**3. Actuator testing: a case study**

The flight actuator under test is a brushless DC motor. A permanent magnet synchronous motor (PMSM) is used as the loading motor. For simplicity, the load calculator is used in proportional mode. In proportional mode, the output of the load calculator is directly pro‐ portional to the reference command of the flight actuator. Let *θ<sup>a</sup>* represents the reference command of flight actuator, then the output of the load calculator *Tr* is *Cθa* where "*C*" represents the proportionality constant. The parameter *Tr* is set as reference command of the torque loading motor. From **Figure 2**, the reference command signal of the flight actuator *θa*is constructed from a real test scenario of a flight vehicle.

**Figure 2.** Reference signal command of flight actuator.

Before discussing the control problem, basic understandings of system dynamics is a pre requisite. A PMSM is used as a loading motor. The voltage and torque balance equations of PMSM loading motor are written as

$$\begin{aligned} \mu\_d &= i\_d R\_S + L\_{sd} \frac{di\_d}{dt} - PL\_{sj} i\_q \mathbf{w}\_m \\ \mu\_q &= i\_q R\_S + L\_{sq} \frac{di\_q}{dt} + PL\_{sq} i\_d \mathbf{w}\_m + P \boldsymbol{\nu}\_m \mathbf{w}\_m \\ T\_e &= \frac{3P}{2} [\boldsymbol{\nu}\_m \dot{i}\_q + \left(L\_{sd} - L\_{sq}\right) i\_d i\_q = J \frac{d \boldsymbol{\nu}\_m}{dt} + B \boldsymbol{\nu}\_m + T\_f + T\_L \end{aligned} \tag{1}$$

In Eq. (1) [*id iq*] is the d-axis and q-axis current vector, [*ud uq*] represents d-axis and q-axis voltage vector, *wm* is the angular velocity of torque motor, [*Lsq Lsd Rs*] represents inductances and resistances of ELS motor,[*P* , *ψm*] represents the number of pole pairs and magnetic flux of rotor, [*J*, *B*] represents inertia and damping coefficient, *kt* <sup>=</sup> <sup>3</sup>*<sup>P</sup>* <sup>2</sup> *ψm*is torque constant, *kb* = *Pψm* is back emf constant, and [*Te* , *Tf* , *TL*] is the electromagnetic torque, friction torque and output load torque. Assuming that inertia and damping coefficient of torque sensor is very small, thus the dynamics can be written as

$$T\_L = K\_s \left(\theta\_m - \theta\_a\right) \tag{2}$$

Here, [*θ<sup>m</sup> θa*] represents the angular position of loading motor and actuator, *Ks* is the total stiffness of torque sensor and connecting shaft. To achieve largest torque operation and to eliminate coupling effect between speed and currents, we set d-axis reference current *i d* ′ equal to zero. Considering the effect of PWM driver and the current feedback, Eq. (2) can be written as given in [4]. For simplicity, we set *Lsq* = *L* and *Rs* = *R*

portional to the reference command of the flight actuator. Let *θ<sup>a</sup>* represents the reference command of flight actuator, then the output of the load calculator *Tr* is *Cθa* where "*C*" represents the proportionality constant. The parameter *Tr* is set as reference command of the torque loading motor. From **Figure 2**, the reference command signal of the flight actuator *θa*is

Before discussing the control problem, basic understandings of system dynamics is a pre requisite. A PMSM is used as a loading motor. The voltage and torque balance equations of

*<sup>m</sup> e m q sd sq d q mfL*

In Eq. (1) [*id iq*] is the d-axis and q-axis current vector, [*ud uq*] represents d-axis and q-axis voltage vector, *wm* is the angular velocity of torque motor, [*Lsq Lsd Rs*] represents inductances and resistances of ELS motor,[*P* , *ψm*] represents the number of pole pairs and magnetic flux of rotor,

torque. Assuming that inertia and damping coefficient of torque sensor is very small, thus the

Here, [*θ<sup>m</sup> θa*] represents the angular position of loading motor and actuator, *Ks* is the total stiffness of torque sensor and connecting shaft. To achieve largest torque operation and to

 ( ) *L sm a T K* = q q

= + - = + ++

*dt*

, *TL*] is the electromagnetic torque, friction torque and output load

(1)

<sup>2</sup> *ψm*is torque constant, *kb* = *Pψm* is back

(2)

y

*<sup>P</sup> dw T i L L i i J Bw T T*

constructed from a real test scenario of a flight vehicle.

