*2.3. Control System Model Construction*

The operation of the main liquid manure spraying system decreased the response time of the operating system. This established the liquid manure spraying operating system transfer function. The liquid manure spreading input operating channel model was collected in real-time using an angular velocity sensor. The controller output the current signal to the solenoid valve after conversion. A solenoid valve was used to control valve opening. The final output of the system was the instantaneous current speed of liquid manure. Current speed was coupled back to the control system through the flowmeter in the flow chart of the operate channel, as exhibited in Figure 2. Finally, control via the inverse back-coupling closed cycle was conducted through the control system.

**Figure 2.** Control system diagram of liquid manure spreading.

In the light of input/output relationships in the operating system flow chart, the system input/output relationships are presented (Equation (1)):

$$V = f(q, v) = q \cdot 10^{-4} \cdot w \cdot v \tag{1}$$

where V (L/min) is liquid manure volume exportation from the precision spray operating channel, F (q, v) is the transversion function, vehicle running speed is v(m/s), Q is the target fertilization amount input into the system according to the amount of liquid fertilizer used in corn fields in Tai'an, Shandong Province (q was set as 55 L/hm2), and W (m) is the width of fertilization. According to the vehicle conditions in this study, w = 6 m.

According to the liquid manure spraying operating channel in Figure 2, the system's input feedback data were from the instantaneous discharge registration through the flowmeter. The current signal was the control system signal exported through the retroaction passageway. The control system switched signals and compared them with the target fertilizer amount and vehicle speed input to implement the inverse back-coupling control.

Thus, the control model feedback link use indicates Equation (2):

$$H(s) = \frac{v(s)}{V(s)} = \frac{700}{s \cdot 5 \cdot w} \cdot e^{-\tau s} \tag{2}$$

where t (s) is the delay time of preference transmission in the feedback link. The transfer function aleatory variable after the Laplace transform is S, and the transfer function inverse back-coupling chain is H.

The discharge monitoring of the operating channel via flowmeter in this study was real-time online. The retroaction latency of link time could be left out in the light of the actual hardware situation.

In the light of actual demand for a liquid manure precise spraying operating channel, the solenoid valve was our major control aim. We chose a Shanghai Kaiweixi Co. VB7200 solenoid valve. Figure 3 shows the signal control chart.

**Figure 3.** Signal control chart of solenoid valve.

In Figure 3, propel module input/output signals were electricity semaphores. Transfer function was a ratio and delay segment. The relationship was as follows (Equation (3)):

$$G\_1(\mathbf{s}) = \frac{I\_{\rm out}}{I\_{\rm in}} = k\_\mathbf{s} \cdot e^{\mathbf{r}\mathbf{s}} \tag{3}$$

where k*<sup>s</sup>* is the magnify quotient of the translator. *Iin* as well as I*out* are current signal input/output with the propel module, respectively, and G1(s) is the propel module's transfer function.

Drive module current signal transmission delay (τ) was set to τ < 0.02 s. Therefore, τ was left out in the system, such that the propel module's transfer function indicated a proportional sector [19].

The solenoid valve had a DC motor. The electricity signal was the dominant input signal. The angle of the electrical machinery's axle was the output. The DC motor's electrocircuit of semaphore control included an armature loop balance and induction of the rotor, as well as the balance of the axle couples of electrical system. Equation (4) demonstrates the balance equation:

$$\begin{cases} \dot{\iota}\_a(t) - E = R\_d \cdot \iota(t) + \frac{d\iota(t)}{dt} L\_d\\ E = \omega\_n \cdot k\_{\rm cp} = \dot{\theta}\_n \cdot k\_{\rm cp} \\ \iota(t) \cdot k\_T - M\_1 = \dot{\omega}\_n \cdot l\_n = \ddot{\theta}\_n l\_n \end{cases} \tag{4}$$

