**2. System Modeling**

#### **2.1. System analysis**

The schematic diagram of the force loader unit is shown in **Figures 1** and **2**. The loader unit consists of one loader and one steel bar, which is connected to the load cell and linear motor actuator. The linear motor actuator, used to apply force, consists of a motor with linear motion and an encoder. The loader is attached to the steel bar and then through a load cell to the linear motor. In short, the force is applied on the material plate by a loader unit which is connected to a linear motor actuator through a steel bar, and the applied force is recorded by a force sensor.

The load applied on the material plate varies as a ramp function. The user can select the slope of the ramp function by setting the maximum force in a finite time period on the touch screen, and can perform the test under different forces ranging from 0 to 300 lbs. The control objective is to ensure the applied force track the reference force command for measuring the material plate's yield stress qualified or not.

#### **2.2. Mathematic model of the motor-loader unit**

DC motors are widely used as actuators for high-precision servo control owing to its good working characteristics and simple mathematical model. Mechanical resonance phenomena are ubiquitous because the transmission shaft is not completely rigid and will be distorted under force. For the servo-motor-drive system, considering the mechanical resonance phenomena, the double-mass structure model is commonly used to describe such dynamical systems.

The electrical equilibrium equation can be written as [27]

*i* + *La* \_\_*di dt* <sup>+</sup> *Ke θ* ̇

The motor output torque *Tm* is determined by the armature current as

is the counter-electromotive force coefficient, *θm* is the motor rotating angle and *um* is the

with the motor speed. **Figure 3** illustrates the mechanical parameter of the motor with a load.

*<sup>m</sup>* reflects that the back electromotive force (EMF) has a linear relationship

A CMAC-Based Systematic Design Approach of an Adaptive Embedded Control Force Loading…

http://dx.doi.org/10.5772/intechopen.71420

259

*<sup>m</sup>* = *um* (1)

*i* (2)

is the armature inductance, *i* is the armature current,

*Ra*

**Figure 3.** Motor and load double mass diagram.

**Figure 2.** Photo of the loader unit.

*θ* ̇

is the armature resistance, *La*

*Tm* = *Kt*

where *Ra*

motor voltage. *Ke*

*Ke*

**Figure 1.** Schematic drawing of the loader unit.

A CMAC-Based Systematic Design Approach of an Adaptive Embedded Control Force Loading… http://dx.doi.org/10.5772/intechopen.71420 259

**Figure 2.** Photo of the loader unit.

**2. System Modeling**

plate's yield stress qualified or not.

**2.2. Mathematic model of the motor-loader unit**

Loader

**Figure 1.** Schematic drawing of the loader unit.

Steel bar

Load cell

The schematic diagram of the force loader unit is shown in **Figures 1** and **2**. The loader unit consists of one loader and one steel bar, which is connected to the load cell and linear motor actuator. The linear motor actuator, used to apply force, consists of a motor with linear motion and an encoder. The loader is attached to the steel bar and then through a load cell to the linear motor. In short, the force is applied on the material plate by a loader unit which is connected to a linear motor actuator through a steel bar, and the applied force is recorded by a force sensor. The load applied on the material plate varies as a ramp function. The user can select the slope of the ramp function by setting the maximum force in a finite time period on the touch screen, and can perform the test under different forces ranging from 0 to 300 lbs. The control objective is to ensure the applied force track the reference force command for measuring the material

DC motors are widely used as actuators for high-precision servo control owing to its good working characteristics and simple mathematical model. Mechanical resonance phenomena are ubiquitous because the transmission shaft is not completely rigid and will be distorted under force. For the servo-motor-drive system, considering the mechanical resonance phenomena, the double-mass structure model is commonly used to describe such dynamical systems.

Support Force Force Support

Linear motor

Force sensor

Material plate

**2.1. System analysis**

258 Adaptive Robust Control Systems

**Figure 3.** Motor and load double mass diagram.

The electrical equilibrium equation can be written as [27]

$$R\_{\dot{a}}\dot{\imath} + L\_{\dot{a}}\frac{d\dot{\imath}}{dt} + K\_{\dot{\imath}}\dot{\Theta}\_{m} = \omega\_{m} \tag{1}$$

where *Ra* is the armature resistance, *La* is the armature inductance, *i* is the armature current, *Ke* is the counter-electromotive force coefficient, *θm* is the motor rotating angle and *um* is the motor voltage. *Ke θ* ̇ *<sup>m</sup>* reflects that the back electromotive force (EMF) has a linear relationship with the motor speed. **Figure 3** illustrates the mechanical parameter of the motor with a load.

