*5.1.3 Travel axis model*

The free body diagram of the helicopter system dynamics based on the travel axis is presented in **Figure 7**. In this model, when *ρ* 6¼ 0, the main forces acting on the helicopter dynamics are the thrust forces of propeller motors (*F <sup>f</sup>* , *Fb*). These forces have a component that generates a torque on the travel axis. Assume that the helicopter body has roll up by an angle *ρ* as shown in **Figure 7**. Then the dynamics of travel axis for 3DOF helicopter system is modeled as follows:

#### **Figure 6.**

*Schematic diagram of the pitch axis model for 3DOF helicopter scheme.*

**Figure 7.** *Schematic diagram of the travel rate axis model for 3DOF helicopter scheme.*

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

The thrust forces of the two propeller motors *F <sup>f</sup>* þ *Fb* are required to keep the helicopter in flight case and is approximately *W:*

$$J\_r \dot{r} = -\mathbf{W} \sin(\rho) l\_1 - \mathbf{M}\_{f,r} \tag{29}$$

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:

$$J\_r \dot{r} = -W \rho l\_1 - M\_{f,r} \tag{30}$$

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].

$$
\ddot{\varepsilon} = \frac{K\_c l\_1}{J\_c} V\_s \tag{31}
$$

$$
\ddot{\rho} = \frac{K\_c l\_\rho}{J\_\rho} V\_d \tag{32}
$$

$$
\dot{r} = \frac{Wl\_1}{J\_r} \rho \tag{33}
$$

### **5.2 Helicopter state space model**

*Jρ*€*ρ* ¼ *Kcl<sup>ρ</sup> V <sup>f</sup>* � *Vb*

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

of travel axis for 3DOF helicopter system is modeled as follows:

*Schematic diagram of the pitch axis model for 3DOF helicopter scheme.*

*Schematic diagram of the travel rate axis model for 3DOF helicopter scheme.*

*Jrr*\_ ¼ � *F <sup>f</sup>* þ *Fb*

The free body diagram of the helicopter system dynamics based on the travel axis is presented in **Figure 7**. In this model, when *ρ* 6¼ 0, the main forces acting on the helicopter dynamics are the thrust forces of propeller motors (*F <sup>f</sup>* , *Fb*). These forces have a component that generates a torque on the travel axis. Assume that the helicopter body has roll up by an angle *ρ* as shown in **Figure 7**. Then the dynamics

*5.1.3 Travel axis model*

**Figure 6.**

**Figure 7.**

**26**

� *<sup>M</sup> <sup>f</sup>*,*<sup>ρ</sup>* (26)

*Jρ*€*ρ* ¼ *KclρVd* � *M <sup>f</sup>*,*<sup>ρ</sup>* (27)

*sin* ð Þ*<sup>ρ</sup> <sup>l</sup>*<sup>1</sup> � *<sup>M</sup> <sup>f</sup>*,*<sup>r</sup>* (28)

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. Let *x nx* ð Þ¼ 1 ½ � *x*1,*x*2, *x*3,*x*4, *x*5,*x*6, *x*<sup>7</sup> *<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 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 ½ � *u*1,*u*<sup>2</sup> <sup>T</sup> <sup>¼</sup> *<sup>V</sup> <sup>f</sup>* ,*Vb* <sup>T</sup> and the elevation angle, pitch angle and travel rate are assumed the output's vector such that, *y t*ð Þð Þ¼ *px*1 ½ � *ϵ*, *ρ*,*r T*.

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

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \boldsymbol{\rho} = \mathbf{x}\_2 \\ \dot{\mathbf{x}}\_2 &= \dot{\boldsymbol{\varepsilon}} = \mathbf{x}\_3 \\ \dot{\mathbf{x}}\_3 &= \ddot{\boldsymbol{\varepsilon}} = \frac{K\_c l\_1}{J\_c} \left( V\_f + V\_b \right) \\ \dot{\mathbf{x}}\_4 &= \ddot{\boldsymbol{\rho}} = \frac{K\_c l\_\rho}{J\_\rho} \left( V\_f - V\_b \right) \\ \dot{\mathbf{x}}\_5 &= \dot{\boldsymbol{r}} = \frac{W l\_1}{J\_r} \mathbf{x}\_2 \\ \dot{\mathbf{x}}\_6 &= \dot{\mathbf{y}} = \mathbf{x}\_1 \\ \dot{\mathbf{x}}\_7 &= \dot{\boldsymbol{r}} = \mathbf{x}\_4 \end{aligned} \tag{34}$$