Time (s) is expressed as t. Entering electricity in the DC motor is *ia*(*t*)(mA). The motor's electromotance is expressed by E (V). The armature voltage is expressed by u(t) (V). Total armature resistance is expressed by *Ra* (Ω). The armature's inductance is expressed by *La*(H). *k ep* is the back electromotive force coefficient. The angle of electrical machinery's axle is expressed by *θ <sup>n</sup>* ( ◦). The motor torque coefficient is displayed as *KT*. Motor load torque is displayed as *M*<sup>1</sup> (N··· m, *M* <sup>1</sup> = *f* · .. *θn*, where f is the friction coefficient). The rotational inertia of the rotor's moment of inertia is expressed by *Jn* (kg··· m2). Finally, angular speed of the motor rotor is displayed as *ω <sup>n</sup>* (rad/s).

The DC motor transfer function in the solenoid valve was obtained by transforming Equation (4) using a Laplace transformation, as shown in Equation (5):

$$\begin{aligned} \text{G}\_{2}(\text{s}) \quad &= \frac{\theta\_{\text{a}}(\text{s})}{\text{I}\_{\text{a}}(\text{s})} = \frac{k\_{\text{T}}}{k\_{\text{tp}}k\_{\text{T}}\text{s} + \text{R}\_{\text{a}}(\text{/}\text{s} + \text{f})\text{s}^{2} + \text{L}\_{\text{a}}(\text{/}\text{s} + \text{f})\text{s}^{3}} \\ &= \frac{0.51}{2.2 \times 10^{-3} \cdot \text{s} + 5.8 \times 10^{-5} \cdot \text{s}^{2} + 8.7 \times 10^{-5} \cdot \text{s}^{3}} \end{aligned} \tag{5}$$

where *θn*(*s*) is the angle of the electrical machinery's axle with the Laplace transform function. *Iα*(*s*) is the Laplace transform function of the motor input current, and *G*2(*s*) is the motor's transfer function. We can see Table 1 for simulation parameters [9].

**Table 1.** Simulation parameters.


The gear set was the main component of the reducer. The displacement output of the valve element was the axis speed of DC motor after deceleration. The valve element displacement, 0–18 mm, is expressed as X. The valve's opening was translocation of the valve element.

The transmission relevance of decelerator input/output is expressed as the reduction ratio, which adopts the proportional dominant mode. The transfer function is expressed as follows (Equation (6)):

$$G\_3(s) = \frac{X(s)}{\theta\_{\text{fl}}(s)} = \frac{Y}{2\pi p} = 8.19 \times 10^{-6} \tag{6}$$

where the decelerator gear ratio is displayed as p. The lead of the drive rod is expressed with *Y* (mm). The spool traversed by the Laplace function is expressed by *X*(*s*). The reducer transfer function is demonstrated with *G*3(*s*).

During this research, the flow and opening of the solenoid valve were linear under the same-pressure operating mode. Therefore, Equation (7) shows the relevance within flow and opening:

$$G\_4(s) = \frac{V(s)}{X(s)}\tag{7}$$

The flow as well as the valve opening's transfer function is *G*4(*s*).

In Figure 3, the plant of the front path is guided by the solenoid valve. Equation (8) shows its transfer function:

$$\mathbf{G}(\mathbf{s}) = \frac{V(\mathbf{s})}{I\_{\mathbf{d}}(\mathbf{s})} = \mathbf{G}\_1(\mathbf{s}) \cdot \mathbf{G}\_2(\mathbf{s}) \cdot \mathbf{G}\_3(\mathbf{s}) \cdot \mathbf{G}\_4(\mathbf{s}) \tag{8}$$

where the transfer function of solenoid valve is defined by *G*(*s*).

As can be seen from the control model as well as the functions of each control activity, in this research, Equation (9) is the closed–cycle retroaction control transfer function of the precision broadcast application control system:

$$G\_z(s) = \frac{G(s)}{H(s)G(s) + 1} = \frac{4.17}{64.3s + 16.9s^2 + 25.4s^3 + 2.92} \tag{9}$$

The liquid manure spreading system's dominant transfer function is *Gz*(*s*).