80 Recent Progress in Some Aircraft Technologies

**Figure 2.** Reference signal command of flight actuator.

( ) <sup>3</sup> [ <sup>2</sup>

y

*d d d s sd s q m*

*di u i R L PL i w dt di u i R L PL i w P w dt*

=+ -

[*J*, *B*] represents inertia and damping coefficient, *kt* <sup>=</sup> <sup>3</sup>*<sup>P</sup>*

*q q q s sq sq d m m m*

=+ + +

PMSM loading motor are written as

emf constant, and [*Te* , *Tf*

dynamics can be written as

$$\frac{d\dot{q}\_q}{dt} = -\frac{R}{L}\dot{i}\_q - \frac{K\_b}{L}\omega\_m + \frac{K\_\nu K\_{\dot{l}}}{L} \tag{3}$$

Here, *ki* is the current controller gain. Current feedback is assumed to be unity. Assuming that load and friction torque are zero and taking Laplace transform of Eq. (3) and eliminating *I*(*s*) , we get the transfer function from input voltage to output position as

$$\frac{\theta\_m}{\mu\_q} = \frac{\frac{K\_v K\_i}{K\_b}}{S \left(t\_m t\_c S^2 + t\_m S + 1\right)}\tag{4}$$

In Eq. (4), *tm* <sup>=</sup> *RJ kt kb* is the electromechanical time constant, *te* <sup>=</sup> *<sup>L</sup> <sup>R</sup>* is the electromagnetic time constant. Replace Eq. (2) into Eq. (2), the simplified relation can be written as

$$\begin{aligned} T\_L &= \frac{k\_v k\_l k\_l \mu - \text{s} [(Ls + R)(Js + B) + k\_b k\_t] K\_s \theta\_a}{D(s)} \\ D(s) &= \text{s} \left[ (Ls + R)(Js + B) + K\_b K\_t \right] + T \left(Ls + R \right) \end{aligned} \tag{5}$$

From the numerator of Eq. (5), it is concluded that extra torque is caused by the effect of*θa*, *θ*˙ *<sup>a</sup>* and *θ*¨ *<sup>a</sup>*. If the reference input command of loading torque motor is zero, i.e.*u* = 0, then Eq. (5) is reduced to the following simplified relation

$$T\_L = \frac{-s[(Ls+R)(Js+B) + k\_bk\_t]K\_s\theta\_a}{D(s)}\tag{6}$$

From Eq. (6), it is concluded that extra torque is acting on the loading torque motor even if the reference input command *u* is zero. Extra torque is a function of the acceleration and velocity components of the actuators movement. After some simplifications, the state equation representation of electrical load simulator is written as

$$\ddot{T}\_L = -a\dot{T}\_L + bu - c\mathcal{f}\left(T\_{\text{extra}}, T\_f, T\_L\right) \tag{7}$$

In Eq. (7) the parameters are defined as

$$\begin{aligned} a &= \frac{k\_t k\_b}{J R} + \frac{B}{J} \\ b &= \frac{K\_s k\_t}{J R} \\ c &= \frac{K\_s}{J} \end{aligned}$$

$$f\left(T\_{\text{extra}}, T\_f\right) = \frac{K\_s}{J} \left(T\_{\text{gft}} + T\_f\right)$$

In Eq. (7) the nonlinear friction is represented using LuGre model, which is written as

$$\begin{aligned} T\_f &= a\_0 Z + a\_1 \dot{Z} + a\_2 \\ \dot{Z} &= \nu - \frac{a\_0 \left| \nu \right|}{\text{g}(\nu)} \\ \text{g}(\nu) &= f\_\mathcal{C} + (f\_\mathcal{C} - f\_\mathcal{S})e^{\left[\frac{\mathcal{V}}{\mathcal{V}\_s}\right]^2}\nu \end{aligned} \tag{8}$$

In Eq. (8) the parameter *g*(*v*) is the Stribeck effect, *vs* is the Stribeck velocity, *fc* is coulomb friction, *fs* is static friction, *z* is the average bristle defection,*a*0 is the stiffness of the bristles, *a*1 is the damping term and *a*2 is the viscous friction coefficient. Now to realistically apply the loading torque on flight actuator, a feedback torque control system plays vital role. In this study, adaptive fuzzy sliding mode control system is used to formulate the torque control system.

