*5.4.2 GA-PID controller*

Based on Eqs. (50), (60) and (71), the absolute values of PID, PD and PI gain parameters for elevation, pitch and travel axis model respectively for helicopter

system are listed in **Table 3** [21]. Using the values in **Tables 1** and **3**, the closedloop transfer function of elevation, pitch and travel axis Eqs. (56), (65) and (70) become as in Eqs. (72), (73) and (74) respectively:

$$\frac{\varepsilon(s)}{\varepsilon\_{\varepsilon}(s)} = \frac{1445s^3 + 247s + 674.9}{1.8145s^3 + 1445s^2 + 2474s + 674.9} \tag{72}$$

*p s*ð Þ

*Number of generation of GA-LQR parameters Q and R.*

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

Krp k15

Kri k17

*Values of gain parameters for PID, PD and PI controllers.*

*r s*ð Þ

provide a stable output response.

**Figure 11.**

**Table 3.**

**35**

*pc*ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>69</sup>*:*<sup>08</sup> *<sup>s</sup>* <sup>þ</sup> <sup>269</sup>*:*<sup>3</sup>

**PID parameters Relationship Absolute Value** Kϵ<sup>p</sup> 2k11 10.6463 Kϵ<sup>d</sup> 2k13 2.3438 Kϵ<sup>i</sup> 2k16 0.3302 Kρ<sup>p</sup> 2k12 5.3634 Kρ<sup>d</sup> k14 1.4799

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

k12

k12

*rc*ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>2</sup>*:*<sup>878</sup> *<sup>s</sup>* <sup>þ</sup> <sup>1</sup>*:*<sup>634</sup>

Based on bounded input signal, the elevation, pitch and travel axis model of 3DOF helicopter system are unstable as they give unbounded outputs. The output responses for elevation, pitch and travel angle are illustrated in the **Figure 12**. It can be say that the open loop helicopter system without control action is unable to

<sup>0</sup>*:*0319*s*<sup>2</sup> <sup>þ</sup> <sup>69</sup>*:*<sup>08</sup> *<sup>s</sup>* <sup>þ</sup> <sup>269</sup>*:*<sup>3</sup> (73)

0.7678

0.3230

<sup>1</sup>*:*<sup>815</sup> *<sup>s</sup>*<sup>2</sup> <sup>þ</sup> <sup>2</sup>*:*<sup>878</sup> *<sup>s</sup>* <sup>þ</sup> <sup>1</sup>*:*<sup>634</sup> (74)


**Table 2.** *Parameters of GA tuning method.* *A Hybrid Control Approach Based on the Combination of PID Control with LQR Optimal Control DOI: http://dx.doi.org/10.5772/intechopen.94907*

**Figure 11.**

elements of the LQR weight matrices *Q* and *R* through iteration based on GA

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

matrices *Q* and *R* obtained based on the GA tuning approach are given by:

The feedback gain matrix of the LQR controller can be mathematically calculated using Eq. (13), where P matrix is the stabilizing solution of the Riccati

Based on the proposed fitness function stated in Eq. (40), the LQR weighting

*Q* ¼ *blkdig*ð Þ 26*:*258, 0*:*869, 0*:*431, 0*:*475, 1*:*87, 0*:*026, 0*:*705 ,

In this application, by using the state matrix ð Þ *A* , input matrix ð Þ *B* and the tuned weighting matrices (*Q*, *R*Þ, the optimized feedback gain matrix *K* stated below is

*K* ¼ *lqr A*ð Þ , *B*, *Q*, *R*

Based on the feedback gain matrix and using Eq. (11), the LQR control effort

¼ � <sup>5</sup>*:*323 2*:*682 1*:*172 0*:*739 2*:*059 0*:*165 0*:*<sup>866</sup> 5*:*323 �2*:*682 1*:*172 �0*:*739 �2*:*059 0*:*165 �0*:*866 

Based on Eqs. (50), (60) and (71), the absolute values of PID, PD and PI gain parameters for elevation, pitch and travel axis model respectively for helicopter system are listed in **Table 3** [21]. Using the values in **Tables 1** and **3**, the closedloop transfer function of elevation, pitch and travel axis Eqs. (56), (65) and (70)

**GA property Value/Method** Population Size 20 Max No. of Gen. 100

Selection Method Normalized Geo. Selection

Crossover Method Scattering Mutation Method Uniform Mutation

<sup>3</sup> <sup>þ</sup> <sup>247</sup> *<sup>s</sup>* <sup>þ</sup> <sup>674</sup>*:*<sup>9</sup>

<sup>1</sup>*:*8145*s*<sup>3</sup> <sup>þ</sup> <sup>1445</sup>*s*<sup>2</sup> <sup>þ</sup> <sup>2474</sup>*<sup>s</sup>* <sup>þ</sup> <sup>674</sup>*:*<sup>9</sup> (72)

*xT*

(71)

vector for the 3DOF helicopter system is dertermined as follows:

5*:*3232 �2*:*6817 1*:*1719 �0*:*7399 �2*:*0590 0*:*1651 �0*:*8661 

*<sup>K</sup>* <sup>¼</sup> <sup>5</sup>*:*3232 2*:*6817 1*:*1719 0*:*7399 2*:*0590 0*:*1651 0*:*<sup>8661</sup>

optimization method are presented in **Figure 11**.

*R* ¼ *blkdig*ð Þ 0*:*469, 0*:*469 ,

determined using the Matlab software instruction:

become as in Eqs. (72), (73) and (74) respectively:

*<sup>ϵ</sup>c*ð Þ*<sup>s</sup>* <sup>¼</sup> <sup>1445</sup>*<sup>s</sup>*

*ϵ*ð Þ*s*

equation stated in Eq. (14).

*u*1 *u*2 

**Table 2.**

**34**

*Parameters of GA tuning method.*

**Figure 10.**

*Definition of GA chromosome.*

*5.4.2 GA-PID controller*

*Number of generation of GA-LQR parameters Q and R.*


**Table 3.**

*Values of gain parameters for PID, PD and PI controllers.*

$$\frac{p(s)}{p\_c(s)} = \frac{69.08 \, s + 269.3}{0.0319s^2 + 69.08 \, s + 269.3} \tag{73}$$

$$\frac{r(s)}{r} = \frac{2.878 \, s + 1.634}{1} \tag{74}$$

$$\frac{r(s)}{r\_c(s)} = \frac{2.878\,\text{s} + 1.634}{1.815\,\text{s}^2 + 2.878\,\text{s} + 1.634} \tag{74}$$

Based on bounded input signal, the elevation, pitch and travel axis model of 3DOF helicopter system are unstable as they give unbounded outputs. The output responses for elevation, pitch and travel angle are illustrated in the **Figure 12**. It can be say that the open loop helicopter system without control action is unable to provide a stable output response.

**Figure 12.** *Open loop response of Helicopter system.*

In this study, in order to achieve a stable output, a hybrid control system using LQR based PID controller for 3DOF helicopter system is proposed to control the dynamic behaviour of the system. To validate the proposed helicopter stabilization system, the controller is simulated using Matlab programming tool. Three axis, elevation, pitch, travel rate, are considered in the simulation process of the control system. The performance of the helicopter balancing system is evaluated under unit step reference input using rise, settling time overshoot and steady state error parameters for the elevation, pitch and travel angles to simulate the desired command given by the pilot.