The motor output torque *Tm* is determined by the armature current as

$$T\_w = Kj\tag{2}$$

where *Kt* is the electromagnetism-torque constant.

The torque equilibrium equation of the motor is

$$J\_m \ddot{\Theta}\_m + B\_m \dot{\Theta}\_m = T\_m - T\_l \tag{3}$$

where *Jm* is the moment of inertia of the motor, *Bm* is the viscidity damping coefficient of the motor extremity and *Tl* is the torque of the load extremity. *Tl* can be represented as

$$T\_l = K\_s(\Theta\_m - \Theta\_l) \tag{4}$$

*GH*(*s*) <sup>=</sup> *uh*

which is a low-pass filter.

**Figure 4.** Open-loop block diagram.

where *Ka*

*Ks*

*θl*

where *Ja*

can be simplified as

**2.4. Simplified model**

(*s*) \_\_\_\_ *<sup>u</sup>*(*s*) <sup>⋅</sup> \_\_1

drivers [28]. The PWM power amplifiers can be represented as

*GA*(*s*) <sup>=</sup> *<sup>K</sup>* \_\_\_\_\_*<sup>a</sup>*

is the amplifying multiple and *f*

*GS*

Without considering the disturbance torque *Td*

 = ∞ and *θ<sup>l</sup>*

 = *Bm* + *Bl*

*Kt d*3 *θ* \_\_\_\_*<sup>m</sup> dt*<sup>3</sup> <sup>+</sup> *Ra J <sup>a</sup>* + *La <sup>B</sup>* \_\_\_\_\_\_\_\_*<sup>a</sup> Kt*

If the motor armature inductance is regarded as *La*

*l θ*¨ *<sup>l</sup>* + *Bl θ* ̇

following dynamical relationship

 ≈ *θm*.

*J*

, *Ba*

*Tl* = *J*

(*s*)/*θm*(*s*) ≈ 1, *θ<sup>l</sup>*

By considering *Ks*

 = *Jm* + *Jl*

motor can be obtained as

*La <sup>J</sup>* \_\_\_*<sup>a</sup>*

*<sup>T</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>*Ts* \_\_\_\_\_ *<sup>s</sup>* <sup>⋅</sup> \_\_1

In a high-precision servo system, PWM-based amplifiers are commonly used as the motor

The load cell signal conditioner linearly converts the force into a voltage signal, and its model

*s*

*<sup>l</sup>* = *Ks*(*θ<sup>m</sup>* − *θ<sup>l</sup>*

*a θ*¨ *<sup>m</sup>* + *Ba θ* ̇

> *d*2 *θ* \_\_\_\_*<sup>m</sup> dt*<sup>2</sup> <sup>+</sup> (*Ke* <sup>+</sup>

) ⇒ *θl* (*s*) \_\_\_\_\_

 ≡ *θm*, the motor-loader model can be simplified as

*Ra B* \_\_\_\_*<sup>a</sup> Kt* ) *dθ*\_\_\_\_*<sup>m</sup>*

is very large and the time constant is very small, therefore in the lower frequency range

1 + \_\_\_1 2 *f s* *<sup>T</sup>* <sup>≈</sup> \_\_\_\_ *<sup>T</sup>* \_\_ *T* <sup>2</sup> *s* + 1

A CMAC-Based Systematic Design Approach of an Adaptive Embedded Control Force Loading…

is the modulation frequency.

⋅ \_\_1

*<sup>T</sup>* (9)

http://dx.doi.org/10.5772/intechopen.71420

261

*<sup>s</sup>* (10)

(*s*) = *KSen* (11)

, the angles of the motor and the load have the

*<sup>m</sup>* = *Tm* (13)

 ≈ 0, the system can be further simplified as

*dt* <sup>=</sup> *um* (14)

(12)

*<sup>θ</sup>m*(*s*) <sup>=</sup> *<sup>K</sup>* \_\_\_\_\_\_\_\_\_ *<sup>s</sup> J*

. Deleting the intermediate variables, the dynamical equation of the

*<sup>l</sup> s* <sup>2</sup> + *Bl s* + *Ks*

where *θ<sup>l</sup>* is the rotating angle of the load shaft and *Ks* is the mechanical rigidity of the rotating shaft.