#### **3.1. Adaptive fuzzy sliding mode control for electrical load simulator system**

Sliding mode is a robust control method which has been widely applied to many nonlinear systems [10–15]. This section provides an overview of derivations of torque control system for electrical load simulator's system. Let *TL*be the output load torque and *Tr*be the desired torque signal, we define tracking error vector as

$$\begin{array}{l}\mathbf{e} = T\_L - T\_r \\ \dot{\mathbf{e}} = \dot{T}\_L - \dot{T}\_r \\ \ddot{\mathbf{e}} = \ddot{T}\_L - \dddot{T}\_r \end{array} \tag{9}$$

Error surface vector is defined as

*T aT bu cf T T T L L* =- + - ( *extra f L* , , ) && & (7)

( ) ( )

*<sup>K</sup> fT T T T <sup>J</sup>*

*<sup>s</sup> extra f sft f*

= +

2

(8)

[ ]*s*

*v v*

*t b*

*k k <sup>B</sup> <sup>a</sup> JR J*

= +

*s t*

*K k <sup>b</sup> JR*

=

*K c J*

=

*f*

&

signal, we define tracking error vector as

*s*

,

In Eq. (7) the nonlinear friction is represented using LuGre model, which is written as

012 0 | | ( )

&

*T a Z aZ a*

= ++

*a v Z v g v*

= -

() ( )

=+ -

**3.1. Adaptive fuzzy sliding mode control for electrical load simulator system**

*c cs*

*gv f f f e v*

In Eq. (8) the parameter *g*(*v*) is the Stribeck effect, *vs* is the Stribeck velocity, *fc* is coulomb friction, *fs* is static friction, *z* is the average bristle defection,*a*0 is the stiffness of the bristles, *a*1 is the damping term and *a*2 is the viscous friction coefficient. Now to realistically apply the loading torque on flight actuator, a feedback torque control system plays vital role. In this study, adaptive fuzzy sliding mode control system is used to formulate the torque control system.

Sliding mode is a robust control method which has been widely applied to many nonlinear systems [10–15]. This section provides an overview of derivations of torque control system for electrical load simulator's system. Let *TL*be the output load torque and *Tr*be the desired torque

> *L r L r L r*

& & & (9)

*eT T eT T eT T*

é ù = ê ú = - ê ú ê ú ë û = - && && &&

In Eq. (7) the parameters are defined as

82 Recent Progress in Some Aircraft Technologies

$$\begin{bmatrix} \mathbf{s} = \dot{\mathbf{e}} + \lambda \mathbf{e} \\ \dot{\mathbf{s}} = \ddot{\mathbf{e}} + \lambda \dot{\mathbf{e}} \end{bmatrix} \tag{10}$$

Assuming that the nominal parameters of the system are known, then by combining Eqs. (7), (9) and (10) yields

$$\dot{s} = \left(-a\dot{T}\_L + bu - f\left(T\_{\text{extra}}, T\_f\right)\right) - \ddot{T}\_r + \lambda \dot{\varepsilon} \tag{11}$$

The control law is given by

$$u = \frac{1}{b} \left( a\dot{T}\_L + \hat{f} \left( T\_{\text{extra}}, T\_f \rfloor \theta \right) + \ddot{T}\_r - \lambda \dot{e} \right) - \frac{1}{b} \left( K\_d \, s - \text{w.sgn}(s) \right) \tag{12}$$

From Eq. (12), it can be analyzed that the total control effort *u* is the sum of three terms

$$
\mu = \mu\_T + \mu\_f + \mu\_{\text{extra}} \tag{13}
$$

Here, *uT* is the control effort for torque tracking, *uf* is the friction compensation control and *uextra* is the extra torque compensation control. The unknown function *f* ˜(*Textra*, *Tf* |*θ*)is the estimated output of fuzzy logic for friction and extra torque.

#### *3.1.1. Stability analysis*

To prove stability of the closed loop, the Lyapunov function is chosen as

$$\begin{aligned} V &= \frac{1}{2} \left( \mathbf{s}^2 + \sum\_{i=1}^n \eta\_i \boldsymbol{\varTheta}^2 \right) \\\\ \bullet \bullet &= \mathbf{s} \mathbf{s} + \sum\_{i=1}^n \eta\_i \boldsymbol{\varTheta}\_i \dot{\varTheta}\_i \end{aligned} \tag{14}$$

