**2.1 Calssical PID controller**

A PID is the most popular controller technique that is widely used in industrial applications due to the simplicity of its structure and can be realized easily for various control problems as the gain parameters of the controller are relatively independent [5, 6]. Basically, the controller provides control command signals *u t*ð Þ based on the error *e t*ð Þ between the demand input and the actual output of the system. The continuous time structure of the classical PID controller is as follows:

$$u(t) = K\_p e(t) + K\_i \int\_0^t e(\tau)d\tau + K\_d \frac{de(t)}{dt} \tag{1}$$

where *Kp*,*Ki* and *Kd* are the proportional, integral and differential components of the controller gain. These controller gain parameters should be tuned properly to enable the output states of the system to efficiently follow the desired input.

### **2.2 FOPID controller**

FOPID is a special category of PID controller with fractional order derivatives and integrals. Its concept was introduced by Podlubany in 1997. During the last decade, this controller approach has attracted the attention of control engineers in both academic and industrial fields. Compared with the classical PID controller, it offers flexibility in dynamic systems design and more robustness.

### *2.2.1 Fractional order calculus*

Fractional order calculus is an environment of calculus that generates the derivatives or integrals of problem functions to non-integer (fractional) order. This fractional order mathematical operation allows to establish a more accurate and concise model than the classical integer-order method. Moreover, the fractional order calculus can also produce an effective tool for describing dynamic behavior for control systems [7].

*A Hybrid Control Approach Based on the Combination of PID Control with LQR Optimal Control DOI: http://dx.doi.org/10.5772/intechopen.94907*

$$aD\_t^a = \begin{cases} \frac{d^a}{dt^a} & a > 0\\ 1 & a = 1\\ \int (dt)^{-a} & a < 0 \end{cases} \tag{2}$$

Fractional order calculus is a generalization of differentiation and integration to non-integer order fundamental operator which is denoted by *aD<sup>α</sup> <sup>t</sup>* where *a* and *t* are the operation limits and *α α*ð Þ ∈ **R** is the order of the operation. The formula of continuous differ-integral operator (*aD<sup>α</sup> <sup>t</sup>* ) is defined as in Eq. (2) [8]. There are two commonly used definitions for general fractional differ-integral *aD<sup>α</sup> t* � �, which are used for realization of control problem algorithm:

**Grunwald – Letnikov (GL) definition:**

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

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

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.

In this section, basics and theory of integer and fractional order PID controllers

A PID is the most popular controller technique that is widely used in industrial applications due to the simplicity of its structure and can be realized easily for various control problems as the gain parameters of the controller are relatively independent [5, 6]. Basically, the controller provides control command signals *u t*ð Þ based on the error *e t*ð Þ between the demand input and the actual output of the system. The continuous time structure of the classical PID controller is as follows:

ð*t*

*e*ð Þ*τ dτ* þ *Kd*

*de t*ð Þ

*dt* (1)

0

where *Kp*,*Ki* and *Kd* are the proportional, integral and differential components of the controller gain. These controller gain parameters should be tuned properly to enable the output states of the system to efficiently follow the desired input.

FOPID is a special category of PID controller with fractional order derivatives and integrals. Its concept was introduced by Podlubany in 1997. During the last decade, this controller approach has attracted the attention of control engineers in both academic and industrial fields. Compared with the classical PID controller, it

Fractional order calculus is an environment of calculus that generates the deriv-

atives or integrals of problem functions to non-integer (fractional) order. This fractional order mathematical operation allows to establish a more accurate and concise model than the classical integer-order method. Moreover, the fractional order calculus can also produce an effective tool for describing dynamic behavior

are presented. Theory of an intelligent LQR controller, which is used with PID

controller to combine a hybrid control system, is also introduced.

*u t*ðÞ¼ *Kpe t*ðÞþ *Ki*

offers flexibility in dynamic systems design and more robustness.

proposed to improve a nonlinear quarter car suspension system.

**2. Controller theory**

**2.1 Calssical PID controller**

**2.2 FOPID controller**

*2.2.1 Fractional order calculus*

for control systems [7].

**18**

$$aD\_t^af(t) = \frac{d^af(t)}{dt^a} = \lim\_{h \to 0} h^{-a} \sum\_{j=0}^{|\mathbf{x}|} (-1)^j \binom{a}{j} f(t - jh) \tag{3}$$

where ½ � *<sup>x</sup>* is integer part of *x, x* <sup>¼</sup> *<sup>t</sup>*�*<sup>a</sup> <sup>h</sup> , h* is time step and *<sup>α</sup> j* � � is binomial coefficients, its expression is given by:

$$
\binom{a}{j} = \frac{a(a-1)\dots \dots (a-j+1)}{j!} \tag{4}
$$