The general state and output matrix equations describing the dynamic behavior of the linear-time-invariant helicopter system in state space form are as follows:

$$\mathbf{x}(t) = A\mathbf{x}(t) + B\mathbf{u}(t) \tag{35}$$

$$\mathbf{y}(t) = \mathbf{C}\mathbf{x}(t) + D\mathbf{u}(t) \tag{36}$$

**Figure 8**. The control system is analysed mathematically and then simulated using Matlab software tool to validate the proposed hybrid controller. Based on the desired performance parameters, which include rise and settling time, overshoot and error steady state, the fitness function of the control problem is formulated as

**Symbol Physical unit Numerical values** *J<sup>ϵ</sup>* kg m<sup>2</sup> 1.8145 *Jr* kg m<sup>2</sup> 1.8145 *J<sup>ρ</sup>* kg m<sup>2</sup> 0.0319 *W* N 4.2591 *lm* m 0.88 *lb* m 0.35 *l<sup>ρ</sup>* m 0.17 *Kc* N/V 12

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

where, *S* is closed loop transfer function of the helicopter system, *S:tr*, *ts*, *O*,*ess* are the rise time, settling time, maximum overshoot and error steady state of the closed-loop control system. It is worth considering that the control input effort is considered in the evaluation process of the proposed stabilizing helicopter system. In this study, the design of the controller is effectively optimized by using GA tuning method which is adopted to obtain optimum elements values for LQR weighting matrices *Q* and *R*. These optimized matrices are used to calculate the optimum controller gain matrix by using Eqs. (13) and (14). However, in this study,

the LQR gain matrix is determined by using the Matlab command *'lqr'*.

In this subsection, the gain parameters *Kp*, *Ki*,*Kd* of the PID controller are calculated approximately from the feedback gain *K* of LQR controller based on GA tuning method. For this application, analyzing Eq. (15) yields the following

¼ � *<sup>k</sup>*<sup>11</sup> *<sup>k</sup>*<sup>12</sup> *<sup>k</sup>*<sup>13</sup> *<sup>k</sup>*<sup>14</sup> *<sup>k</sup>*<sup>15</sup> *<sup>k</sup>*<sup>16</sup> *<sup>k</sup>*<sup>17</sup>

helicopter system, it can express the form of PID controllers used to meet the

*Vs* ¼ *Kϵpe<sup>ϵ</sup>* þ *Kϵde*\_

If *ϵd*, *ρ<sup>d</sup>* and *rd* are the desired pitch angle, elevation angle and travel rate of the

In this study, for elevation angle, the control equation is based on the following

*<sup>ϵ</sup>* þ *K<sup>ϵ</sup><sup>i</sup>*

ð

*eϵdt* (42)

.

*<sup>k</sup>*<sup>11</sup> �*k*<sup>12</sup> *<sup>k</sup>*<sup>13</sup> �*k*<sup>14</sup> �*k*<sup>15</sup> *<sup>k</sup>*<sup>16</sup> �*k*<sup>17</sup> � �*x<sup>T</sup>* (41)

*F* ¼ 0*:*3*S:tr* þ 0*:*3*S:ts* þ 0*:*2*S:O* þ 0*:*2*S:ess* (40)

follows:

**Table 1.**

*Values of physical parameters of 3DOF helicopter system.*

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

*5.3.1 PID approximation*

control effort [22]:

*u*1 *u*2 � �

PID control equation:

where *xT* <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>*

desired output states as follows: [7, 8].

Where *A nxn* ð Þ is the system matrix, *B nxm* ð Þ is the input matrix, *C pxm* ð Þ is the output matrix, and *D mxp* ð Þ is feed forward matrix, for the designed system. Based on Eqs. (34)–(36) are rewritten as follows [21].

$$
\begin{bmatrix}
\dot{e} \\
\dot{\rho} \\
\dot{e} \\
\dot{\rho} \\
\dot{r} \\
\dot{s} \\
\dot{\gamma}
\end{bmatrix} = \begin{bmatrix}
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{Gl\_1}{J\_1} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0
\end{bmatrix} \begin{bmatrix}
e \\
\rho \\
\dot{\rho} \\
\dot{\rho} \\
r \\
3 \\
\dot{\mathcal{I}}
\end{bmatrix} + \begin{bmatrix}
0 & 0 \\
0 & 0 \\
\frac{K\_c l\_1}{J\_c} & \frac{K\_c l\_1}{J\_c} \\
\frac{K\_c l\_p}{J\_p} & -\frac{K\_c \rho}{J\_p} \\
0 & 0 \\
0 & 0 \\
0 & 0
\end{bmatrix} \begin{bmatrix}
V\_f \\
V\_b
\end{bmatrix} \tag{37}
$$

In this study, for the purpose of control system design, the model of the system is formulated in state space form using the physical parameters values listed in **Table 1** [21]. Based on Eq. (37) and using the parameters values in **Table 1**, the state equation of the system is given by Eq. (39):