Here*θ*˜*<sup>i</sup>* <sup>=</sup>*<sup>θ</sup>* ^ *<sup>i</sup>* −*θ<sup>i</sup>* . Combine Eq. (11) and Eq. (14)

$$\stackrel{\star}{V} = \text{s} \left( \left( -a \stackrel{\star}{T}\_{L} + bu - f \left( T\_{extu'} \stackrel{\star}{T}\_{f} \right) \right) \stackrel{\star}{-} \stackrel{\star}{T}\_{n} \right) + \sum\_{i=1}^{n} \eta\_{i} \mathcal{D}\_{i} \dot{\mathcal{D}}\_{i} \tag{15}$$

Define *T*˙ *<sup>n</sup>* <sup>=</sup> <sup>−</sup>*<sup>T</sup>* ¨ *<sup>r</sup>* + *λe*˙ and combine Eq. (12) into Eq. (15)

$$\begin{aligned} \overset{\bullet}{W} &= s \begin{pmatrix} \overset{\bullet}{\cdot}\_{\cdot L} & \overset{\bullet}{\cdot}\_{\cdot L} + \overset{\bullet}{f} \{ T\_{extn'} \,^\*T\_f \} \\\\ -f \{ T\_{extn'} \,^\*T\_f \} - \overset{\bullet}{T}\_n \end{pmatrix} + \overset{\bullet}{T}\_n \overset{\bullet}{\cdot}\_{\cdot L} \dot{\sigma}\_i \dot{\sigma}\_j \end{aligned} \tag{16}$$

The fuzzy approximation error is defined as [3]

$$\begin{aligned} e\_f &= f\left(T\_{\text{extra}}, T\_f\right) - \tilde{f}\left(T\_{\text{extra}}, T\_f\right) \mid \boldsymbol{\theta}^\*\\ \theta\_i \xi\_i \left(\dot{\boldsymbol{\theta}}\right) &= \hat{f}\left(T\_{\text{extra}}, T\_f\right) / \boldsymbol{\theta} - \tilde{f}\left(T\_{\text{extra}}, T\_f\right) \mid \boldsymbol{\theta}^\* \end{aligned} \tag{17}$$

Combining Eq. (16) and Eq. (17) yields [16]

$$\hat{V} = \operatorname{s}\left\{ \hat{f}\left(\mathbf{T}\_{\text{extr}\nu} \mid \mathbf{T}\_{f}\right) \right\} \boldsymbol{\Theta} - f\left\{ \mathbf{T}\_{\text{extr}\nu} \mid \mathbf{T}\_{f}\right\} \mathbf{-K}\_{d} \mathbf{s} - \boldsymbol{w} \operatorname{sgn}\{\mathbf{s}\} \right\} \\ + \sum\_{i=1}^{n} \eta\_{i} \boldsymbol{\Theta}\_{i} \dot{\boldsymbol{\Theta}}\_{i} \tag{18}$$

$$\begin{aligned} \overset{\bullet}{W} &= \left\{ \overset{\bullet}{f} \{ T\_{extn'} \, ^\*T\_f \} \, \middle| \, \Theta - \overleftarrow{f} \{ T\_{extn'} \, ^\*T\_f \} \, \middle| \, \Theta^\* - \left\{ \left( f \left( T\_{extn'} \, ^\*T\_f \right) - \overleftarrow{f} \left( T\_{extn'} \, ^\*T\_f \right) \mid \Theta^\* \right) \right\} \end{aligned} \tag{19}$$
  $\begin{aligned} -K\_d s - w \text{.sgn}(s) \Big\} + \sum\_{i=1}^n \eta\_i \partial\_i \dot{\varTheta}\_i \end{aligned} \tag{19}$ 

Using Eq. (19) the following adaptive law is derived

$$\dot{\boldsymbol{\Theta}}\_{i} = -\eta\_{i}\,^{\ast}\mathbf{s}\_{i}\,^{\xi}\_{i} \begin{pmatrix} \boldsymbol{\Theta}\_{\prime} \ \dot{\boldsymbol{\Theta}} \end{pmatrix} \tag{20}$$

By replacing Eq. (20) in Eq. (19) and simplifying

$$\dot{V} = \mathbf{s} \left( -e\_f - K\_d \mathbf{s} - \mathbf{w}. \text{sgn}(\mathbf{s}) \right) \tag{21}$$

It is assumed that ideally fuzzy compensating error *ef* is approaching zero, and by choosing *Kd* > 0 it can be shown that

$$
\dot{V} = -sK\_d s \le 0 \tag{22}
$$

#### *3.1.2. Results and discussion*

Define *T*˙ *<sup>n</sup>* <sup>=</sup> <sup>−</sup>*<sup>T</sup>*

¨

84 Recent Progress in Some Aircraft Technologies

*V* .