The torque equilibrium equation of the load is

$$J\_l \ddot{\boldsymbol{\Theta}}\_l \star \mathbf{B}\_l \boldsymbol{\Theta}\_l = \boldsymbol{T}\_l - \boldsymbol{T}\_d \tag{5}$$

where *J l* is the moment of inertia of the load, *Bl* is the damping coefficient on the load side and *Td* is the disturbance torque, including the friction torque, the coupling torque and the external disturbance torque. The displacement of the loader mass generated from the linear motor has a linear relationship with the rotating angle of the load shaft, which can be represented as

$$\text{Dis} = \text{K}\_{\text{D}} \cdot \Theta\_{l} \tag{6}$$

The force generated on the solar panel glass has an almost linear relationship with the deformation of the glass, which can be represented as

$$F\_m = K\_p \cdot D \text{is} \tag{7}$$

#### **2.3. Mathematical models of other components**

The control objective is to track the input command through feedback control based on the signal measured by the force sensor. In addition to the motor-loader and the micro controller unit (MCU), other components included in the control loop are a digital to analog converter (DAC), an amplifier and a load cell. **Figure 4** illustrates the open-loop plant structure.

The DAC converter is basically a zero-order hold. Assume the controller's output is *u*(*t*), the amplifier's input is *uh* (*t*) and the system's sampling time is *T*. The zero-order hold can be represented as

$$
u\_h(t) = \mu(kT), kT \le T \le (k+1)T\tag{8}$$

Considering the sampling process, by replacing *e*<sup>−</sup>*Ts* with *<sup>e</sup>* <sup>−</sup>*Ts* <sup>≈</sup> (−*<sup>s</sup>* <sup>+</sup> \_\_2 *<sup>T</sup>*)/(*<sup>s</sup>* <sup>+</sup> \_\_2 *T*), the DAC converter and the zero-order hold can be described as

**Figure 4.** Open-loop block diagram.

where *Kt*

where *θ<sup>l</sup>*

shaft.

where *J l*

*Td*

is the electromagnetism-torque constant.

*mθ*¨ *<sup>m</sup>* + *Bmθ* ̇

> *l θ*¨ *<sup>l</sup>* + *Bl θ* ̇

where *Jm* is the moment of inertia of the motor, *Bm* is the viscidity damping coefficient of the

 is the disturbance torque, including the friction torque, the coupling torque and the external disturbance torque. The displacement of the loader mass generated from the linear motor has a linear relationship with the rotating angle of the load shaft, which can be represented as

*Dis* = *KD* ⋅ *θ<sup>l</sup>* (6)

The force generated on the solar panel glass has an almost linear relationship with the defor-

*Fm* = *KF* ⋅ *Dis* (7)

The control objective is to track the input command through feedback control based on the signal measured by the force sensor. In addition to the motor-loader and the micro controller unit (MCU), other components included in the control loop are a digital to analog converter

The DAC converter is basically a zero-order hold. Assume the controller's output is *u*(*t*), the

(*t*) and the system's sampling time is *T*. The zero-order hold can be

(*t*) = *u*(*kT*), *kT* ≤ *T* < (*k* + 1)*T* (8)

*<sup>T</sup>*)/(*<sup>s</sup>* <sup>+</sup> \_\_2

*T*), the DAC con-

(DAC), an amplifier and a load cell. **Figure 4** illustrates the open-loop plant structure.

Considering the sampling process, by replacing *e*<sup>−</sup>*Ts* with *<sup>e</sup>* <sup>−</sup>*Ts* <sup>≈</sup> (−*<sup>s</sup>* <sup>+</sup> \_\_2

is the torque of the load extremity. *Tl*

*<sup>m</sup>* = *Tm* − *Tl* (3)

can be represented as

) (4)

is the mechanical rigidity of the rotating

*<sup>l</sup>* = *Tl* − *Td* (5)

is the damping coefficient on the load side and

The torque equilibrium equation of the motor is

*Tl* = *Ks*(*θ<sup>m</sup>* − *θ<sup>l</sup>*

is the moment of inertia of the load, *Bl*

mation of the glass, which can be represented as

**2.3. Mathematical models of other components**

verter and the zero-order hold can be described as

amplifier's input is *uh*

*uh*

represented as

The torque equilibrium equation of the load is

*J*

is the rotating angle of the load shaft and *Ks*

*J*

motor extremity and *Tl*

260 Adaptive Robust Control Systems

$$G\_{\rm H}(\rm s) = \frac{u\_{\rm s}(s)}{u(s)} \cdot \frac{1}{T} = \frac{1 - e^{-\gamma\_s}}{s} \cdot \frac{1}{T} \approx \frac{T}{\frac{T}{2}s + 1} \cdot \frac{1}{T} \tag{9}$$

which is a low-pass filter.

