞ� <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

ð*t*

*f*ð Þ*τ*

t

*a*

ð Þ �<sup>1</sup> *<sup>j</sup> <sup>α</sup> j*

*<sup>h</sup> , h* is time step and *<sup>α</sup>*

(2)

*<sup>t</sup>* where *a* and *t* are

*t* � �, which are

� �*f t*ð Þ � *jh* (3)

� � is binomial

*<sup>t</sup>* ) is defined as in Eq. (2) [8]. There are two

*j*

*<sup>j</sup>*! (4)

ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>α</sup>*�*n*þ<sup>1</sup> *<sup>d</sup><sup>τ</sup>* (5)

*dz* (6)

� � is given by expected form:

*<sup>t</sup> f t*ð Þ*dt* (7)

<sup>0</sup>*D<sup>α</sup>*�*m*�<sup>1</sup> *<sup>t</sup> f t*ð Þ (8)

D<sup>μ</sup> was proposed by Igor

The intent of this study is to design a new hybrid PID controller based on an optimal LQR state feedback controller for stabilization of 3DOF helicopter system. To this end an improvement in the system performance has been achieved in both the transient and steady-state responses. In the proposed system the classical PID and optimal LQR controller have been combined to formulate a hybrid controller system. Simulations were implemented utilizing Matlab programming environment to verify the efficiency and effectiveness of the proposed hybrid control method.
