*5.1.2 Pitch axis model*

In this chapter, a hybrid PID controller based on LQR optimal technique is designed to stabilize 3DOF helicopter system. The proposed hybrid LQR-PID controller is optimized using GA optimization method, which is used to tune its gain

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

The conceptual platform of 3DOF helicopter scheme is presented in **Figure 4**. It consists of an arm mounted on a base. The main body of the helicopter constructed of propellers driven by two motors mounted are the either ends of a short balance bar. The whole helicopter body is fixed on one end of the arm and a balance block

The balance arm can rotate about the travel axis as well as slope on an elevation axis. The body of the helicopter is free to roll about the pitch axis. The system is provided by encoders mounted on these axes used to measure the travel motion of the arm and its elevation and pitch angle. The propellers with motors can generate an elevation mechanical force proportional to the voltage power supplied to the motors. This force can cause the helicopter body to lift off the ground. It is worth considering that the purpose of using a balance block is to reduce the voltage power supplied to the propellers motors. In this study, the nonlinear dynamics of 3DOF helicopter system is modelled mathematically based on developing the model of the

The free body diagram of 3DOF helicopter system based on elevation axis is shown in **Figure 5**. The movement of the elevation axis is governed by the following

<sup>2</sup> *cos*ð Þ*<sup>ϵ</sup> sin* ð Þþ *<sup>ϵ</sup> <sup>l</sup>*1*Fm cos*ð Þ� *<sup>ρ</sup> <sup>l</sup>*1*Fm* � *MG* � *Mf*,*<sup>ϵ</sup>* (17)

parameters.

**5. Case study: helicopter control system**

**5.1 Helicopter structure and modeling**

system behavior for each of the axes.

*τ*\_

installed at the other end.

*5.1.1 Elevation axis model*

differential equations:

**Figure 4.**

**24**

*Prototype model of 3DOF helicopter system.*

*Jϵ*€*ϵ* ¼ *Jp* � *J<sup>τ</sup>* 

> Consider the pitch schematic diagram of the system in **Figure 6**. It can be seen from the figure that the main torque acting on the system pitch axis is produced from the thrust force generated by the propeller motors. When *ρ* 6¼ 0, the gravitational force will also generate a torque *Mw*,*<sup>ρ</sup>* acts on the helicopter pitch axis. The dynamics of the pitch axis can be modeled mathematically as follows:

$$J\_{\rho}\ddot{\rho} = F\_{f}l\_{\rho} - F\_{b}l\_{\rho} - \mathbf{M}\_{w,\rho} - \mathbf{M}\_{f,\rho} \tag{23}$$

Where *Mf*,*<sup>ρ</sup>* is the friction moment exerted on the pitch axis.

$$M\_{w, \rho} = m\_h g l\_h \sin\left(\rho\right) \cos\left(\varepsilon\right) \tag{24}$$

Based on the assumption that the pitch angle *ρ* ¼ 0*, Mw*,*<sup>ρ</sup>* ¼ 0, then Eq. (23) becomes as follows:

$$J\_{\rho}\ddot{\rho} = l\_{\rho} \left( F\_f - F\_b \right) - M\_{f,\rho} \tag{25}$$

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

$$J\_{\rho}\ddot{\rho} = K\_{c}l\_{\rho}\left(V\_{f} - V\_{b}\right) - M\_{f,\rho} \tag{26}$$

$$J\_{\rho}\ddot{\rho} = K\_{c}l\_{\rho}V\_{d} - \mathbf{M}\_{f,\rho} \tag{27}$$

The thrust forces of the two propeller motors *F <sup>f</sup>* þ *Fb*

helicopter in flight case and is approximately *W:*

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

**5.2 Helicopter state space model**

Let *x nx* ð Þ¼ 1 ½ � *x*1,*x*2, *x*3,*x*4, *x*5,*x*6, *x*<sup>7</sup>

the output's vector such that, *y t*ð Þð Þ¼ *px*1 ½ � *ϵ*, *ρ*,*r*

<sup>T</sup> <sup>¼</sup> *<sup>V</sup> <sup>f</sup>* ,*Vb*

system state space model:

½ � *u*1,*u*<sup>2</sup>

**27**

are required to keep the

*Jrr*\_ ¼ �*Wsin*ð Þ*ρ l*<sup>1</sup> � *M <sup>f</sup>*,*<sup>r</sup>* (29)

*Jrr*\_ ¼ �*Wρl*<sup>1</sup> � *M <sup>f</sup>*,*<sup>r</sup>* (30)

*Vs* (31)

*Vd* (32)

*ρ* (33)

*<sup>T</sup>* <sup>¼</sup> *<sup>ϵ</sup>*, *<sup>ρ</sup>*, *<sup>ϵ</sup>*\_, *<sup>ρ</sup>*\_,*r*, *<sup>ʓ</sup>*, *<sup>γ</sup> <sup>T</sup>* be the state vector of the

(34)

Where *M <sup>f</sup>*,*<sup>r</sup>* is the friction moment exerted on the travel axis. As *ρ* approaches to zero, based on sinc function, *sin* ð Þ¼ *ρ ρ*, the above equation becomes as follows:

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

Based on the assumption that the coupling dynamics, gravitational torque (*Mw*,*<sup>ϵ</sup>*) and friction moment exerted on elevation, pitch and travel axis are neglected, then the dynamics modeling equations Eqs. (22), (27) and (30) for 3DOF helicopter system can be simplified as in Eqs. (31), (32) and (33) respectively [21].

> €*<sup>ϵ</sup>* <sup>¼</sup> *Kcl*<sup>1</sup> *Jϵ*

€*<sup>ρ</sup>* <sup>¼</sup> *Kcl<sup>ρ</sup> Jρ*

> *<sup>r</sup>*\_ <sup>¼</sup> *Wl*<sup>1</sup> *Jr*

In order to design a state feedback controller based on LQR technique for 3DOF helicopter system, the dynamics model of the system should be formulated in state space form. In this study, the proposed hybrid control algorithm is investigated for the purpose of control of pitch angle, elevation angle and travel rate of 3DOF helicopter scheme by regulating the voltage supplies to the front and back motors.

system, the state variables are chosen as the angles and rate and their corresponding angular velocities, and *ʓ*\_ ¼ *ϵ*, *γ*\_ ¼ *r* . The voltages supplied to the front and back propellers motors are considered the input's vector such that, *u t*ð Þð Þ¼ mx1

Based on Eqs. (31)-(33), choosing these state variables yields the following

*Jϵ*

*Jρ*

*Jr x*2

*x*\_<sup>1</sup> ¼ *ρ* ¼ *x*<sup>2</sup> *x*\_<sup>2</sup> ¼ *ϵ*\_ ¼ *x*<sup>3</sup> *<sup>x</sup>*\_<sup>3</sup> <sup>¼</sup> €*<sup>ϵ</sup>* <sup>¼</sup> *Kcl*<sup>1</sup>

*<sup>x</sup>*\_<sup>4</sup> <sup>¼</sup> €*<sup>ρ</sup>* <sup>¼</sup> *Kcl<sup>ρ</sup>*

*<sup>x</sup>*\_<sup>5</sup> <sup>¼</sup> *<sup>r</sup>*\_ <sup>¼</sup> *Wl*<sup>1</sup>

*x*\_<sup>6</sup> ¼ *ʓ*\_ ¼ *x*<sup>1</sup> *x*\_<sup>7</sup> ¼ *γ*\_ ¼ *x*<sup>4</sup>

<sup>T</sup> and the elevation angle, pitch angle and travel rate are assumed

*T*.

*V <sup>f</sup>* þ *Vb* 

*V <sup>f</sup>* � *Vb* 