In a high-precision servo system, PWM-based amplifiers are commonly used as the motor drivers [28]. The PWM power amplifiers can be represented as

$$G\_{\lambda}(\mathbf{s}) = \frac{K\_{\mathbf{s}}}{1 + \frac{1}{2f\_{\mathbf{s}}}s} \tag{10}$$

where *Ka* is the amplifying multiple and *f s* is the modulation frequency.

The load cell signal conditioner linearly converts the force into a voltage signal, and its model can be simplified as

$$G\_{\rm s}(\mathbf{s}) = \mathbf{K}\_{\rm sw} \tag{11}$$

#### **2.4. Simplified model**

Without considering the disturbance torque *Td* , the angles of the motor and the load have the following dynamical relationship

$$T\_l = J\_l \ddot{\theta}\_l + B\_l \dot{\theta}\_l = K\_s (\theta\_m - \theta\_l) \implies \frac{\theta\_l(s)}{\theta\_n(s)} = \frac{K\_s}{J\_l s^2 + B\_l s + K\_s} \tag{12}$$

*Ks* is very large and the time constant is very small, therefore in the lower frequency range *θl* (*s*)/*θm*(*s*) ≈ 1, *θ<sup>l</sup>*  ≈ *θm*.

By considering *Ks*  = ∞ and *θ<sup>l</sup>*  ≡ *θm*, the motor-loader model can be simplified as

$$J\_a \ddot{\Theta}\_m + B\_a \dot{\Theta}\_m = T\_m \tag{13}$$

where *Ja*  = *Jm* + *Jl* , *Ba*  = *Bm* + *Bl* . Deleting the intermediate variables, the dynamical equation of the motor can be obtained as

$$\frac{L\_s l\_s}{K\_i} \frac{d^3 \theta\_m}{dt^3} + \frac{R\_s l\_s + L\_s B\_s}{K\_i} \frac{d^2 \theta\_m}{dt^2} + \left( K\_e + \frac{R\_s B\_s}{K\_i} \right) \frac{d \theta\_m}{dt} = \mu\_m \tag{14}$$

If the motor armature inductance is regarded as *La*  ≈ 0, the system can be further simplified as

$$\frac{R\_s l\_s}{K\_t} \frac{d^2 \theta\_m}{dt^2} + \left( K\_\ell + \frac{R\_s B\_s}{K\_t} \right) \frac{d \theta\_m}{dt} = \ \mu\_m \tag{15}$$

Define an objective function:

estimated parameters *b*<sup>0</sup>

late the values of *A*<sup>f</sup>

*y*(*t*) = *Af*

follows.

*<sup>Ψ</sup>* <sup>=</sup> [

where *A*̂ *f*

After *Af*

*A*f

*J* = ∑

The output signals can be decomposed as

sin(*ω*0) sin(*h*) … sin(*nh*)

[

Third, *Af /Am* and *Φ* are calculated as

is the estimation of *Af* .

*<sup>A</sup>*̂

Minimize the objective function, and prompt ∂*J*/∂*aj*

, *b*1

, ⋯, *bm*, *a*<sup>1</sup>

, *a*2 , ⋯, *an* .

*<sup>y</sup>*(*t*) <sup>=</sup> *Af* sin(*<sup>t</sup>* <sup>+</sup> *<sup>Φ</sup>*) <sup>=</sup> *Af* cos*<sup>Φ</sup>* sin*<sup>t</sup>* <sup>+</sup> *Af* sin*<sup>Φ</sup>* cos*<sup>t</sup>*

First, we select the sampling interval *t* = 0, *h*, 2*h*, …, *nh*, where *h* is a step time.