*ϵ ρ r* ¼ *ϵ ρ ϵ*\_ *ρ*\_ *r* ʓ *γ* þ 0 0 0 0 0 0 *V <sup>f</sup> Vb* � � (38) *ϵ*\_ *ρ*\_ €*ϵ* €*ρ r*\_ ʓ\_ *γ*\_ 0 0 10000 0 0 01000 0 0 00000 0 0 00000 0 2*:*065 0 0 0 0 0 1 0 00000 0 0 00100 *ϵ ρ ϵ*\_ *ρ*\_ *r* ʓ *γ* þ 0 0 0 0 *:*8197 5*:*8197 *:*949 �63*:*949 0 0 0 0 0 0 *V <sup>f</sup> Vb* � � (39)

### **5.3 Helicopter control system design**

Based on step input, a hybrid controller is designed for the following desired performance parameters: rise time (*tr*) less than 10 ms, settling time (*ts*) less than 30 ms, maximum overshoot percentage, (*MO*) less than 5%.

Under the assumption that the desired system states are zero the block diagram of the proposed helicopter control system based on the LQR controller is shown in

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


**Table 1.**

The general state and output matrix equations describing the dynamic behavior of the linear-time-invariant helicopter system in state space form are as follows:

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

Where *A nxn* ð Þ is the system matrix, *B nxm* ð Þ is the input matrix, *C pxm* ð Þ is the output matrix, and *D mxp* ð Þ is feed forward matrix, for the designed system. Based

In this study, for the purpose of control system design, the model of the system

 

Based on step input, a hybrid controller is designed for the following desired performance parameters: rise time (*tr*) less than 10 ms, settling time (*ts*) less than 30

Under the assumption that the desired system states are zero the block diagram of the proposed helicopter control system based on the LQR controller is shown in

*ϵ ρ ϵ*\_ *ρ*\_ *r* ʓ *γ*

þ

*ϵ ρ ϵ*\_ *ρ*\_ *r* ʓ *γ*

þ

 

 

*V <sup>f</sup> Vb* � �

is formulated in state space form using the physical parameters values listed in **Table 1** [21]. Based on Eq. (37) and using the parameters values in **Table 1**, the state

*ϵ ρ ϵ*\_ *ρ*\_ *r* ʓ *γ*

þ

*Kcl<sup>ρ</sup> Jp*

on Eqs. (34)–(36) are rewritten as follows [21].

0 0 10000 0 0 01000 0 0 00000 0 0 00000

1 0 00000 0 0 00100

 

0 0 10000 0 0 01000 0 0 00000 0 0 00000 0 2*:*065 0 0 0 0 0 1 0 00000 0 0 00100

ms, maximum overshoot percentage, (*MO*) less than 5%.

*ϵ*\_ *ρ*\_ €*ϵ* €*ρ r*\_ ʓ\_ *γ*\_

*ϵ ρ r*

 

**5.3 Helicopter control system design**

 

*ϵ*\_ *ρ*\_ €*ϵ* €*ρ r*\_ ʓ\_ *γ*\_

 *Gl*<sup>1</sup> *Jt*

equation of the system is given by Eq. (39):

\_ *x t*ðÞ¼ *Ax t*ðÞþ *Bu t*ð Þ (35) *y t*ðÞ¼ *Cx t*ðÞþ *Du t*ð Þ (36)

*Kcl*<sup>1</sup> *Je*

*V <sup>f</sup> Vb* � �

(37)

(38)

*V <sup>f</sup> Vb* � �

(39)

� *Kc<sup>ρ</sup> Jρ*

*Values of physical parameters of 3DOF helicopter system.*

**Figure 8**. The control system is analysed mathematically and then simulated using Matlab software tool to validate the proposed hybrid controller. Based on the desired performance parameters, which include rise and settling time, overshoot and error steady state, the fitness function of the control problem is formulated as follows:

$$F = \mathbf{0.3S.t}\_r + \mathbf{0.3S.t}\_t + \mathbf{0.2S.O} + \mathbf{0.2S.e\_{it}} \tag{40}$$

where, *S* is closed loop transfer function of the helicopter system, *S:tr*, *ts*, *O*,*ess* are the rise time, settling time, maximum overshoot and error steady state of the closed-loop control system. It is worth considering that the control input effort is considered in the evaluation process of the proposed stabilizing helicopter system. In this study, the design of the controller is effectively optimized by using GA tuning method which is adopted to obtain optimum elements values for LQR weighting matrices *Q* and *R*. These optimized matrices are used to calculate the optimum controller gain matrix by using Eqs. (13) and (14). However, in this study, the LQR gain matrix is determined by using the Matlab command *'lqr'*.