=*s*( −*aT* . *L* + *aT* .

− *f* (*Textra*, *Tf* ) −*T*

The fuzzy approximation error is defined as [3]

qx q

Combining Eq. (16) and Eq. (17) yields [16]

(*Textra*, *Tf* )/ *θ* – *f*

*i*=1 *n ηi θ*˜*i θ*˜*i* ˙

Using Eq. (19) the following adaptive law is derived

By replacing Eq. (20) in Eq. (19) and simplifying

It is assumed that ideally fuzzy compensating error *ef*

<sup>−</sup> *Kd <sup>s</sup>* <sup>−</sup>*w*. sgn(*s*)) <sup>+</sup> ∑

*V* .

*Kd* > 0 it can be shown that

*V* . =*s*( *f* ^ =*s*( *f* ^

*<sup>r</sup>* + *λe*˙ and combine Eq. (12) into Eq. (15)

*L* + *f* ^

> ¨ *<sup>n</sup>*) <sup>+</sup> ∑ *i*=1 *n ηi θ*˜*i θ*˜*i* ˙

= -

=

(*Textra*, *Tf* )<sup>|</sup> *<sup>θ</sup>* <sup>+</sup> *<sup>T</sup>*

( ) ( )

%

*i i extra f extra f*

*f extra f extra f*

*e fT T fT T*

˜(*Textra*, *Tf* ) <sup>|</sup> *<sup>θ</sup>* \*

*θ*˜*i* ˙ <sup>=</sup> <sup>−</sup>*η<sup>i</sup>* −1 *si ξi* (*θ*, *<sup>θ</sup>* .

( ) ( ) ( )

(*Textra*, *Tf* )/ *<sup>θ</sup>* <sup>−</sup> *<sup>f</sup>* (*Textra*, *Tf* ) <sup>−</sup> *Kd <sup>s</sup>* <sup>−</sup> *<sup>w</sup>*.*sgn*(*s*)) <sup>+</sup> ∑

, , |

*fT T fT T*

q

, /– , | ˆ

¨

\*

q

−(( *f* (*Textra*, *Tf* ) − *f*

*<sup>n</sup>* − *Kd s* −*w*.*sgn*(*s*)

\*

*i*=1 *n ηi θ*˜*i θ*˜*i*

˜(*Textra*, *Tf* ) <sup>|</sup>*<sup>θ</sup>* \*

) (20)

is approaching zero, and by choosing

*V s e Ks w s* =- - - ( *f d* .sgn( )) & (21)

<sup>0</sup> *<sup>d</sup> V sK s* =- £ & (22)

˙ (18)

))

q

& % (17)

(16)

(19)

For simulations and validity of the proposed control scheme, the following parameters are used. Total inertia of the system is given as *J* = 0.04*Kg*/*m*<sup>2</sup> , resistance *R* = 7.5*Ω*, inductance *L* = 1*mH*, motor torque constant *kt* = 5.7325*Nm*/*A*, back emf constant *kb* = 5.7325*Nm*/*V*, viscous coefficient *B* = 0.244*Nm*/*rad*/*s*, torque sensor stiffness *Ks* = 950*Nm*/*rad*, static friction *fs* = 3*Nm*, coulomb friction *fc* = 2.7*Nm*, *σ*0 = 260*Nm*/*rad*, *σ*1 = 2.5*Nm* − *s*/*rad*, *σ*0 = 0.022*Nm* − *s*/*rad*and Stribeck velocity *α* = 0.001*rad*/*s*. The parameters of the controller are given as fuzzy learning rate *ηi* = 0.0001, amplifier gain *ku* = 10, *kd* = 10, *w* = 1.5, *λ* = 15.

#### *3.1.3. Loading motor torque tracking performance*

The testing of actuator is performed under the loading torque *Tr* = 16*θa* where "*C* = *10*". From **Figure 3**, it is concluded that the output torque applied by the loading motor is exactly the same as the reference loading torque.

**Figure 3.** Loading motor closed loop performance.

**Figure 4.** Flight actuator closed loop performance with aerodynamics load.

## *3.1.4. Flight actuator angle tracking performance under load*

**Figure 4** presents the testing results and the qualification of the autopilots of the flight actuators under the aerodynamic loading shown in **Figure 3**, which is mechanically transmitted from loading motor. From the results provided, it is concluded that the flight actuator under can withstand the non-linear profile of the aerodynamic load supplied. Moreover, the autopilot position controller is also robust and the position tracking errors are small enough.