*c*1

\_\_\_\_\_ *c*1 <sup>2</sup> + *c*<sup>2</sup> 2 \_\_\_\_ *Am*

/*Am* and *Φ* are obtained, it is easy to estimate the parameters *b*<sup>0</sup>

based on Levy's method. Finally, we get the identified transfer function:

**Table 1.** Actual frequency response (*ω* is angular frequency, rad/s; *Φ* is phase delay, rad).

, *Φ* = *tg*<sup>−</sup><sup>1</sup>

*<sup>G</sup>*(*s*) <sup>=</sup> <sup>−</sup>38.48*<sup>s</sup>* <sup>−</sup> 141.4 \_\_\_\_\_\_\_\_\_\_\_\_\_ 1.523 *<sup>s</sup>* <sup>2</sup> <sup>+</sup> 8.324*<sup>s</sup>* <sup>+</sup> <sup>1</sup> (26)

*ω* **1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1**

/*A*<sup>m</sup> 16 22 25 27 30 31 40 46 64 122 16 *Φ* 1.71 1.82 1.78 1.77 1.89 1.92 1.90 2.00 1.94 2.23 1.71

*<sup>f</sup>* \_\_\_ *Am* = <sup>√</sup>

cos(*ω*0) cos(*h*) … cos(*nh*)] , the least square solutions of *c*<sup>1</sup>

[

*Af* cos*Φ Af* sin*Φ*]

= [sin*t* cos*t*]

Second, by defining *Y*<sup>=</sup> [*y*(0) *<sup>y</sup>*(*h*) … *<sup>y</sup>*(*nh*)] *<sup>T</sup>*

*i*=1 *L*

‖*D*(*j ω<sup>i</sup>*

m), then two matrix equations will be obtained. By solving the matrix equations, we can get the

**Table 1** shows actual frequency response obtained by experiments. We employ different sinusoidal input *r*(*t*) = *Am* sin *ωt* with different angular frequency *ω* (from 0.1 to 1 rad/s) to excite the open-loop system. By theoretical analysis, the output signals are in the form

 sin(*ωt* + *Φ*). Comparing the output with input sinusoidal signals, we can calcu-

/*A*m and phase delay angle *Φ* based on the least square estimation as

, *c*<sup>1</sup>

 = *Af*

 cos *Φ*, *c*<sup>2</sup>

*<sup>c</sup>*2] <sup>=</sup> (*Ψ<sup>T</sup> <sup>Ψ</sup>*)−<sup>1</sup> *<sup>Ψ</sup><sup>T</sup> <sup>Y</sup>* (24)

and *c*<sup>2</sup>

(*c*<sup>2</sup> /*c*1) (25)

, *b*1 , *a*1 and *a*<sup>2</sup>

 = *Af*

)*ε*(*j ω<sup>i</sup>*

A CMAC-Based Systematic Design Approach of an Adaptive Embedded Control Force Loading…

)‖<sup>2</sup> (22)

http://dx.doi.org/10.5772/intechopen.71420

 = 0 (*k* = 1, 2, …,

263

(23)

 sin *Φ* and

in Eq. (18)

can be derived as

 = 0 (*j* = 1, 2, …, n) and ∂*J*/∂*bk*

And the transfer function between the input voltage and output rotational angle is

$$G\_{\rm M}(\mathbf{s}) = \frac{\theta\_n(\mathbf{s})}{\mu\_n(\mathbf{s})} = \frac{K\_n}{T\_n s^2 + \mathbf{s}} \tag{16}$$

where *Tm* <sup>=</sup> *Ra <sup>J</sup>* \_\_\_\_\_\_\_\_ *<sup>a</sup> Ke Kt* + *Ra Ba* is the time constant of the motor system, and *Km* <sup>=</sup> *<sup>K</sup>* \_\_\_\_\_\_\_\_ *<sup>t</sup> Ke Kt* + *Ra Ba* is the gain coefficient of the motor system.

#### **2.5. Open-loop transfer function**

The mathematical model of the open-loop plant is the cascade of the units described previously:

$$\mathbf{G}\_p(\mathbf{s}) = \mathbf{G}\_{\dot{H}}(\mathbf{s}) \cdot \mathbf{G}\_A(\mathbf{s}) \cdot \mathbf{G}\_M(\mathbf{s}) \cdot \mathbf{G}\_S(\mathbf{s}) \tag{17}$$

Because the system bandwidth is much lower than the system sampling rate and the modulation frequency of the PWM amplifier, the time constant of the DAC converter and the amplifier is very small. In the lower frequency range, the effect of the DAC converter and the amplifier can be omitted. Therefore, the open-loop plant can be simplified as

$$G\_p(s) = \frac{b\_i s + b\_0}{a\_i s^2 + a\_i s + 1} \tag{18}$$

where *b*<sup>0</sup> , *b*<sup>1</sup> , *a*1 and *a*<sup>2</sup> are final parameters.

Obvious, it is a second-order model. Only four parameters need to be identified.
